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Louisiana State UniversityLSU Digital Commons
LSU Historical Dissertations and Theses Graduate School
1990
Quantum Monte Carlo Simulations of Hubbardand Anderson Models.Yi ZhangLouisiana State University and Agricultural & Mechanical College
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QUANTUM MONTE CARLO SIMULATIONS OF HUBBARD AND ANDERSON MODELS
A Dissertation
Submitted to the Graduate Faculty of the
Louisiana State University and
Agriculture and Mechanical College
in partial fulfillment of the
requirement for the degree of
Doctor of Philosophy
in
The Department of Physics and Astronomy
by Yi Zhang
B.S., University of Science and Technology of China, 1984
M.S., Louisiana State University, 1988
May, 1990
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ACKNOWLEDGEMENTSa n e w / ia
I wish to express my most sincere gratitude to my advisor Pro
fessor Joseph Callaway for his continued valuable guidance and help
during my years of studies at Louisiana State University and for the
criticism of the manuscript
Help from all members of the theoretical solid state group, in
particular, Drs. Han Chen and Duanpin Chen, is mostly appreciated.
I also extend my appreciation to the Physics Department and all its
members for their friendness and help throughout my graduate study
period.
I am indebted to Dr. Han Chen for critical readings of portions
of this manuscript I am also grateful to Drs. Goodrich and Browne
for their useful suggestions and criticisms of this dissertation.
My final thanks go to Fei Xu for her constant encouragement,
love and support
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TABLE OF CONTENTS
CHAPTER 1. INTRODUCTION.............................................. 1
CHAPTER 2 . MONTE CARLO SIMULATIONS ................. 14
2.1 Classical Monte Carlo .................................................... 15
2.2 Simulation Procedures ..................................................... 20
2.3 Quantum Monte Carlo .................................................... 23
CHAPTER 3. EXTENDED HUBBARD MODEL................... 36
3.1 Introduction ...................................................................... 37
3.2 The model ........................................................................ 41
3.3 Results .............................................................................. 48
3.4 Summary .......................................................................... 83
CHAPTER 4. PERIODIC AJNDERSON MODEL ................... 85
4.1 Introduction ...................................................................... 86
4.2 The model ........................................................................ 88
4.3 Results.............................................................................. 92
4.4 Summary .......................................................................... 103
CHAPTER 5. CONCLUSIONS ................................................. 104
REFERENCES ............................................................................ 107
APPENDIX. COMPUTER PROGRAMS ................................. 117
iv
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ABSTRACT
We present) in this dissertation, a numerical simulation method
to study interacting fermion systems. The general simulation pro
cedures are discussed in connection with a description of a Quantum
Monte Carlo simulation algorithm for interacting electrons on lat
tices.
The algorithm presented here has been used to simulate interact
ing electrons on lattices, and it makes possible the study of substan
tially larger systems than can be studied by other numerical
methods. As long as certain limits of applicability are respected,
model Hamiltonians of interacting electrons can be studied without
resort to uncontrolled approximations. The method then provides
nearly exact solutions to model Hamiltonians of many-body systems,
in the sense that the degree of error can be controlled.
We also discuss some results obtained from simulations of the
extended Hubbard model and the periodic Anderson model.
For the extended Hubbard model in two dimensions, different
regions are identified where electron correlation produces antifer
romagnetism, charge-density-waves, and singlet pairing supercon
ducting behaviors. We also find regions where transitions from one
type of correlation to another occur.
v
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For the symmetric periodic Anderson model, we observe the
formation of local spin momenta at high temperatures and their
quenching at low temperatures, as in the single impurity Anderson
model. In addition, we find antiferromagnetic interaction between /
local momenta at low temperatures which are not present in a single
impurity problem.
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CHAPTER 1. INTRODUCTION
We present in this dissertation a Monte Carlo method for simu
lations of model Hamiltonians of interacting electrons in solid state
physics. We also present some resulte obtained from the simulations
of the Hubbard (1) and Anderson (2) models.
Theoretical solid state physics is concerned with the description
of systems of interacting electrons. However, exact solutions to the
complicated many-body problems are almost impossible to obtain.
In general, two approaches have been adopted: 1) approximate
single-particle description of many-electron systems; 2)
simplification by replacement of the physical Hamiltonian by a
model which retains some aspects of the dynamic electron interac
tions . In the first approach, interactions are approximated by aver
age potentials, and electrons are described as independent particles
moving in the average field of other electrons and ions. The
independent particle model has been widely used to describe many
phenomena in the solid. The successful classification of many
metals, insulators and semiconductors and explanation of their pro
perties by band theories is a major success. Although the indepen
dent particle model has provided us great understanding of the phy
sics of many solid state phenomena, other phenomena in solids are
intrinsically many-body effects and cannot be explained in the
1
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2
framework of a single particle picture. Among these many-body
effects are electron correlations and collective excitations such as
spin waves and plasmons. Many-body interactions may also lead to
a ground state which can not be described in a single particle pic
ture, as in the case of superconductivity. Model manybody Hamil
tonians have to be considered in such cases.
We are interested in the simulations of models of interacting
electrons having the following general Hamiltonian,
H ” 2 d" U ijnifinjn' ■ (1*1)ijfi linn'
The models are defined here on some convenient single particle
basis states | * > in the second quantized form. In this basis set
c,.+ (cifJ) denotes the creadon(destruction) of an electron of spin fi
( t 4 ) in state |i > and nifi is the corresponding electron occupation
number. The state |i> is generally chosen as either the single parti
cle Wannier state (3) localized at lattice site i or an atomic orbital
centered at site i. With this choice of basis set, the model is
described as defined on a lattice. The first term of the Hamiltonian
(1.1) describes the electron motion in the solid determined by the
hopping integral t^, in which the average electron-electron interac
tion has been taken into account (as in electronic band structure cal
culations). The parameters I/,;- represent short range Coulomb
interactions. The long range electron interaction is not included
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explicitly as justified by Hubbard (1) for the Hubbard model.
Eq. (1.1) defines a class of model Hamiltonians which are appli
cable only to systems for which a one band picture is appropriate.
However, the utility of Eq. (1.1) is greater than might at first be
realized, and many systems have been modeled successfully in terms
of Eq. (1.1). Two widely used models - the Hubbard and Anderson
models are simple cases of (1.1). The antiferromagnetic Heisenberg
model (4) also can be shown to be a limiting case. The many-body
model incorporates electron interactions into the description of sys
tems of electron by explicitly retaining the electron-electron interac
tions in the model Hamiltonian, and thus is expected to explain
some many-body phenomena in the solid. The inclusion of many-
body terms, even in their simplest form, usually makes the problem
much more complicated to solve. In this dissertation, we will
present calculations based on Quantum Monte Carlo simulations for
the Hubbard and Anderson models which provide some partial solu
tions of the two models. These two models have attracted consider
able attention in connection with studies of the magnetism of transi
tion metals, metal-insulator transitions, heavy fermions, and recently
high temperature superconductivity.
The ferromagnetism of 3d transition metals (Fe, Ni, Co) and
their compounds has been long a fascinating problem (5), which has
been considered from two opposite starting points. The localized
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model (Heisenberg model) describes the electrons as localized on
atoms. The interatomic exchange interactions lead to the magnetic
order in these materials. In the band model (6 ), each magnetic
cairier(electron) is itinerant, and the ordered magnetic state is sta-
blized by the weak electron-electron interaction. Both models are
capable of describing some phenomena in magnetic systems (7 ). The
low temperature spin waves (8,9) and high temperature Curie-Weiss
susceptibilities (10) are qualitatively explained by the localized
model, whereas the low temperature specific heat and the famous
non-integral magnetic moments in 3d transition metals can be
explained by the band model (7). A unified theory is needed that
would interpolate between the two extreme limits.
As an attempt to provide a theory of itinerant magnetism and
electron correlations, the Hubbard model was introduced in 1963 (1)
to describe electron interactions in narrow band systems, in particu
lar the electron correlations and magnetism of d-electrons (11). The
simplest form of Eq. (1.1) is
H = t ^ CipCjn 4" U > (1*2)<b'>p *
where the electron hopping is taken as nearest-neighbor hopping
<$j>, and electron interactions are represented by a short range
repulsion U between electrons of different spins at same site.
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Since the model was proposed, various approximate techniques
have been applied, including the Green’s function decoupling
scheme (1), mean field theory (12), functional integral (13), and
variational approaches (14). However these techniques often give
conflicting results except in the weak interaction limit Most of the
approximate techniques used are uncontrolled and there is no gen
eral agreement on the properties of the model. Even though the
model is simple in appearance, it is solvable exactly only in a few
limiting cases. In one dimension, the model was solved by Lieb
and Wu (15) using the Bethe-ansatz technique at zero temperature,
and in three dimensions in large U limit by Nagaoka (16) when
there is one extra electron in an otherwise half-filled band for certain
lattices.
The Hubbard model has also been used to describe metal-
insulator and magnetic phase transitions in compounds of transition
metals (FeO, NiO, CoO, MnO, ...) (17). Mott-Hubbard insulators are
usually identical to ordinary magnetic insulators which cannot be
explained by the band theory (they would be metals in the band
theory). A metal-insulator transition is a transition from a Mott-
insulator to a metallic state (18). In Mott’s theory, a crystalline array
of hydrogen-like atoms, or more generally atoms in which there is
an incomplete shell, may make a transition from the metallic to the
non-metallic state as the interatomic distance is increased. For NiO,
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the transition from the insulating to the metallic state occurs at room
temperature under pressures of about 2 Mbar (19). Mott pointed out
the importance of electron correlations in such materials and the
related breakdown of the conventional Bloch-Wilson band theory.
Early results (1,13) on the Hubbard model show that the ground
state of itinerant electrons in a narrow half filled band becomes
insulating and antiferromagnetic when the electron interaction U is
larger than the band width W.
The second model in which we are interested is the periodic
Anderson model (20-23). Tins model is a natural generalization to
the well-known Anderson impurity model (2), which was originally
introduced to describe the formation of localized magnetic moments
on dilute magnetic impurities in a nonmagnetic host For the single
impurity model, extensive effort finally leads to the renormalization
group (24) and Beihe-ansatz (25) solutions to the problem in some
limiting cases. The results show that the model also exhibits both
Kondo effect and mixed-valence phenomena.
The periodic Anderson model has been widely used to describe
heavy fermion systems and mixed-valence phenomena in / electron
systems (20,21). A heavy fermion system is characterized by its
large specific heat at low temperatures (22). The electronic specific
heat coefficient of a heavy fermion metal 7 (T) — [s two or
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three order of magnitude larger than in ordinary metals, and 7 is so
large that the phonon contribution to the low temperature specific
heat can be ignored (T<2QK). Comparing with a free-electron for
mula for 7 (proportional to the electronic mass m), the large
enhancement of 7 can be interpreted as due to a large effective *fThelectron mass — of order 103. Heavy fermion systems are also
unusual in their magnetic, electric and superconducting properties.
The periodic Anderson model has the following form for a two
orbital system when defined on a lattice:
H =-< S «%. + v S ( 4J/*. +/£<*»)<&■>(* <#>**
+E/ S. »/.> + E«/.tn/n » f1-3)»/* »
where the conduction electron band is formed by the d electron via
the nearest-neighbor hopping, and / refers to the localized / orbi
tals with single particle energy level E j. V hybridizes the conduc
tion band and the localized / levels.
Various approximate approaches and elegant techniques have
been used to study the Anderson model, such as the real-space
renormalization group (26) and the large N expansion (27,28).
These investigations shed some light on the nature of the systems.
Recent exact diagonalization studies provide further understanding
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of the model in the strong interaction and weak hybridization limit
(29). five different regions have been specified in terms of the aver
age number of / electrons.
An important approach to the study of model Hamiltonians is
the finite cluster approach (29-36). The exact results to the model
Hamiltonians on clusters provide solutions to the model without
uncontrolled approximations. They often provide the first step to
the basic understanding of the general properties of the models, and
guide more elegant theoretical and analytical approaches to the prob
lem. Generally, two methods have been used in the cluster calcular
tions. The first one is the exact diagonalization method (29-31). This
method deals explicitly with the matrix representation of a model in
a suitable basis set, and diagonalizes the Hamiltonian. It is, how
ever, limited by the size of the matrix that can be diagonalized by
available computers in a reasonable amount of computing time.
Shiba and Fincus (33) applied this approach to the one dimensional
half-filled Hubbard model, and studied the thermodynamical proper
ties of the model. Recently Callaway and Chen (29,30,34,35) further
extended the limit of this method by incorporating the symmetrizar
don of basis sets, and studied the Hubbard and Anderson models on
different clusters. Details of the method and some results can be
found in Chen’s dissertation (36).
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Our principal method for the study of model Hamiltonians is
the Monte Carlo simulation method (37). This method has been
widely used to study models of High- Te superconductivity (38), The
Hubbard and similar models (39-42) are used to describe the new
oxide superconductors, particularly in regard to the Cu02 layers.
Calculations (43) of the electron-phonon interaction and the experi
mental observation of weak or negligible isotope effects (44) in
these materials suggest that phonons alone cannot explain high- Te
superconductivity according to the approach of conventional BCS
theory. The antiferromagnetism and spin fluctuations (45,46) in
these systems also indicate the possible importance of magnetic
interactions for superconductivity. It is believed that the new
mechanism of high- Te may originate from electronic degrees of
freedom. Since analytic methods are usually unable to provide
satisfactory solutions to the models of these complicated systems at
this time, Monte Carlo simulation studies of the models on finite
clusters will certainly provide valuable information about the proper
ties of the models. The Monte Carlo method can deal with much
larger systems than is possible with the exact diagonalization
method. The results are exact in the sense that the degree of error
can be controlled, in contrast with other uncontrolled analytic or
approximate techniques.
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We have studied the half-filled extended Hubbard model (one
electron per site) on a two dimensional square lattice (47), and
investigated properties of the model in different regions of its
parameter space. We found a phase transition between an antifer
romagnetic and a charge density wave state. We also identified the
region where the superconducting pairing is stronger than antifer
romagnetic and charge density wave correlations.
We have also applied the Monte Carlo method to the symmetric
Anderson model (48). The systems under consideration are two
dimensional square clusters. We find various behaviors of the sys
tems in different regions characterized as free orbitals, local
moment, and Kondo regions, similar to those observed in the impur
ity Anderson model (24,49,50). In addition we find the system
develops f —f antiferromagnetic correlations at low temperatures,
which contribute to the partial screening to the local moments, as
compared to the complete screening of the local moment by the con
duction electrons in the single impurity model.
The development of the Monte Carlo simulation technique
began more than thirty years ago when Metropolis et al. (51) intro
duced the idea of importance sampling, which makes the method
practical and efficient The simulation technique together with
importance sampling have been applied to many topics (52), includ
ing classical spin systems, classical fluid dynamics, the one
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component plasma, and surface properties. However the direct appli
cation of the technique to the fermion systems is still a very chal
lenging problem (53).
The major difficulty for fermion systems has been the incor
poration of the Pauli principle and fermion statistics into an
efficient simulation algorithm. Development of algorithms remains
an open topic and active area. The Green’s function Monte Carlo
method (54), a generalization of a Monte Carlo method suggested
by von Neumann and Ulam (55), has been widely used to simulate
real material systems. At this point* studies of the ground state have
been restricted to free electrons and light atom system. The fixed
node approximation (56) partially overcomes the difficulty of nega
tive probability in the simulations.
For fermions and quantum spins on lattice, various algorithms
have been proposed and used (37,57-63). The sizes of the systems
we can study utilizing the existing algorithms are substantially larger
than can be handled by exact diagonalization. In one dimension, an
algorithm (57) exists for which the simulation time scales linearly
with the size of the system. Otherwise, systems of modest sizes can
be studied and results can be extrapolated to larger systems. In
higher dimension, no algorithm existe for fermion systems that
scales like (3N at low temperatures (where /? is the inverse tempera
ture for a finite temperature simulation and N is the number of
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space sites). The exact updating procedure proposed by Blanken-
becler, Scalapino and Sugar (BSS) requires /3N3 operations (37).
The fermion statistics and Pauli principle have been incorporated
explicitly in the algorithm. As originally proposed and used, the
BSS algorithm becomes unstable at low temperature. Hirsch and Fye
(HF) (60) developed an algorithm based on the BSS algorithm in
which the number of operation is proportional to (where N0 is
the impurity or interaction sites). This algorithm is stable at low
temperature and suitable for the study of dilute magnetic impurity
systems. In addition, Hirsch (61) has recently developed an algo
rithm which generalizes the BSS algorithm and the HF algorithm
with time proportional to (where f3o<0 is a scale factor).
The most recent advances in regard to algorithms came with the
development of a zero temperature algorithm (62) for simulations of
fermions, and stablizadon procedures for the BSS algorithm at low
temperatures (63), which make possible the simulation of fermion
systems at low temperatures.
The organization of the rest of the dissertation is as follows: In
chapter 2 we describe the method of calculation. Section 2.1 reviews
the basics of the Monte Carlo simulation method. The simulation
procedures are explained in Section 2.2. In section 2.3 the quantum
Monte Carlo simulation method is illustrated with a description of
the BSS algorithm.
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Chapter 3 presents the resultB from the simulation studies of the
extended Hubbard model in a two dimensional square lattice. Fol
lowing the introduction (section 3.1), we describe the basic proper
ties of the model in section 3.2. In section 3.3 we discuss the pro
perties of the model in different regions of itB phase diagram. This
chapter is summarized in section 3.4.
Chapter 4 starts with a brief introduction to the Anderson model
and its general properties (sections 4.1 and 4.2), the results of the
simulations are discussed in section 4.3 and concluded in section
4.4.
Chapter 5 summerizes all chapters and provides some sugges
tions to further studies. Computer programs are listed in the appen
dix. Some explanation about the simulation code as well as a flow
diagram of the computer programs is also included.
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CHAPTER 2. MONTE CARLO SIMULATIONS
This chapter presents the basic aspects of Monte Carlo simular
tion methods. Starting from the classical Monte Carlo method, the
basic simulation procedures are illustrated with a description of the
widely used Metropolis importance sampling algorithm. We then
discuss a Quantum Monte Carlo algorithm for simulations of
interacting fermions on a lattice, which has been used in the current
study of the Hubbard and Anderson models. The results of these
simulations will be presented in the next two chapters.
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2.1 Classical Monte Carlo
In classical statistical mechanics, the thermodynamical average
of an observable A at finite temperature T is given by
fA(x)e~PH(x)dx
< 1 > = /e -W > dx ’ (2,1)
where f3=\jkB T , is the inverse temperature, and H(x) is the Ham
iltonian of the system described by a complete set of dynamic vari
ables x. In principle, an observable quantity A could be calculated
through the evaluation of Eq. (2.1) once the forms of A(x) and
H(x) are known. However, in practice, the direct evaluation of Eq.
(2 .1) is very difficult, because an extremely complicated integral is
encounted. For many particle systems, the integration (2.1) is high
dimensional. Standard numerical integration routines can not be
applied here since they are normally designed for lower dimensional
integrations.
The Monte Carlo method approximates the integral (2.1) as the
average of A on M independent sampling configurations of { x } in
phase space. The central limit theorem (64) ensures that the sam
pling error decreases as the inverse of the square root of sampling
points, independent of the dimension of the integral space or
phase space.
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A direct but naive method to evaluate Eq. (2 .1) is to sample
randomly chosen values of the variables x,- uniformly in the phase
space, and calculate <&>as the average of A(x,-) weighted by the
factor
£ e W ‘>A(Xi)<4>=ii-vj-------------, (2.2)
g e^ « M1=4
where A(xt) is the value of A in the state or configuration x,-. How
ever this inefficient simple scheme is not of great use. For large sys
tems, the entire phase space is very big, but the important part of
the phase space that contributes significantly to the integral (2 .1) is
very small, which means that most of the randomly generated
configurations { xi } will not contribute significantly to the average
<A> Since there are not enough configurations generated in the
important region, the statistical errors will be very large unless an
enormous number of sampling points are used.
Metropolis et al. (50) introduced the idea of importance sam
pling. Here the M configurations xlt x2, ... , xM are not chosen
completely at random, but are constructed according to a probability
P(x) proportional to e~&H(x\ Hie integral (2 .1) is then reduced to a
simple arithmetic average
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£ ^ HMA{Xi)/P(xt)<4 > - 1=1_______________________
S t'4a{',)lP(xi)«=1
1 ^ ^= (2.3)
iW 1=4
Hie generation of configurations according to e~^H^ ensures that
the configurations are distributed over the important part of the
phase space, therefore provides an efficient sampling.
The desired distribution of configurations {a;} can be con
structed from a random walk of variables x through the phase space
via a Markov process (52,64). A Markov chain of variables ( x x ,
x2, xn, ..., ), a trajectory through phase space, is generated by
specifying a transition probability P(x{ —► x}-) from one point a?,- to
another point x}- in the phase space, for which the distribution of
element of the Markov chain xn+l depends only on the
element xn. It is sufficient that the transition probability P(x{ —^Xj)
satisfies the detailed balance condition:
= _ „ . ) (2.4)
in order that the Markov process converges to the distribution
e-pH{x) gjjj jjgg eventua] access to every configuration in the phase
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space.
There are many ways to specify P(xi —► a: ■) that satisfy Eq.
(2.4), which correspond to different algorithms for generating ran
dom configurations. One commonly used algorithm to generate a
Markov chain is the Metropolis rejection method which specifies the
transition probability as: (51)
if H ixfi-H ixi) > 0
1 otherwise
In Metropolis algorithm a proposed configuration change
Xf is always accepted if the change lowers the energy of the
system, e.g. H{xi ) < # (# ,). Otherwise, the change is accepted with
the probability e~ ^^ , where AH
The heat bath algorithm is very similar to the Metropolis
method, in which the transition probability is specified as:
P(xi -► */) = j + ^ | % ) ^ ) ] (2‘6^
It can be easily shown that P satisfies the detailed balance condition
Eq. (2.4) for both Metropolis and heat bath algorithms. Using Eq.
(2.5) for the transition probability sometimes makes the system
reach the desired distribution faster than choosing Eq. (2 .6 ). We
shall refer the desired distribution of configurations of {a:} as the
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thermal equilibrium distribution.
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2.2 Simulation Procedures
The key aspect of the Monte Carlo method described in the pre
vious sections is the generation of a set of random configurations of
variables x distributed according to which can be achieved
by the use of Metropolis algorithm through acceptance or rejection
of proposed changes of configurations x. A general simulation pro
cedure consists of the repetition of the following steps for construct
ing a Markov chain of variables {a}, after an initial configuration x0
is specified: (52)
(I) From configuration a?,*, one proposes a random change
x,- —►a;/, and calculates the energy change associated
with that change AH' = H (xi,) —H(xi).
(II) Calculate the transition probability P =e~P*H for the
proposed change of configuration.
(HI) Generate a random number r uniformly distributed in
[0,1].
(IV) If P > r , the proposed change of x{ is accepted, and the
new configuration x- is counted as the (z-HL) element of
the Markov chain.
(V) If P < r , then a;/ is rejected, and the old configuration a:,-
is counted again as the (z'-K)A element of the Markov chain.
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(VI) Calculate and record the value of observable A in the
configuration A(a;l+l).
For most systems, the number of degrees of freedom N is very
large, and each configuration of a: is a vector (®l, x2, • • *, x N) in
phase space. In proposing a change of a;, it is possible to change the
entire vector, e.g. proposing a random change for each elements of
x:
xk —+x* = x k-t£kAa (2.7)
where £* is uniformly distributed in [—% , % ] , and Ac is the step
size of the changes. Here the variables of x are assumed to have
continuous values. Hie step size Ac must be chosen carefully. If
Ar is too small, a change of configuration ®f will almost always be
accepted, due to the small changes of H{x). On the other hand, if
Ar is too large, it is very likely that a change of will be rejected,
since x, is likely to be moved outside the important region of the
phase space. A good choice of the step size Ar is such that about
50% of the proposed changes are accepted.
In practical simulations, a commonly used method is to update
one or a few elements of a: at a time, and leaving other elements
fixed. This procedure is advantageous because the change of H in
tins case can be evaluated much easier and faster than changing the
full vector of x. If H(x) is local in x and when one element of x
is changed, the change of H(x), AH, usually involves only a few
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operations. All elements may be selected for updating either sequen
tially or randomly.
Since subsequent configurations of {a:} differ only by one ele
ment their physical properties are very strongly correlated (52). To
ensure that a set of independent configurations {a;} distributed
according to e~^H^ is used in calculating the averages of physical
quantities, it is necessary to select only a subset of configurations
from the constructed Markov chain, each separated by a number of
Monte Carlo sweeps. Therefore, step (VI) of the simulation pro
cedure needs only to be performed after every a few Monte Carlo
sweeps. The evaluation of physical quantities in step (VI) is referred
to as a Monte Carlo measurement The quantity M in Eq. (2.3) is
then the total number of Monte Carlo measurements. M should be
large enough to keep the statistical errors within allowed values, and
the number of Monte Carlo sweeps between measurements should
also be large enough to ensure that the selected configurations are
uncorrelated and independent In addition, a large number of
configurations (warm-up sweeps) in the earlier simulation iterations
are always affected by the initial configuration, they are not charac
teristic of the desired thermal equilibrium distribution. They should
not be used in the evaluations of physical quantities. Only those
configurations that reached thermal equilibrium can be used to cal
culate the desired thermodynamical averages.
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2.3 Quantum Monte Carlo
Hie major difference between a Quantum Monte Carlo simular
tion and a classical Monte Carlo simulation lies in the evaluation of
the quantity e~$H, where H is the Hamiltonian. In classical mechan
ics the Hamiltonian H(x) is simply a c-function of a set of dynamic
variables x of the system, and the Boltzmann weight e~^H^ is
known once the configuration x is specified. For a quantum system,
H is an operator, and in general we do not know how to evaluate
e~&H. Fortunately, the Monte Carlo simulation procedures described
in the previous sections can still be used for pure boson systems
with some slight modifications. The starting point is analogous to
the path-integral formulation of the quantum field theory, in which
the average value of an operator A is given by: (65)
[A< *> as ' ( 2 *8 )
where S is the action defined at imaginary time r = it, and tp is the
field on which S and A depend. Eq. (2.8) has the same form as in
Eq, (2.1) except (3H is replaced by the action S, and the integration
is over the boson field i}>, which are commuting c-numbers. Hence
can still serve as a relative probability, and the simulation pro
cedures of section (2 .2 ) can be applied to pure boson system
directly. For fermion systems, the direct application of previous pro
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cedures falls because of the anti-commuting fermion fields. In this
case, e-5 is not a c-number, and cannot serve as a relative probabil
ity. To get around with this difficulty, one approach is to introduce
some auxiliary boson fields and formally integrate out the fermion
degrees of freedom at first (37), leaving an effective action S<ff
which is generally a complicated determinant and non-local in the
auxiliary boson fields.
To illustrate more clearly the method dealing with interacting
fermion systems, we consider here the extended Hubbard model,
defined on lattices:
H = t £ Ci+ citl + t / £ n ^ » (2-9)» <? > i
where i,j denote lattice sites, c,J is the electron creation (des
truction) operator at site i (j) with spin fx, and
'ft ~ C ip ^ift J n t ~ ^ » ' t *
Hie first term of H describes electron motion and t is the electron
hopping parameter. In this model, electrons are allowed to hop
between nearest neighboring sites <ij> U denotes the short range
Coulomb interaction between electrons with different spins at the
same site, V measures the interaction between electrons at nearest
neighboring sites, and jxc is the chemical potential of the system
which implicitly controls the electron density of the system. The
system under consideration is a finite cluster with N sites.
Page 32
25
To perform Montse Carlo simulation for this model, we write the
partition function of the system as
Z = Tr e ^ H , (2.10)
where /? = l/kB T is the inverse temperature. The problem here is,
in general, the inability to exponentiate the quantum Hamiltonian
and perform the required trace. However, we can transform the par
tition function Z into a form which can be more easily handled.
Rewriting e~&H as a product of L factors, the partition function Z
becomes
Z = T y , (2.11)i=i
where there are L imaginary time slices (65) and =(3fL. To for
mally trace out the fermion degrees of freedom, we first break each e-&rH into two terms using the Trotter approximation (66)
e -* H + 0(Ar2) , (2.12)
where H0 consists of the hopping part of Hamiltonian H, and Hx is
the remaining part which contains the electron interactions U and V.
H 0 ~ t C»'/2 Cj(i
X) Cijt cjn > (2.13)
Page 33
26
where K is the hopping matrix, and
= ^ i ; w tTn(i + y £ n ,n y —mc£ * V (2.14)« <a> t
Hie error of the break-up is of the order which can be con
trolled if the number of the time slices L is made large enough. The
electron-electron interactions in H l can be eliminated through the
use of the discrete forms of the Hubbard-Stratonovich transforma
t i o n : ^ ^ )
= 4 E e 1 (215)
with cosh(X{/) = e * u* for U >0. For U <0,
___1_ ^ {*>.>+«,y)+*[/K>+»,y) (2 16)
with e~^rf/ =cosh(2\(/)yboshs( \ p ) , and Xy =ln(coshXf/).
One auxiliary Ising variable a0(t,/) is introduced at each space-time
lattice site (i,l) for the interaction U, while for terms involve theinteraction V,
Vrijrij- = V £ n {/injy , (2.17)nn'
four auxiliary Ising variables <r1(<^’>/), c 3(<2i>/) and
a 4(<^y>/) are introduced between each pair of nearest neighboring
Page 34
27
sites <%j >(in a two dimensional lattice ) at each time slice I .
Using Eq. (2.15) and (2.17), we have now for U >0 and V >0,
e-*rHx{i) = J _ exp j Xtf<r0(i%f -n .-J— ^ ( n , - f +n(J) ]
<#>
-A rV x j(n ,-+ n y) + ArV cZ ni}<(j> i
i S c*m c»/i- y e ** • (2-18)
where is the coefficient of the quadratic term c,J ci(l. Substi
tute Eq. (2.18) and (2.13) into Eq. (2.11). The partition function becomes
L ci,»iZ = £ 21- n « c • (2.19)
£70. <7l ) t r 2i<T8i ° 'A ^=1
The quartic terms in the original H have been replaced by
quadratic terms after introducing auxiliary Ising fields, and the prob
lem is now reduced to non-interacting fermions coupled to auxiliary
Ising fields.
Since all terms in Eq. (2.19) are now quadratic, the trace over
the fermion degrees of freedom can be taken directly (32,37), leav
ing a summation over the auxiliary Ising variables,
Page 35
28
z = £ ndetlz+n^l
= s n detM,, (2 .2 0 )<T0 <71 2 8 =44
with B f = e~ ^Ke Vti , and = / 4- B£B£_y * • 'B$B£ , where K
and VJ^l) are N x N matrices (N is the number of space sites of the
system), and
(K)v (2.21)
W o : = i y ( ) '% - (2.22)
With the fermion degrees of freedom removed, the d (in this case 2)
dimensional quantum simulation problem is now transformed into a
d + 1 dimensional classical simulation problem, and the summation
over the Ising spin variables can be evaluated by standard Monte
Carlo methods described in the previous sections. The Boltzmann
weight e~$H is replaced in this case by a product of two determinants:
P =detM'|'detM| (2.23)
The Monte Carlo procedures for evaluations of the summation over
these Ising spins now consists of generating a set of Ising spin
configurations cr = (a 0 , cr1, a2, er3, cr4) with probability proportional
to detA/t(<7)-detMj(cr). In deciding whether to accept or reject a
Page 36
29
proposed change of a Ising spin, it is necessary to calculate the
ratios of the determinants between the new and old configurations,
detlWJo7)R“ = d e t '
The total probability for acceptance is P =R^ 'R^. Unfortunately,
detAf^ is nonlocal in the Ising spin a fields, and evaluating R^
directly can be very time consuming, since a direct evaluation of the
determinants would require of order IV3 operations. The exact updat
ing procedures introduced by Blankenbecler, Scalapino and Sugar
(BSS) (37) requires only TV2 operations per update when a change of
Ising spin is accepted. Using this algorithm, we sweep through the
space-time lattice many times, updating one Ising spin variable at a
time. A t time slice I , a single flip of spin changes B f to
^ ( / - b Y ) with
^ = e V,(/)(<r')-V,(/}(<T)_/ i (2>25)
Since each spin cr0 appears only in one diagonal element of
1fp(l) , and spins cr1,o,2,£73,<74 connect to two diagonal elements of
V ^l) , at most two elements of are non-zero. If a flip of one
Ising spin involves sites p and q , then
(4% .
Page 37
30
+ W M . y (2-26)
The ratio of the determinants is
detM„ (a1) R» = detitf„ (<t)
= det[ / +B£ —Bf‘ (o') —B f ] det[ I +B£ Bi* (<r) — B f ] '
We rearrange the factors in the determinants cyclically and have
_ det[ / + B ^ —B(B£ - B f i l +A f ) ] det [ / - B ( B[ - B f \
= d e t [ / - K / - ? f ( / ) m (2.28)
where
»<•(<) = [ ^ + % - S f —B/1]”1, (2.29)
is related to the equal-time single particle Green’s function (37)
(0 (2-30)
Since Zs/1 has at most two non-zero diagonal elements, the deter
minant of (2.28) is reduced to a 2>2 determinant, and can be
evaluated, which gives to
Page 38
31
- 9 V \ q9 V ) (2.31)
The ratio of determinants between the new and old configurations
can be easily calculated to determine whether to accept or reject
proposed change of Ising spins, provided that the Green’s function
gP(l) at time / is known. If the Green’s function g ^ l) is known,
and a proposed change of Ising spin at time / is accepted, the
values of g^il) must be updated to its new value for the next
update. The Green’s function for the new Ising spin configuration
satisfies the Dyson’s equation:
J W = [ / + %
= g'‘( l) " (O IA /V W ■ (2.32)
With only two non-zero elements in Af1, the new values of g^{l)f
are easily found to give:
»W = 9 H i k ~ { [(**-*"((),>) - W ] X-**'ft
+ [(Siq-9l‘(l)yN,V)]X
[?',( 0 „ ( i + i ( i - j ' ,(0 w )-^ ',(0 ) + s V ) qP9“(‘)PjN>‘( t)} }(2.33)
With above information, the procedure for the simulation should be
Page 39
32
clear initially a Ising spin configuration is specified
( °o > ai t * a% i a 4 )o the initial equal-time Green’s function attime 1 is directly calculated using its definition (2.29). We then
sweep through the space-time lattice, updating one Ising spin at a
time. For a proposed change of Ising spin at time / , the probabil
ity
detM / detA/i'P Rj detA/|
is calculated using Eq (2.31) to determine whether to accept or
reject the new configuration. If the new configuration is accepted,
the Green’s function is updated via Eq (2.33). After all spins of a
given time slice I have been updated, the Green’s function for the
next time slice can be obtained through the following relartion
»<■(<-*)=% (2-35)
Ihis process should in principle go on forever, after the initial
configuration is chosen and the Green’s function ff^l) is calculated.
In practice, however, as one proceeds through the lattice, round-off
error builds up due to the finite precision of computers, and it is
necessary to recalculate gp directly from its definition, once the
accuracy of g*1 drops below acceptable level. When the thermal
equilibrium is reached after some warm-up steps, we calculate the
average value of quantities of interest We evaluate the value of
Page 40
33
operator A in a particular Ising spin configuration for every several
MCS. Total M measurements are taken to calculate the final averages.
The main resulte obtained from the simulations are the single
particle Green’s function < c1/l(/1)cyJ(/2) > from which we are able
to calculate all quantities of interest Here the average < > is only
restricted to the trace over fermions, not to the Ising variables. The
single particle equal-time Green’s function is related to <//i(/) via the
following relations:
< « . > ( 0 ^ ( 0 > = g V ) < j • (2.36)
For time-dependent Green’s function, we have
« v > U )e ,i(0 > = [ % - B f »"(!)]o
={[1-9'‘(1)1B/-Brtfy, (2.37)
and
«^ ('K -„(l) > = {[l-9 '‘(l))Bf -B ,V ‘ };,
=[B/‘- B / ?'*(l)]/( . (2.38)
Many-particle Green’s function can be constructed directly from the
single-particle Green’s function using Wick’s theorem (69),
Page 41
34
<cl /)c //)> C c jfc-t( /)Cm( / ) > + < c ^ /)Cm(0><cy(/)cjr t0 > -(2.39)
The average value of an operator A can be obtained in two
steps: First the operator A is averaged over the fermion trace in each
different Ising configuration used for measurements according to a
Monte Carlo probability. Wick’s theorem is generally used to decou
ple the many-body Green’s function into single-particle Green’s
function which is simply ^(Z). In the second step, values of A from
different measurements are averaged to give a final result and the statistical error.
So far we have implicitly assumed that the weight used in the
simulations, P =detMf -detMj, is always non negative. However,
the product of two determinants need not be positive except in some
special cases. When the product is not positive, an additional aver
age over the sign of the probability should be performed to take the
negativity into account This problem could become serious if the
average probability goes to zero (70).
Define:
( d e tM f(< 7) -d e tM j(o - )8%gn{a)- detM^ ay detM^ j »
and
Page 42
35
P f((r) = IdetMfcrJ’detM^cr) | . (2.41)
We have
X) A(<r)detMj (o)-detMj (<r)^ A s = _£______________________________
XJ detM| (<r)-detA/ (<r)a
YtA{a)-s%gn[oryP,(a) a
~ J]'sign(a)’P f{a)
<A-sign <sign 'Zpi (2.42)
Here the probability is chosen as die absolute value of the two
determinants, and the average is with respect to die positive probar
bility. Hie negative sign has been incorporated into the operator A
In addition, the average <$ign must be calculated too.
Page 43
CHAPTER 3. EXTENDED HUBBARD MODEL
In this chapter, we will discuss some resulte obtained from the
simulation of the Hubbard model. The system under consideration is
a cluster representing a square lattice. We are interested in the pro
perties of the system as the electronrelectron interaction parameters
( U, V) vary. The phase diagram of the system in the ( U, V) plane
for the half-filled band case is studied. We find a phase transition
between an antiferromagnetic ordered state and a charge-density
ordered state. We also investigate the transition to a superconducting
state in the negative U case .
36
Page 44
37
3.1 Introduction
The extended Hubbard model has been discussed in chapter 2 in
connection with the development of the quantum Monte Carlo simu
lation technique for the description of interacting fermions. For con
venience, we write down the Hamiltonian again,
H (3.1)i <? > i
where t is the nearest neighbor hopping integral, <Sj> denotes
nearest neighbors, and fx, electron spins ( t 4) ; UandV are on-site
and nearest-neighbor interactions respectively. The chemical poten
tial n e implicitly determines the electron density of the system.
The extended Hubbard model (47) we study here is a direct
extension of the simple one band Hubbard model, in which the
interaction V is set to be zero. Even for the simple Hubbard model,
exact solutions are scare, despite much effort in the past 26 years.
Only in a few instances, do we have exact solutions to the problem.
In one dimension, the exact ground state property were obtained by
Lieb and Wu (15) using Beth-Ansatz technique. They have shown
that the ground state for a half-filled band has short range antifer
romagnetic correlations and the system is an insulator for all posi
tive U. Away from half-filling, numerical results indicate that a
Fermi surface may exist in the ground state. (71) The finite
Page 45
38
temperature properties of the ID half-filled Hubbard model were
studied by Shiba and Pincus (33) using the exact diagonalizafion
method for finite clusters. In three dimensions, Nagaoka (16)
proved that the ground state is ferromagnetic in large U limit, when
there is an extra electron or a single hole in an otherwise half-
filled-band for certain lattices. This has been seen in the exact diag-
onalization of clusters (30). However, it has been argued that
Nagaoka’s theorem can not be applied in the thermodynamic limit
where one must consider a finite fraction of holes in an infinite sys
tem. In the case of half-filled band, the model in strong coupling
limit is equivalent to the antiferromagnetic Heisenberg model (4).
To second order in t, the effective Hamiltonian is
H ,„ = j £ (3-2)<#>
in which is a spin operator on site i. The spins are coupled anti-
4ferromagnetically with J — The large U suppresses double
occupancy of the same sites.
In this paper, we study the extended Hubbard model in two-
dimensions, exploring the effect of the interactions U and V. We
find that the ground state phase diagram shows several regions in
which the properties of the system are qualitatively different
Depending on the values of U and V, the ground state of the
Page 46
39
system can show antiferromagnetic correlations, charge orders, pair
ing correlations characteristic of superconductivity, or condensations.
Emery (72) obtained a ground state phase diagram for the one
dimensional extended model in the weak coupling limit using renor
malization group techniques. A transition from an antiferromagnetic
to a charge density wave state was found on the boundary U = 2 V .
Recent exact diagonalization (34) and Monte Carlo simulation (74)
studies are in general agreement with the early results (75). How
ever, there are still questions about the exact location of the transi
tion and the order of the transition. The solutions from the exact
diagonalization (34) study indicate that the transition is located
slightly above the line U =2V, and is a sharp one.
The discovery of high Tc superconductivity in cuprates has
attracted much attention to the Hubbard model in general, with spe
cial emphasis on the 2D square lattice. Anderson (41) suggested that
the half-tilled 2D Hubbard model in the strong coupling limit may
be able to describe the new high temperature superconductivity, for
which the conventional BCS theory is unable to predict the observed
high transition temperatures. The existence of long range antifer-
romagnetic order in the ground state of the half-filled band Hubbard
model, and the effect of holes on magnetic correlations has been a
major interest Anderson argued that the 2D Hubbard model does
not show long range antiferromagnetic order it its ground state,
Page 47
40
instead, the system is in a singlet resonating valence bond (RVB)
state. Recent simulation studies (75-77), however, have overwhelm
ing evidence to indicate that the 2D half-filled Hubbard model and
the antiferromagnetic Heisenberg model do have antiferromagnetic
long range order in their ground states.
Page 48
41
3.2 The Model
The system under consideration is a square lattice with periodic
boundary conditions, as shown in Fig. 1. The number of sites,
N =4X4. A larger cluster (6X6) has been considered in some
cases. The electron motion in the model is described by t in the Eq.
(3.1). When restricted to nearest-neighbor hopping, the single parti
cle energy level of non-interacting electron corresponds to a band,
E(7c) = —2t(coskx + cosky), (3.3)
with band width W =8t. The density of states for free electron is
shown in Fig. 2, where a logarithmic singularity occurs at E = 0
due to the topology of the two dimensional system. In addition to
the translational symmetry (with periodic boundary condition) and
point group symmetry, electron-hole symmetry exists in the half
filled case. The chemical potential is given at all temperatures by
}j>e = ~ + 4 V as will be shown below.
Consider an electron-hole transformation in the square lattice
with two sublattices A and B:
<*,■„ = ( - 1)% + (3.4)
Page 49
42
Fig. 1. A 4X4 cluster on the square lattice. Periodic boundary
conditions are used here.
O O O
O • •
O • •
O • •
O • •
O O O
O O O
• • O
• • O
• • O
• • O
O O O
Page 50
43
Fig. 2. Density of states for free electrons with single particle
energy band Eq. (3.3).
0.4
0.3
^ 0.2
0.1
0.00 2 4-4E
Page 51
44
where
0 if * is on sub lattice A1 if % is on sublattice B * (3.5)
We have
and
d~+d' = 1 —c ~c-
—&ij •
(3.6)
(3.7)
Under the transformation, the Hamiltonian in the hole-representation
is
H = - t £ + tf£ (l-« |/ 't ) ( l - < )i
+V£ (2-f!/)(2-n/)-Me £(2-n,0<tj> i
=-*S *Vt nu +<0>* * < ? >
- ( U+2 VZ-nc )Y^nf + (U+2 VZ-2/xc) W (3.8)I
where Z is the number of nearest neighbor sites. (Z=4 for two-
dimensional square lattices.) When
Page 52
45
U + 2 VZ =fxc
or
= ~ m = ~ H V , (3.9)
R=-* S *&<tj,+USnlWl +VZnfnf-li''£nf. (3.10)<fj P i <i}> i
Therefore H is invariant under the electron-hole transformation . As
a consequence, the average of electron number equals N when
We have used Monte Carlo simulation method (32,37)
described in chapter 2 to study this model in the parameter space
(U,V) . To control the systematic errors within a few percent, the
time slice parameter L is taken so that ( U + 4 V)Ar~0.5. A typi
cal Monte Carlo run for a square (4X4) cluster involved 2000—4000
measurements separated by two Monte Carlo sweeps {MCS)f and
proceeds by 1000 warm-up sweeps. Calculations were made on
FPS-264 and IBM-3084 computer systems.
As a test of the program, some comparisons were made with
the results of exact diagonalization calculations for some small clus
ters. Fig. 3 shows the results of one of these comparisons in which
the first and second nearest neighbor charge correlation functions are
shown for a ring of eight sites in the case U =2.0 V = 1.5 . The
Page 53
46
Fig. 3. First (lower curve) and second neighbor (upper curve)
charge correlation functions as functions of temperature for a ring of
eight sites. Solid curves are the results of an exact diagonalization
calculation; dots (•) and open circles (o) are the Monte Carlo
results. Parameters: U =2, V =1.5.
2.0
1.0
0.5
0.0 L_ 0.0 t.O 2.0 3.0 4.0
keTt
Page 54
47
agreement is reasonably good, particular at low temperatures. Hie
small discrepancies (about 5%) at higher temperature are attributed
to the fact that the thermal properties in the exact diagonalization
calculations are obtained using a canonical ensemble while the
current Monte Carlo calculation employs a grand canonical ensem
ble (30).
In the case of a four site system (a square) we find good agree
ment between Monte Carlo results and those obtained from exact
diagonalization using a grand canonical ensemble at all temperar
hires. This comparison indicates that the systematic error in the
Monte Carlo calculation is less than 5% for the ZV we used. Hie
statistical error is usually smaller than the size of dots used in our
graphs. For this reason we do not supply error bars in most cases.
Generally we found that statistical errors in correlation functions are
larger than errors in susceptibilities.
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48
3.3 Results
We now present the ground state phase diagram in the parame
ter space of U and V. Various thermodynamic quantities are
obtained, including correlation functions and susceptibilities. To
study the magnetic properties of the system, we calculate the spin-
spin correlations, defined as
<SizSjz > = <(nlt “Wli)(nyf-nyi)> , (3.11)
where the on-site spin correlation function <(5,-z)2 > is the local
magnetic moment The magnetic structure factor is obtained from
the Fourier transform of the spin-spin correlation functions,
S ( t) >ij
= 4 » ,r< ^ < ( n , . f X*Vt ) > (3-12)
A peak of 5(F) at F = (7r,7r) would indicate the formation of antifer
romagnetic ordering. The k -dependent magnetic susceptibility is
the response function of spin-spin correlations, which is given by
= 4 / i r •,(<))>0 ij
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49
= 4?/<*r S e ‘* ^ " ^ )< [ re.t ( r ) “ n. iW ]‘[» it (° ) - « ; i ( 0 ) l >*JV0 ij
= 4 f s s « 'r‘('5’'^ )<[",',(0 -M O] -Kt(l) ] >JV M l;
(3.13)Similarly, charge-charge correlation functions are defined as
<n,ny > = < K T + « t i )(nyr + n ; i ) > , (3.14)
where <jrc,2> measures the double occupancy or on-site charge
correlation. The charge structure factor which measures the change-
density-wave order is given by
S,(H) = 4 < (« ,-« ,)(ny^ , . ) > , (3.15)ij
where n{ is the average number of electron on site t. The charge
susceptibility is
Xc(£) = 4 ? J d r ^ e lki^ - Rf)<lni{T) -«;-][ny(0) - n y]> .JV o ij
(3.16)
To explore the possibility of electron pairing in the model, we also
consider the pairing correlation functions and susceptibilities. The
local singlet pairing susceptibility is defined as
Page 57
50
a,(*) =4 I dT Se W /5)<Sit(r)ca(r)e/l(°)<!jt(0) >■JV o ij(3.17)
For lc =0,
xp(° ) =4 Jdr S<c,rWcaWcjt (o)cyf (0) >JV 0 i;
1 fi= a? / rfr (°)c^ 't (°) >>(3-18)O k k '
where is the Fourier transform of cf/i.
In the following, we will confine our attention to k = 0 (uni
form case) and k = (7T,7r) (staggered case). We will refer S{jf) and
yfif) as the staggered structure factor and susceptibility, respectively.
The staggered magnetic susceptibility will diverge at low tempera
ture if the system is in an antiferromagnetic state. Similarly, we
expect the staggered charge susceptibility to become divergent if the
system is in a charge-density wave state. A divergence in the paring
susceptibility indicates the possibility of forming a superconducting
state.
Page 58
51
A. U>0, V=0
This is the ordinary single band Hubbard model, which has
been studied in the square geometry considered here, by Hirsch
(32), and recently by White et al. (77) for system up to 12X12 in
size. They used a new stabilization method (63) to achieve lower
temperature than previous obtained. There has recently been consid
erable controversy concerning the possibility that the ground states
of the spin % two-dimensional antiferromagnetic Heisenberg model
and the half-filled single band Hubbard model might lack long-range
order (41). Although a formal analytic proof has not been found,
there is overwhelming evidence from numerical simulations (75-77)
that these two models do in fact exhibit long range antiferromag
netic order. As the systems we consider are not large, we do not
obtain conclusive results on this point However, our results are in
agreement with those of Hirsch (32) and White (78), showing the
building up of strong antiferromagnetic correlations at low tempera
ture, and a strongly divergent staggered magnetic susceptibility. We
take the opportunity to introduce the reader to our approach to the
presentation and analysis of data beginning with this rather well stu
died case.
As temperature decreases, electrons in the half-filled band
become more localized, as a result, local moments form gradually
on each site. In addition, spin-spin correlations between these local
Page 59
52
moments start to develop, resulting in an antiferromagnetic ordered
state.
Fig. 4 shows the spin-spin correlation functions as functions of
temperature for a 4X4 square cluster. The alternation of signs in the
correlation functions on neighboring sites clearly shows the antifer
romagnetic type of correlations. Correlations begin to develop
around /3 =1 and nearly reach saturation about j3 = 8 or T = 1/8.
We have used a low temperature algorithm of Hirsch (61) to reach
this low temperature. In Fig. 5 we plot the staggered magnetic
structure factor Sijt) for 17=4 . The building up and saturation of
S(t?) at low temperature again indicate that the system is in an anti
ferromagnetic state at low temperature. Fig. 6 shows the reciprocal
of both the uniform and staggered magnetic susceptibility for U=4 .
The straight lines are linear least-squares fits to the data points with
kB T / t> 1.5. It is apparent that above this temperature, the Curie-
Weiss law applies reasonably well to describe the susceptibilities,
X"1 (3.19)
For the uniform susceptibility, 6 is negative (the least-squares fit
value is $= —0.17±0.07). This is typical of antiferromagnets. On
the other hand, 0 is positive (0.39±0.01) for the staggered suscepti
bility, indicating a possible divergence and a phase transition. The
behavior of the staggered susceptibility here is similar to the
Page 60
53
Fig. 4. First (•), second (0 ) and third (zS) neighbor spin-spin
correlation functions as functions of the inverse temperature /?.
Parameters: U =4, V =0.
AN
t/TN
inv0
0f t
6 8
Page 61
54
Fig. 5. Staggered magnetic structure factor S(lc) for
U =4, V =0. Notice saturates at low temperature around
/? «8 . (•) for F = ( 7 r , 7 r ) ; (A) for F=(0,0).
4
3
2
1
00 2 4 6 8
P
Page 62
55
Fig. 6. Uniform (•) and staggered (o) reciprocal magnetic sus
ceptibility for the case U =4, V =0. The straight lines are least-
squares fits to the data points for kB T/t> 1.5. The intercepts are at
kB0/t = -0.17±0.07 and 0.39±0.01, respectively. The negative 0
for x(0) is typical of antiferromagnets.
8.0
6.0
4.0
2.0
0.0 1.0 2.0 4.03.0kBT
t
Page 63
56
uniform susceptibility for bulk ferromagnets, in which 6 would be
approximately the Curie temperature. However, one sees a tail on
the susceptibility so that does not reach zero at finite tem
perature. This is, in part, a result of the finite size of the sample we
considered. In addition it has been demonstrated many years ago
that the two-dimensional Heisenberg antiferromagnetic model with
finite-range interactions can not have long-range two sublattice order
at finite temperature (78). We expect the same conclusion is appli
cable here to the Hubbard model, which is equivalent to the antifer
romagnetic Heisenberg model in large U limit Long range order
probably exists at T =0.
The reciprocal of the uniform susceptibility ^ (O ) deviates from
linear behavior in the range of temperatures where the linear fit to
X-1C5f) is approaching zero. This behavior is similar to that observed
in exact diagonalization calculations for the Hubbard model (30) on
small clusters. In a bulk three-dimensional antiferromagnet, x -1(0)
has a sharp minimum (cusp) at the Neel temperature and approaches
a finite limit as T—>0. Some recent results for the square lattice
Heisenberg antiferromagnet (79) imply that x-1(0) should also have
a minimum at finite temperature and approach a finite limit at T=0 .
Monte Carlo simulations for the Hubbard model at larger sizes and
lower temperature (77) do show the minimum of X"10) at finite tem
perature. The present results are only partly consistent with these
Page 64
57
expectations, but it is possible that the deviations are due principally
to finite size effects.
We do not show the charge correlation function and susceptibil
ities, nor the pairing correlations and susceptibilities here, because
they are in this case, generally small, and not dramatically tempera
ture dependent
Page 65
58
B. 17=0, V>0.
We turn our discussion to the other limit, where the on-site
interaction U is zero. In this case, the system behaviors completely
different from the previous case A. The system appears to approach
a state [usually described as a charge-density-wave (CDW)j in
which sites are alternately almost doubly occupied or almost empty.
The nearest neighbors of a doubly occupied site would be empty,
and the second neighbors occupied, and so on, conceptually similar
to a two sublattice antiferromagnet
We show in Fig. 7 the development of double occupancy with
decreasing temperature. Three different values of V: 0.5, 0.75, and
1.0 are considered. The tendency toward double occupancy obvi
ously increases with increasing V and decreasing temperature. For
the two larger values of V , there are indications of saturation of
< n2> at low temperature. This behavior is similar to the local
moment formation in the antiferromagnetic case. However the local
moment < S '/> and spin-spin correlation functions in this case do
not show significant increase at low temperature, and remain small.
At first sight, it may appear surprising that the high temperature
limit of < i2> is not 1. However, a simple argument shows that the
correct value is 3 /2 when the electron are uncorrelated. (In a large
system in which the number of electrons equals the number of sites,
if one electron is on a given site, the probability that another one
Page 66
59
Fig. 7. The development of partial double site occupancy for
U =0, V >0 as the temperature decreases is illustrated by the plot
of the average of the square of the site occupation number < n 2 >
The curves are guides to the eye only.
(•) V =0.5; (o) 7 =0.75; (A) 7 = 1.0.
2.0
0.0 1.0 2.0 3.0KbT
t
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60
will be present is only 1/2 because only one spin state is available
there, where other sites have two states). If <&2> is calculated for
the case studied in case A above ( U >0, V— 0) the high temperature
limit for < n 2>is approached from below rather than from above.
Fig. 8 shows the dependence of the charge-correlation function
on the separation of sites for V =0.5 and 1.0 at temperature
kB T/t =1/4. It is quite obvious from the graph that the system is in
a CDW state. It is seen that if site "0” is occupied, first and fourth
neighbors have depressed occupancy, and second, third, and fifth
neighbors have enhanced occupancy. Increasing V increases the
magnitude of the charge alternation. Perhaps the most interesting
point about Fig. 7 is that the magnitude of the charge-charge corre
lation function does not show any appreciable decrease with dis
tance over the range considered, in this case, the whole lattice.
In addition, the reciprocal staggered charge susceptibility xT1
will approach zero, similar to the staggered magnetic susceptibility
in case A above. The staggered charge structure factor shows similar
development and saturation at low temperatures. We will show this
behavior in the next two subsections for charge-density-wave state.
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61
Fig. 8. The charge-charge correlation function is shown for the
central site and the first through fifth neighbors for U =0, and
kB Tft =0.25. (•): V —1.0; (o) V =0.5.
2.0
< n 0 nj>1.0
o . o 1.0 2.0 3.0 4.0 5.0
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62
c . u > o, v>o.
This is the region between the two limiting cases A and B, and
the interesting physics in this region is the competition between U
and V leading to a transition between spin-ordered and charge-
ordered states, as we would expect In one dimension, this crossover
occurs near the line U= 2 V (34,72,73). In the present case of a
two-dimensional square lattice, we find a similar transition near
U=4V. Unfortunately, we can not carry out this calculation to low
enough temperatures to determine accurately the exact location and
order of the transition. The primary difficulty is due to the negative
probability problem we mentioned in chapter 2. The average sign
becomes small at low temperature as the condition U = 4 V is
satisfied.
Fig. 9 shows the average sign [defined in Eq. (2.40)] as parame
ter V increases for U—4 at temperature kB T =1/2. For small
V< U /4 or V<1, the average sign is well behaved, ie. <Sign>is
close to X. At the other end V' U (4, the average sign behaves simi
larly, and we don’t have a problem with negative probability. How
ever, as the system crosses the boundary U «4U , the average sign
drops away from 1 to about 0.6 at temperature kB T—\ f l ,
As the temperature becomes lower, the problem associated with
the negative sign becomes more severe, and the average sign is very
Page 70
<Sig
n>
63
Fig. 9. Average sign as a function of parameter V for
U —4, kB T/t = l/J. The average sign drops well below 1.0 as the
condition U —4 V is satisfied.
1
.8
.6
.4
2
0.8.6 1.21 1.4
V
Page 71
64
close to zero. As a result of this, the statistical errors become very
large, hence it is very difficult (almost impossible) to calculate phy
sical quantities accurately. Loh et al. pointed out that the average
sign decays exponentially to zero as temperature decreases, prevent
ing us from reaching very low temperature (70). Despite this
difficulty in the simulation, we are still able to see the transition
from antiferromagnetic state to charge-density-wave state as the
interaction V increases and acrosses V ~ U /i .
Fig. 10 shows the behavior of the charge-density-wave order
parameter m defined by
" > = < ( ] f E « iF' \ ) ! > (3-20)
for A = (7r,7r) as a function of V for the cases U—2 and 17=4 . For
small V , m is small (in the range of 0.1 and 0.2) indicating the
absence of charge-density-wave order. On the other hand, m begins
a rapid rise near U—A.V to approach 1 for large V.
Figs. 1 1 - 1 3 compare the behavior of staggered magnetic and
charge structure factors for U= 4 and 17=0.5, 1.0, and 1.25 as func
tions of temperature. For V—0.5, the charge structure factor remains
small and is nearly temperature independent, but the magnetic struc
ture factor is developing to become dominant at low temperature.
This is similar to the U >0, V==0 case, where it has been shown
Page 72
65
Fig. 10. Behavior of the CDW order parameter m as a function
of V. m begins a rapid rise near U = 4 V.
(o): U = 2 and kgT/t =1/5; (•): U = 4 and kB T /t =0.5.
m 0.5
o.o 0.5
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66
Fig. 11. Staggered magnetic (•) and charge (o) structure factors
as functions of temperature. U = 4 , V =0.5. Staggered magnetic
structure factor is dominant and increases at low temperature , while
the staggered charge structure factor remains small.
2.0
1.0s
0.5
0.0 1.0 2.0 3 .0
t
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67
Fig. 12. Same as Fig. 11. Parameters are U =4 , ^ = 1.0.
Both structure factors rise at low temperature.
2.0
S
0.5
0.02.0 3 .0
koTt
Page 75
68
Fig. 13. Same as Fig. 11. Parameters are V =4 , V = 1.25.
The staggered charge structure factor develops rapidly at low tem
perature, but the staggered magnetic structure factor is suppressed.
10.0 r
5.0
___ i3 .00.0 2.0
kBTt
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69
that a long-range antiferromagnetic order exists at low temperature.
When V is increased to 1.0 so that the condition 17=4 V is satisfied,
both magnetic and charge structure factors increase at low tempera
ture. Although the magnetic structure factor appears to grow faster,
it is not clear what will happen at T==0. In the case of V=1.25,
the charge structure factor is strongly dominant and the magnetic
structure factor is being suppressed as the temperature decreases. It
is consistent with the exact diagonalization calculations to expect
that the transition occurs for V slightly larger than t / / l (34).
Fig. 14 shows the behavior of magnetic and charge susceptibili
ties for the same set of parameters (U, V). The two types of suscep
tibilities behave similar to the structure factors in the sense that for
small V , the magnetic susceptibility is the dominant one and
diverges at low temperature, but for large V , charge susceptibility
dominates and diverges strongly at low temperature. It is evident
that the system changes from an antiferromagnetic to a charge-order
state near U—4 V. Similar transitions are observed at other values of
U .
At high temperatures, Curie behavior (x « T 4) is expected for
all susceptibilities in view of the expressions for the susceptibilities,
for example, assuming,
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70
Fig. 14. Temperature dependence of staggered spin ( — ) and
charge susceptibilities ( — ) for U =4. Curves are labeled by the
values of V. For small V, the spin susceptibility increases rapidly at
low temperatures. For large V, the charge susceptibility is strongly
dominant as the temperature decrease.
20.0
10.0
5.0
2.0 0 . 5
tx t.O
0.5
0.20.5
1.25
.05 L- 0.0 1.0 2.0 3.0 4.0
ksTt
Page 78
71
is finite. Terms of first order in t in Eq. (3.21) then determine the
intercept 6 (in the extrapolated plot of x-4 against T), while high
terms lead to curvature in the plot In many bulk ferromagnetic sys
tems, plotB of x-1(0) versus T are quite linear until one gets close to
the Curie temperature . The $ determined by a straight-line extrapo
lation is often quite close to the actual transition temperature. In an
antiferromagnetB, x~l(0) should be expected to show a minimum,
while the staggered susceptibility, if it could be measured, should
behave similarly to the uniform susceptibility in a ferroraagnet We
are not aware of any demonstration that a similar situation prevails
in regard to either charge-density-wave or superconductive systems
and there is a question as to whether a sharp phase transition at a
finite temperature is to be expected in any of these 2D systems.
However, we think it remains informative to consider the behavior
of 0 as a function of the parameters of the system.
We show in Fig. 15, the intercept 0 from the least squares fits
to the staggered spin and charge susceptibilities for U—A as a func
tion of V. Clearly, for small values of V, the only indication of a
susceptibility divergence at finite temperature, ie. , 0>O, occurs in
the magnetic susceptibility. In the neighborhood of V=1 , Be for
charge susceptibility becomes positive , and rapidly rises to become
large than 0a . We observe that $a does not go to zero as V
approaches 1 from below. This leads to the plausible conjecture
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72
Fig. 15. Intercept 9 in a Curie-Weiss law least-squares fit to the
staggered spin (•) and charge (o) susceptibilities for 17 = 4 as func
tions of V. Lines are guides to the eye only.
0.5
T
-0.5
- 1.00.4 0.6 0.8 1.0 1.2 1.4V
Page 80
73
that the transition between antiferromagnetic and charge-density-
wave states is sharp as V increases as seen from the exact diagonal-
ization study of small clusters.
Page 81
74
D. U < 0 .
Since the parameter U was introduced to describe electron
repulsion on a single site, it seems, at first* unlikely that negative U
values could arise. However, the possibility of negative U was sug
gested by Anderson (80) in regard to localized electronic states in
amorphous semiconductors. It is also possible that an effective
negative U could arise as a result of competing electronic interac
tions, for example, including polarization effects (81). Further, the
negative U case has attracted some interest in that superconductivity
is expected to result (82). Scalettar et al. (83) have discussed the
phase diagram for the negative U Hubbard model on a square lat
tice, and found that away from half-filling there is a transition at a
finite temperature into a superconducting state. We have considered
negative U in our calculations, as well as negative V.
For the positive U case, we have seen that a transition from an
antiferromagnetic state to a charge-density-wave state occurs near
U =4V, we shall see below that a similar transition for negative U
case exists. The transition from eharge-density-wave to singlet
superconducting state occurs at V =0. there is no pairing for small
positive V. On the other hand, charge-density-wave state is unstable
against even a small negative V.
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75
In the presence of an attractive on-site interaction U, electron
pairs form on lattice sites. With a positive nearest-neighbor repul
sion V>0, these pairs avoid each other and form a charge-density-
wave. These electron pairs are quite localized on lattice sites. How
ever, if V is small but negative, pairing correlations develop at low
temperature, and electron pairs become more delocalized. The sys
tem is in a singlet pairing state. The transition between these two
states occurs for V = 0 . In addition, the charge-density-wave order
and singlet pairing order coexist in the ground state of the negative
U Hubbard model.
Consider a transformation that maps the positive U into nega
tive U when V = 0 ,
= c tf > = (~0 ' cit (3.22)
where Si has been defined in Eq. (3.6) ,
H = ~ t £ citcfr+ U £ n,t ni{ ~ £ nt-i i
— * E W i v - V E "n n?l +-ST- S *?■ (3-23)<ij> « «■
The Sg —Sg correlations in the antiferromagnetic case for the posi
tive U half-filled Hubbard model are directly mapped to the CDW
Page 83
76
correlation for negative U Hubbard model.
S<z 'Sjz =Kt - n,i)(n;t_nyi)
— l)(ny “ !)• (3.24)
In addition, the long-range spin-spin correlations in the x direction
are mapped into pairing correlations,
Therefore, the ground state of the pure negative U Hubbard model
( U <0, V = 0 ) exhibits both long-range CDW and singlet super
conducting pairing order. (In the presence of a small first neighbor
interaction V, one of them is unstable and will disappear). We have
done simulations explicitly for the negative U Hubbard model, and
find that indeed the charge-charge correlation maps to the spin-spin
correlation in z direction of antiferromagnetic case,
<(n,‘ —1 )(n}- — 1 )> is the same as <Siz‘SJg > of Fig. 4. The
singlet pairing correlation function maps to the spin-spin correlation
in x direction.
In Fig. 16, we show the reciprocal of the staggered charge sus
ceptibility xc 0*0 88 a function of T at U = -4 for some positive
’ S j x ( C,"!" Cfj + C,’!" C,-|) ( CyJ Cj j + Cyj Cjf )
(3.25)
Page 84
P" Pmi i
Fig. 16. Reciprocal of the staggered charge susceptibility for
U = —4 and V = 0 .2(0 ), 0.5(«), and 1.0(A). Lines are least-squares
fits to the data points at high temperatures.
5.0
4.0
3.0
2.0
0.0 3.0 4,0
t
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78
values of V( 0.2, 0.5, and 1.0). The susceptibilities suggest the pos
sibility of a divergence at low temperatures. The values of the
intercept of a linear least squares fit increases with V , although all
cases show substantial deviations from linear behavior for low tem
perature or small Xc"4 0?) • Fig. 17 shows the development of the
staggered charge structure factor at low temperature in this case,
indicating, we believe, the formation of a CDW state at low tem
perature, as in the case of positive U when V > U / 4. In fact, the
charge correlation seems to be enhanced by negative U. In the case
of V =1.0, the values of Sc(lf} saturates around 14.7, close to the
maximum possible value of 16. The saturation of ^ (n 1) is similar to
what we have seen for S(7f) in the antiferromagnetic case.
Fig. 18 shows the pairing susceptibility Xp(0)t Eq. (3.17), as a
function of temperature for U = —4, and V =0.2, 0.0, and -0.2. It
is seen, that the pairing correlations are suppressed for small posi
tive V, but are enhanced by a small negative V. The pairing state,
hence, is unstable against a small perturbation of positive V. On
the otiier hand, the charge-density-wave disappears in the presence
of a negative V. (We did not show a graph here). Fig. 19 shows
the reciprocal of the pairing susceptibility for U = —4 and V =0.0
and -0.2 . The results are very close. As in the case of other sus
ceptibility it appears that a kind of Curie-Weiss law is reasonably
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79
Fig. 17. Staggered charge structures factor as function of tem
perature for U = - 4 . V =Q.2(*), 0.5(Z^, and 1.0(o). For V = 1.0,
the value of Se (7?) saturates around 14.7 at low temperatures (max
imum possible value 16).
16.0
12.0
8.0
4.0
IJ .0.0 1.0 2.0 4.03.0kBT
t
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80
Fig. 18. Singlet pairing susceptibility xP (0) as functions of tem
perature for U = —4 and V =0.2(A)f 0.0(o), and —0.2(«). The
pairing correlations are suppressed by small positive V, but are
slightly enhanced by a small negative V.
4.0
3.0
Xp2.0
0.0 0.5 1.5 2.0
t
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81
Fig. 19. Reciprocal of the singlet pairing susceptibility as a
function of temperature for U =-A and V = 0 (o), and — 0.2(*).
The straight line indicates linear extrapolation of the high tempera
ture data.
16.0
12.0
4.0
0.0 1.0 2.0k0T
3.0 4.0
t
Page 89
82
well satisfied at high temperatures with a positive intercept 9 .
It would be interesting to explore the entire phase diagram of
the extended Hubbard model, including the region of large negative
V . In the present Monte Carlo simulation the algorithm appears to
become unstable for negative V as the magnitude of V becomes
larger. This maybe an indication of an approach to a condensed
phase as seen from the diagonalization study of small clusters.
Page 90
3.4 Summ ary
We summarize our quantum Monte Carlo simulations performed
for the extended Hubbard model on a square lattice: We have stu
died properties of the model in different regions of its phase
diagram, and observed the formation of antiferromagnetic, charge-
density-wave, and singlet pairing states. An antiferromagnetic to
charge-density-wave transition is found near the line U = 4 V for
positive U, and a transition from charge-density-wave wave to
singlet pairing state occurs at V = 0 for negative £/; Singlet pairing
of the superconducting type becomes significant for negative U and
V <0, but disappears when V >0. On the other hand, charge-
density-wave is unstable in the presence of a negative V. The phase
diagram of the two-dimensional half-filled Extended Hubbard model
is illustrated in Fig. 20.
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84
Fig. 20. The phase diagram of the half-filled extended Hubbard
model in the two dimensional square lattice.
CDW
U = 4V
U
Page 92
CHAPTER 4. PERIODIC ANDERSON MODEL
This chapter presente some results from the simulation of the
periodic Anderson model. The systems under consideration are 4X4
and 6X6 square lattices. Quantum Monte Carlo calculations are
performed for the symmetric case 2Ef +17 =0. Results show a
qualitative sim ilarity to the single-impurity Anderson model in
regard to the formation of local moments, the behavior of magnetic
susceptibilities, and the screening of / moments by conduction elec
trons. At low temperature we find the development of antiferromag
netic correlations between / local moments, which are not present
in the single impurity case.
85
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86
4.1 Introduction
The impurity Anderson model (2) was proposed in 1961 in
order to explain the formation of local moments on dilute transition
metals impurities in nonmagnetic hosts. The properties of the model
proved to be much more complicated and difficult to determine, par
ticularly at low temperatures, than originally anticipated However,
extensive theoretical efforts ultimately led to an elucidation of the
properties of the model via renormalization group techniques (24)
and an exact solution in certain cases through the Bethe-ansatz
approach (25). Ihe model has been found to exhibit both Kondo
and mixed-valence behaviors, as well as the formation of local mag
netic moments. Recent quantum Monte Carlo calculations (60) pro
vide additional information concerning correlation functions.
Hie Anderson lattice Hamiltonian (20,21,22) provides a natural
generalization of the impurity Anderson model, and may be
appropriate for the description of many rare-earth and actinide ele
ments and compounds, due to the fact that these systems are
involved in the mixed valence and heavy Fermion phenomena. The
idea of hybridization of localized / orbitals with extended s, p and d
orbitals in these systems seems to be applicable. Many of the
phenomena described by the impurity Anderson model occur for the
lattice case, but new physics arises from the hybridization-mediated
interaction between electrons occupying localized orbitals (48).
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87
We are interested in the results which can be obtained for the
Anderson-lattice Hamiltonian by numerical techniques. There are
two types of calculations: (1) exact diagonalization for small clus
ters, and (2) quantum Monte Carlo simulations. The exact diagonali
zation procedure gives resultB for the excited state, and thermo
dynamic properties. However, the number of possible states and
hence the dimension of the Hamiltonian matrix which must be diag-
onalized increases extremely rapidly with the number of sites, so
that the method is applicable only to systems which are quite small.
To date the largest system studied is a tetrahedral cluster (29). The
Monte Carlo method can be used for systems which are significantly
larger, but numerical difficulties arise at low temperatures, such as
the negative probability problem.
The Monte Carlo method used here has been described in
chapter 2. The results that can be obtained can be regarded as
almost exact; the qualifier refers both to the use of finite-size sys
tems (although the variation with the size of the system can be stu
died), and to the presence of some statistical error. There are, how
ever, limitations in regard to the applicability of the method both in
regard to temperature and other parameters.
Page 95
88
4.2 The Model
Our specific model is that of a two-dimensional periodic Ander
son square lattice. The geometry of the system is shown in Fig. 1,
where each lattice site has two lands of orbitals which will be called
d and / here. The d orbitals form the d electron conduction band,
while the / orbitals represent the localized / states. The orbital
degeneracy of actual d and / states is not considered here. The Ham
iltonian of the system is
» =-« S <W, +v S (<#/,„+/.}<*,*)
+ # / £ ”/ , > ( 4 . 1 ) •> *
where <5/> denotes nearest neighbor pairs, and /x refers to electron
spins ( t 4). The first term gives rise to the conduction electron band
with energy levels of Eq. (3.3). The second term represents the
hybridization between the conduction d electrons and the / elec
trons. Ej is the energy of the unhybridized single particle / level.
17 the / electron interaction parameter which measures the short
range Coulomb repulsion between two / electrons with different
spins on the same site. The parameter t is the hopping integral and
sets the energy scale of the system. We will choose units in which
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89
The properties of the system turn out to be very complex, and
very different in different ranges of parameter space (U ,V ,E j).
Our calculations are made for the symmetry case where
2Ej +U =0. In this case, electron-hole symmetric existB as in the
half-filled Hubbard model, and we have < n ^ > = < n d > = 0.5 at
all temperatures. The time slice was chosen so that (A t)2 U =
UfPfL2 =0.07. The systematic error introduced by the breakup is
quite small (about 3%). For the symmetric case, it can be shown
that the product detM|(/)*detM^(/) used in the Monte Carlo simula
tion steps is always positive (32) for any configurations of cr's, so it
can be used directly as a Boltzmann weight, and we do not have a
negative probability problem as we have for the extended Hubbard
model.
A typical Monte Carlo run for a fixed set of parameter
(U,V,Ef,{3) on a 4X4 square cluster involved 3000 measurements
separated by two Monte Carlo sweeps through the lattice, and pre
ceded by 1000 warm up sweeps. We also made some calculation
for a large 6X6 system with a smaller number of measurements.
Calculations were mostly made on the FPS-264 floating-point sys
tem.
Fig. 21 compares the present Monte Carlo results with an exact
diagonalization calculation. The results presented show the Monte
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90
Carlo results for the local / moment <5y2>and the first neighbor
/ —/ spin correlation function >■$$*>and exact diagonaliza-
tion results for a small 2X2 cluster. This cluster is, however, the
largest for which we believe exact diagonalization to be practical
(zero temperature properties can be studied for larger cluster using
Lanczos method). The first neighbor spin-spin correlation functions
agree very well, as do the local moments at low temperature. At
high temperatures the calculated local moments show systematic
differences, which we already have in similar Hubbard model calcu
lations (30), and are attributed to the use of the canonical ensemble
in the thermodynamic calculations by the exact diagonalization
method, while the grand canonical ensemble is employed in the
Monte Carlo Calculations. Hence, charge fluctuations are included
in the Monte Carlo results but not in those obtained by exact diago
nalization.
Page 98
91
Fig. 21. First neighbor /-/ spin correlation functions
and / local moment <5/>for a 2 X2 cluster. (•) Monte Carlo results
for < $ /> left-hand scale; (o) Monte Carlo results for <St S ^ P > ,
right-hand scale. The solid curves are results from exact diagonali
zation calculations using a canonical ensemble.
i.o - 0.13
<sf>
0.5 0.002 3
KTt
Page 99
4.3 R esults
92
In this section, we present some results from the simulation of
the symmetric periodic Anderson model. As we will see from the
results, the properties of the model at high temperature are reminis
cent of those of the impurity Anderson model (20). The system will
pass through several different regions as temperature decreases: the
free orbital region, local moment region and strong coupling region.
In addition to behaviors similar to those of single impurity model,
new physics arises due to the correlations between the localized /
moments at low temperature.
The first quantity we present is the / local moment, defined as
< S f > —<( rij^ —rifiJ 2>= < nAt > + <nm > - 2 < n/lT nfi[ > (4.2)
as a function of temperature for a 4X4 system. As the temperature
is lowered, the effective hybridization between / and conduction
electrons is reduced. As a result, local / electron moments begin to
develop. Fig. 22 shows the gradual formation of the local / moment
in the temperature range 2>kB T /t >1. As expected, increasing U
favors local moment formation and increasing V opposes i t For
U = 0 , <Sy2>= 0.5 , and it is evident that the tendency toward
local moment formation is quite weak for U — I. However, in a
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93
Fig. 22. The local moment <$y >for a 4X4 cluster.
(.) U = 6 , 7 =0.8; ( i ) = 4 , 7 = 0 .8 ; (d) U = 2 , 7 = 0 .8 ;
(o) = 1 , V" =0.8. The solid curves are a guide to the eye only.
<s t>
0.50 2 3KT
Page 101
94
finite system, we do not expect a sharp separation between situations
with and without a local moment Below kB T /t = 2 , the local
moment is nearly independent of temperature, and / electrons are
quite localized.
In the single impurity Anderson model, electron correlations
develop at low temperature between the electrons on the impurity
site and the conduction electrons (24,60). The short range correla
tions tend to screen the local moment At temperatures much lower
than the Kondo (24) temperature, the local moment is completely
screened by the conduction electrons, leading to a nonmagnetic
state. The system is effectively a N — 1 electron system when the
impurity site is screened out Somewhat similar behavior occurs in
the lattice model, where the short range correlations between the
conduction electrons and electrons in the localized / orbital also
tend to screen the local moments (29,48).
We show in Fig. 23 through 25 the total uniform susceptibility
(Xt) f electron susceptibility (x /)- The quantities are defined as:
(4.3)
and
(4.4)
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95
where MT and Mf are the total and / electron moments in the sys-
Fig. 23 shows Tx t ^ d Txj as functions of temperature on a
(1.0 and 0.75). The upper-two curves are for T \ / and lower-two
curves show Tx t - The open circles and triangles are the results for
a 4X4 system. The solid dots are results for a 6X6 system with the
same parameters. Increasing the size of the system does not affect
the susceptibility appreciably in the temperature range studied. The
differences between 6X6 and 4X4 systems appear to be within star
tistical errors.
When the temperature is lowered, Tx t and Tx t both increase
first from the high-temperature value (0,25) of the free orbital
region, as the local moments form. In the temperature range
kB T f t> 5, both susceptibilities follow a Curie-Weiss law,
tem:
Mt — £ ( rei/T ~ nif i + n« t ~ nidi )> (4.5)
Mf — S ( n«/t ~ nifl ) (4.6)
logarithmic scale for two values of the hybridization parameter V
(4.7)
Page 103
96
Fig. 23. Susceptibilities Xt and Xf (multiplied by kB T) for
U =5. Upper dashed line (£) for kB Tx/ ( V =0.75); Lower
dashed line (£) for kB T xr {V =0.75); Upper solid line (o) for
kB Txj (V =1.0); Lower solid line (o) for kBT \r (V =1.0); The
dots (•) show the results for a 6 X6 cluster.
0.4
KT X0.2
0.0
KT
Page 104
97
in which 0 is positive, and C «0.25 (in unit of pB2) as expected.
The paramagnetic Curie temperature 9 is different for Xf and Xr>
and depends on U and V, increasing with U for fixed V and
decreasing with increasing V for fixed U. This behavior is con
sistent with that found for the Hubbard model in the high tempera
ture limit (30).
The quantities Txj and Tx t give some measure to the
effective / moments. A t high temperature, or free orbital region,
their values are close to 0.25. As temperature is lowered, local
moments develop, as shown in Fig. 23, the quantities T x/ and
Txf reach maximum close to, but slightly above that at which
moment formation has saturated. The system is in the local moment
region in this temperature range. As the temperature further
decreases, the effective momentB or the quantities T xf and T \ t
start to decrease as correlations build up between the / moments and
the conduction electrons. The decreases in Txf and Tx t as T
decreases is qualitatively similar to that observed in the single
impurity Anderson model. Also, the temperature dependence of the
susceptibility times the temperature resembles those in the single
impurity model. In the strong coupling limit (low temperature
limit), the local moment on the impurity site is completely screened
out, leading to a zero value for Tx& where Xi is the impurity sus-
Page 105
98
ceptibility. In the present case, we were unable to reach a temperar
ture as low as those in the renormalization group studies to see the
complete quench of local moments, but we do see that Txj and
Tx t decrease from their maximum value as temperature is lowered.
Fig. 24 and 25 supplement the resultB shown in Fig. 22 by
showing the effects of variation of U and V on Tx t and Tx/- The
expected tendencies appear. Hie susceptibilities are increased by
larger electron interaction U, and decreased by stronger hybridizar
tion V. In the case of relatively small U (U—l ), Tx t and T \ f do
not have a region of increase with decreasing temperature, below the
free orbital limit This implies that local moments are insignificant
in this case as seen in Fig. 22. The system directly goes into strong
coupling region as temperature is lowered.
We have seen the screening effect on / local moments by con
duction electrons at low temperature in the decrease of Tx/ and
Tx t - However, the screening is only partially due to the conduction
electrons (34,84). At temperature lower than that at which the /
moments form, the local / moments become correlated through their
interaction with the conduction electrons [as in the Ruderman-
Kitde-FCasuya-Yosida (RKKY) approach]. In Fig. 26, we show the
magnetic structure factor of the / electron defined by
Page 106
99
Fig. 24. / electron susceptibility X/ (multiplied by kB T) for a
4X4 cluster, (o) U = 3 , V =0.6; (ty U = 3 , V =0.8; (A) U = 3 ,
V =1.0; (•) U = 1 , V =1.0.
0.4
KT- Xp
0.2
0.0
KTt
Page 107
100
Fig. 25. Total susceptibility x t (multiplied by kB T) for a 4 X4
cluster, V =0.8. (•) U =6; (A) U =4; U =2; (o) U =1;
0.4
K T X
0.2
0.0
KTt
Page 108
101
-■Jr (4-8)
for the cases F = 0 and F = (tt, tt). The results show the building up
of strong antiferromagnetic correlations, characterized by the
increasing 5(7?). The uniform structure factor £(0) remains nearly
temperature independent) and may be suppressed slightly as tem
perature decreases. As expected, both 5(0) and 5(7?) increase with
increasing U. Unfortunately, the saturation of 5(7?) resides at much
lower temperature range, which we were unable to reach at this
point
As in the single impurity model, the ground state is a singlet
state. We expect this to be true for the lattice model, except that the
screening or compensation is only partially by the conduction elec
trons and partially by the correlations between the / local moments
themselves. Zero temperature Monte Carlo simulations for a one
dimensional periodic Anderson model (84) and exact diagonalizadon
for small clusters (34) show that the ground state of the model is
indeed a singlet, and correlation between /-/ local moments persists
at low temperature. The / moments are compensated in part by
conduction electrons, but are also coupled antiferromagnetically.
Similar results were obtained in the finite temperature Monte Carlo
calculations for the one-dimensional periodic Anderson model (85).
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102
Fig. 26. The magnetic structure factor S(lc] for f electrons.
Dashed line (A) U = 6 , V = 1 ; Solid line (£) U = 4 , V =1;
Upper pair of lines at low temperatures k =(7r,7r); low pair of lines
k =(0,0).
2.0
S( K)
1.0
0.0
KT
Page 110
4.4 Sum m ary
The symmetric periodic Anderson model has been studied by
Monte Carlo simulation technique. The system we consider here are
square lattice in size 4X4 and 6X6. We observe behavior qualita
tively similar to that found in the single-impurity Anderson model in
regard to the formation of f local moments, temperature dependence
of the susceptibility, and screening of local moments by conduction
electrons. Three different regions are identified, namely, the free
orbital region, local moment region and strong coupling region. At
low temperature, correlations develop between local moments,
resulting in the partial screening of local moments by conduction
electrons. The antiferromagnetic correlations between local
moments remain strong at low temperature.
Page 111
CHAPTER 5. CONCLUSIONS
In tins chapter we summerize previous chapters and provide
some suggestions for further studies. We also point out what can be
done using tile simulation program we have developed.
A computer simulation program for the study of interacting
electrons is developed. The Quantum Monte Carlo simulation
method provides nearly exact results for many-body model Hamil
tonians where conventional analytic techniques are not very useful
because of uncontrolled approximations. In addition, this method
can be used to study much larger systems than is possible with other
numerical techniques. The general simulation procedures are based
on the Quantum Monte Carlo simulation algorithm proposed by
Blankenbecler, Scalapino and Sugar (37). The program has been
extended to include Hirsch’s algorithm (61) to allow simulations of
quantum many-body systems at low temperatures.
We have applied the Quantum Monte Carlo simulation method
to the extended Hubbard model in two dimensions and considered a
4X4 square lattice. We studied the properties of the system as the
electron-electron interaction parameters (17, V*) vary. For the half-
filled extended Hubbard model, a transition from an antiferromag
netic to a charge-density-wave state is found near the line U = 4V
for positive U. On the other hand, a transition from a charge
104
Page 112
105
density-wave to a singlet pairing state occurs at V = 0 for negative
U. Hie singlet paring of the superconducting type becomes
significant for negative U and V <0, but disappears when V >0. In
the presence of a negative V, charge-density-wave state becomes
unstable
In addition to simulations of the Hubbard model, we have also
done some calculations for the symmetric periodic Anderson model.
Lattices of 4X4 and 6X6 have been considered. Depending on
parameters of the model, three different regions are identified: the
free orbital region, / local moment region and strong coupling
region. Results show a qualitative similarity to the single-impurity
Anderson model at high temperatures in regard to the formation of /
local moments and the behavior of magnetic susceptibilities. At low
temperatures, electron correlations develop between / local moments
and conduction electrons, resulting in the partial screening of local
moments by conduction electrons. We also find the development of
antiferromagnetic correlations between / local moments, which are
not present in the single impurity Anderson model.
There are several additional problems which can be studied by
these methods. One possible and interesting topic is the existence
of extended singlet pairing correlations in the non-half filled
extended Hubbard model. This type of correlation has been sug
gested in small cluster calculations (34). Another important issue is
Page 113
106
the existence of ferromagnetism in the one band simple Hubbard
model, which the model was originally intended to describe.
Although conclusive results on whether the simple one band Hub*
bard model can have a ferromagnetic ground state have not yet been
obtained, it is suggested that the non-half-filled Hubbard model in
FCC (or BCC) structure might be favorable for the ferromagnetism
(86). It is generally believed that geometry plays an important role
in determining properties of the Hubbard model. This problem can
be studied using the existing programs with slight modifications.
Using the density of state program for electronic structure calcular
tion, the parameters of the Hubbard model can be better chosen to
model reasonablely the structure of a FCC band.
The development of an efficient and stable algorithm for simu
lations of fermion systems remains a very challenging topic. An
algorithm that scales like Nj3 in simulation time is a major need in
order to allow simulations of interacting electrons in larger systems.
Meanwhile the negative probability problem needs to be solved in
order to permit the investigation of the low temperature properties of
interacting electron systems.
Page 114
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Page 124
a ppe n d ix : co m pu ter pr o g r a m s
1.0 General Information ............ 119
2.0 Simulation Flow Diagram..................................... 121
3.0 Extended Hubbard M odel..................................................... 124
3.1 Main Program .................................................................. 124
3.2 Subroutines ............................................................. 139
3.2.1 gnpud ....................................................................... 139
3.2.2 gtotal........................................................................ 141
3.2.3 gbtO .......................................................................... 142
3.2.4 g b tl ........................................................................... 145
3.2.5 expekt....................................................................... 148
3.2.6 h e k ............................................................................ 149
3.2.7 links .......................................................................... 150
3.2.8 hev l........................................................................... 151
3.2.9 inhevl........................................................................ 152
3.2.10 bubdl ...................................................................... 154
3.2.11 inbubdl ................................................................... 155
3.2.12 exam ....................................................................... 156
117
Page 125
118
3.2.13 div .......................................................................... 157
3.2.14 matrirw...................................................... 157
3.2.15 deirn........................................................................ 158
3.2.16 results..................................................................... 159
3.2.17 su sll........................................................................ 164
4.0 Periodic Anderson Model ........ 166
Page 126
119
1.0 General Information
This section gives some explanation about the simulation pro
grams. A detail flow diagram is given in the next section as imple
mented for the Hubbard and Anderson models.
The geometry of a system under simulation is specified in the
subroutine hek. In addition, link gives information about the
nearest-neighbor pairs (as used in the extended Hubbard model).
Subroutine expekt is called to calculate the matrix eK as defined in
Eq. (20). Similarly, hevl and inhevl calculate and e-VW. The
B(l) matrices are calculated in subroutines bubdl and inbubdi
gnpud returns the equal-time Green’s function g(l) used in the
updating procedures. The time-dependent Green’s functions are cal
culated through subroutines g total, gbtO, and gbtl. Finally, Monte
Carlo measurements are performed in results, within which susll
calculates susceptibilities.
The programs have been developed mainly on FPS-264 vector
processors, although part of calculations was run on IBM-3084. The
programs can be easily converted to run on the LSU IBM-3090
supercomputer. The following is a summary of the space and time
requirements for the 4X4 extended Hubbard model running on
FPS-264 and IBM-3084. Note that the calculation made on IBM
uses quadruple precision. MC stands for Monte Carlo
Page 127
120
measurements. The units of computer space and time are Mega
words and minutes, respectively.
u V 0 L MC Space Time Machine
4 1 2 24 2000 1 300 IBM(64bits)
4 0 3 24 4000 1 300 FPS(32bite)
Page 128
121
2.0 Simulation Flow Diagram
5)
^ Start Program ^
Read input parameters
ICalculate useful parameters
ISpecify initial Ising spin variables
' " IStart simulation;
Warm_up sweep=0; MC_measurement=0
Begin first time slice, / =1
ICalculate Green's function g(/) using Eq. (2.29)
IBegin first Ising spin, i =1 for time slice I.
VAssuming an Ising spin flipCalculate the probability P using Eq.(2,34)
©
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Generate a random number x between 0 and 1
Compare P > x ?
Accept the proposed flip
Updating Green's function g(/) using Eq.(2.33)
Last Ising spin for time slice / ?Y
Advance to next Ising spin i = / + 1
Last time slice I - L I
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Correct round-off error for g(/) ?
Monte Carlo measurements = NMEMS ?
Warm_up sweep < MMM ? or C_sweep between measurements < NSWEPJ>
Perform Monte Carlo Measurement
Advance to next time slice 1 = 1 + 1
Update Green’s function for newtime slice Eq.(2.35)
Calculate averages and errors of Monte Carlo data. Simulation Ends
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//HBN4P1 JOB ( 1 1 0 3 , 6 6 9 4 5 , 1 5 , 4 ) , 'YZ',MSGCLASS=S,REGION=4096K / / * NOTIFY=PHHANG/*JOBPARM SHIFT=D /*AFTER NONE/ /* S 1 EXEC FPSCL, FTNPARM=’LIST, LIN, SUBCHK1, VOPT=0/ / S I EXEC FPSCL,FTNPARM='LIST,UN,XOFF(ALL)' ,VOPT-3 //FORT.SYSIN DD *C .......................................................................................................................................CCn
EXTENDED HUBBARD MODEL (U > / 0 , V > / 0 ) IN 2-D.
c SYSTEM SIZE: N = ND X ND SITES.cp
PAIRS OF NEAREST NEIGHBORS : N2 = 2*NUc FOR FALF-FILLED CASE : CHEMICAL POTENTIAL UMcn
UM = U/2+4V : AT ALL TEMPERATURESVj
c OTHER PARAMETERS:c LTIME -> INPUT TIME SLICES (FOR EACH PARTITION;c ( TIME SLICES IN BSS ALGORITHM )c IPMAX - > PARTITION NUMBERc ( IN HIRSCH’S ALGORITHM )c ( FOR BSS ALGORITHM, SET IPMAX=1)cp
IPMAX * LTIME - > MAXIMUM TIME SLICES ALLOWED
c NP = N*IPMAX - > GREEN'S FUNCTION MATRIX SIZEc (FOR BSS ALGORITHM NP=N)c MATRICES USED:c EXPEK(I,J), EKIN - > EXP(K), EXP(-K) USED TO CALCULATE B'Sc K: HOPPING MATRIXc EVU(L,I), EVD - > EXP{VU(L){ , EXP{VD(L)jc V: POTENTIAL DUE TO ISING SPINS.c EVUIN(L.I), EVDIN - > EXP(-VU(L)} , EXP{-VD(L)}c E K I(I .J ) - > ITH NEAREST NEIGHBORSc E ( I ,J ) = 1 .0 IF < I ,J >c EQQ(I.J) - > 1, -1 FOR TWO SUBLATTICESc EQ Q (I,J)=1.0 IF I , J ARE INc THE SAME SUBLATTICEc SIGMA(L.I) AND - > ISING SPIN VARIABLES:c S 1 ( L ,K ) . . . S4 SIGMA FOR U --SITE Ic S I . . .S4 FOR V - - LINK-Kc LINKI(K) , LINKJ - > RETURN SITES CONNECTING LINK-Kcr
I=LINKI(K), J=LINKJ(K)V
c GU(NP.NP), GD(NP.NP) - > SUBSET OF GREEN'S FUNCTIONSc USED IN UPDATING PROCEDURESc TIME DEPENDENT GREEN'S FUNCTIONS:c GUT0(L,I, J ) , GDTO - > < CI(L) * CJ+(1) >c GUTL , GDTL - > < CJ+(L) * CI(1) >cc
GUL , GDL - > < CI(L) * CJ+(L) >
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c MEASURED QUANTITIES:G SIG(MES) -> SIGNC TLN, TLNU, TLND -> TOTAL# , TOTAL # UP, TOTAL # DOWN.C UDN -> < NI(UP) * NI(DN) >C ENRY -> TOTAL ENERGYc S S . . . SSS -> LOCAL MOMENT, SPIN CORRELATIONSc SSI: FIRST NEIGHBOR SPIN CORRELATIONc CCO. . . SSS -> DOUBLE OCCUPANCY, CHARGE CORRELATIONSc CC1: FIRST NEIGHBOR CHARGE CORRELATIONc SUS , SUSQ -> SPIN STRUCTURE FACTOR: K=0, K=PIc SUST, SUSQT -> SPIN SUSCEPTIBILITIES: K=0, K=PIc SUSC, SUSCQ - > CHAGRE STRUCTURE FACTOR: K=0, K=PIc SUSCT.SUSCQT -> CHARGE SUSCEPTIBILITIES: K=0, K=PIc PPC, PPCQ -> PAIRING STRUCTURE FACTOR: K=0, K=PIcr*
PPSt PPSQ -> PAIRING SUSCEPTIBILITIES: K=0, K=PILi " ■
cIMPLICIT REAL*8(A-H,0- Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX)
cDIMENSION EK1(N,N),EK2(N,N),EK3(N,N),EK4(N,N),EK5(N,N),EQQ(N,N) DIMENSION LINKICN2) , LINKJ(N2)DIMENSION SP1(LTT,N2) , SP2(LTT,N2) , SIGMA(LTT.N)DIMENSION SP3(LTT,N2) , SP4(LTT,N2)
CDIMENSION EXPEK(N.N) , EKIN(N.N)DIMENSION EVU(LTT.N) , EVD(LTT.N)DIMENSION EVUIN(LTT,N), EVDIN(LTT,N)
CDIMENSION GU(NP,NP), SU(NP,NP) , XU(N.N) , EU(N,N) , YU(N,N)DIMENSION GD(NP,NP), SD(NP.NP) , XD(N,N) , ED(N,N) , YD(N,N)DIMENSION GU1(NP,NP), GD1(NP,NP)DIMENSION KII(NP*NP), KJJ(NP*NP)
CDIMENSION SIG(LB),TLN(LB),TLND(LB),TLNU(LB),UDN(LB)DIMENSION SPIN(LB), ASPIN(LB) ,ENRY(LB)DIMENSION ZM(LB), CZ(LB)
GDIMENSION SS(LB), S S l(L B ),SS2(L B ),S S3(L B )tSS4(LB),SS5(LB) DIMENSION CCO(LB),CCl(LB),CC2(LB),CC3(LB),CC4(LB),CC5(LB)
CDIMENSION SUS(LB), SUSQ(LB)DIMENSION SUST(LB), SUSQT(LB)DIMENSION SUSC(LB), SUSCQ(LB)DIMENSION SUSGT(LB),SUSCQT(LB)DIMENSION PPC(LB), PPCQ(LB)DIMENSION PPS(LB), PPSQ(LB)
CDIMENSION GUTO(LTT,N,N), GDTO(LTT,N,N)DIMENSION GUTL(LTT,N,N), GDTL(LTT,N,N)
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READfl,1112) LTIME, LG READ(1 ,1 1 1 2 ) NMEMS, MMM, NSWEP READ(1,1112) IR,IW READC1,1112 ) ISEED READfl,1112) MID CL0SE(UNIT=1)
1111 F0RMAT(F11,6)1112 F0RMATCI6)C
C INITIALIZING THE PROGRAM:C
NUPDATE=0 NWARMUP=MMM*NSWEP DELTAT=BETA/(LTIME*IPMAX)WRITE(6,*) ’ SITES =' ,N , ' PMAX=’ ,IPMAXWRITE(6,*) ' U=, ,UO,' V = \ V O / UM=',UM,' IU=',IUWRITE(6,*) ' BETA = ' .BETA,' LTIME=', LTIME,' LG=',LGWRITE(6,*)' MONTE CARLO MESUREMENTS:' , NMEMSWRITE(6,*)' MONTE CARLO STEPS (MCS): ' , NSWEPWRITE(6,*)' WARMUP STEPS: \NWARMUPWRITE(6,*)' IR,IW’ ,IR,IW
DLU : LAMBDA(U) ; DLV LAMBDA(V)DLU= DTANH( DELTAT*UO*0. 25D0)DLU=DLOG(( 1 . 0D0+DSQRT(DLU))**2/(1 .ODO-DLU)) DUU=DSINH(- 2 . ODO*DLU)DUD=DCOSH(- 2 . ODO*DLU)- 1 . 0D0 DLV= DTANH( DELTAT*VO*0.2 5 )DLV=DLOG(( 1 . ODO+DSQRT(DLV) ) * * 2 / ( l . ODO-DLV)) DVU=DSINH(- 2 . ODO*DLV)DVD=DCOSH(- 2 . ODO*DLV) - 1 . ODO
C SPECIFY THE INITIAL ISING SPIN CONFIGURATION: C
DO 50 I=1,LTIME*IPMAX C SIGMA'S
DO 511 K=1,N LX=MOD(K,2)IF(LX.EQ.O) THEN
SIGMAfI,K)=-1 . ODO ELSE
SIGMAfl,K)= l.ODO ENDIF
511 CONTINUEC SI THROUGH S4
DO 512 K -l,N 2 X=RAN(ISEED)IF(X.LT.0.5D0) THEN
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DIMENSION GUL(LTT,N,N), GDL(LTT,N,N)C
COMMON /CDELTAT/DELTATCOMMON /CLTIME/LTIMECOMMON /CDL/DLU.DLVCOMMON /CUI/UO.VOCOMMON /CEI/UMCOMMON / CEXPEK/EXFEKCOMMON /CEKIN/EKINCOMMON /CLINK/LINKI,LINKJCOMMON /CSPINS/SIGMA,SP1,SP2, SP3, SP4COMMON /CHEV/EVU.EVDCOMMON /CHEVIN/EVUIN,EVDINCOMMON /GB/GUTO, GDTO, GUTL, GDTL, GUL, GDLCOMMON /CIJX/IJXCOMMON /CEKS/EK1, EK2, EK3, EK4, EK5, EQQCOMMON /CNS/SIG.TLN,TLNU, TLND, UDN, S PIN, ASPIN, ZM, CZ,ENRYCOMMON /CSS/SS, SS1 , SS2, SS3, SS4, SSSCOMMON /CCC/CCO,CC1,CC2,CC3, CC4, CCSCOMMON /CSS1/SUS, SUSQ, SUST, SUSQTCOMMON /CSS2/SUSC, SUSCQ, SUSCT, SUSCQTCOMMON /CSS3/PPC, PPCQ, PPSQ, PPS
COPEN(UNIT=l)
READ INPUTS:T : HOPPING CONSTANCE, USUALLY T=1
UO : INPUT PARAMETER U VO : VUM : CHEMICAL POTENTIAL.
BETA : 1 OVER TEMPERATUREIU : IF U=0 SET IU=1 ; ELSE IF V=0 SET IU=2 ; ELSE SET IU=0 LG : AFTER UPDATING GREEN'S FUNCTION FOR LG TIME SLICES
RECALCULATE G'S FROM SCRACH.WHY : ROUND-OFF ERRORS.GENERALLY: USE LG=4
LTIME: TIME SLICES. FOR BSS ALGORITHM, THIS IS THE TOTAL SLICES. NMEMS: NUMBER OF MONTE CARLO MEASUREMENTS NSWEP: NUMBER OF MC STEPS BETWEEN MEASUREMENTS
MMM : (MMM * NSWEP) IS THE WARM-UP SWEEPS IR : IF NOT EQUAL TO 0 , SIGMA READ FROM CHANNEL 2 IW : IF NOT EQUAL TO 0 , SIGMA WRITTEN TO CHANNEL 3
SET IR AND IW TO 0 IF THE CALCULATION IS ONE SHOT.
MID : AFTER EVERY MID MEASUREMENTS, CALCULATE RESULTS.
READ(1,1111) T,UO,VO READ(1 ,1111 ) UM READ(1 ,1111 ) BETA READ(1 ,1112) IU
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SP1(I,K)= 1 .ODO SP2(I,K)=-1.ODO SP3(I,K)=-1.0DO SP4(I,K )= l.ODO
ELSE SP1(I,K )=-1.0D 0 SP2(I,K )= l.ODO SP3(I,K)= l.ODO SP4(I,K )=-1.0D 0
END IF 512 CONTINUE 50 CONTINUECC ..................................................................... - ............................ - ...........C IF IR IS NOT 0 , ISING SPINS ARE READ FROM CHANNEL 2.
IF (IR.NE.O) THEN OPEN (UNIT=2)WRITE(6,* ) 'SIGMAS READ FROM 2'READ(2,2000) ((SIGMA(K1,K2),K2=1,N) ,K1=1,LTIME*IPMAX) READ(2 ,2 0 0 0 ) ((SP1(K1,K2) ,K2=1,N2),K1=1,LTIME*IPMAX) READ(2,2000) ((SP2(K1,K2) ,K2=1,N2),K1=1,LTIME*IPMAX) READ(2,2000) ((SP3(K1,K2) ,K2=1,N2),K1=1,LTIME*IPMAX)READ(2,2000) ((SP4(K1,K2) ,K2=1,N2),K1=1,LTIME*IPMAX)CL0SE(UNIT=2)
END IFCC ==============— ===============================*====C GEOMETRY: EK1,. , . , EK5 DEFINE THE GEOMETRY OF THE SYSTEMC
CALL HEK(EK1,EK2,EK3,EK4,EK5,EQQ)CC INITIALIZING THE LINKS BETWEEN NEIGHBORS C
CALL LINKS(EK1)CC CALCULATE EXP(K) AND EXP(-K)C
CALL EXPEKT(EKl)CC CALCUATE THE INITIAL GREEN'S FUNCTIONS FOR MC UPDATING.C
CALL GNPUD(GU,GD,1)CC INITIAL DETERMINANTS OF. THE GREEN'S FUNCTIONS C
CALL DETM(GU,DETGU)CALL DETM(GD,DETGD)WRITE( 6 , * ) 'DETGU=', DETGU, ' DETGD=' , DETGD SIGN=DETGU/DABS(DETGU)SIGN=DETGD/DABS(DETGD)*SIGN WRITE(6,* ) 'SIGN=',SIGN
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cc = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
C MONTE CARLO STEP STARTS HERE:C
MES=0DO 66 MT=1.MMM+NMEMS
CC BEGIN MEASUREMENT AFTER (MMM * NSWEP) WARM-UP SWEEPS THROUGHC THE WHOLE SPACE-TIME LATTICEC SUBROUTINE ^RESULTS* CALCULATES THE AVERAGES OF THE QUANTITIES.C
IF (MT.GT.MMM) THEN CALL GTOTAL(GU.GD)MES=MES+1CALL RESULTS(MES, SIGN,GU,GD)
END IFC
DO 64 MS=1,NSWEPCC MC STEPS : NSWEP SWEEPS BETWEEN EACH MEARSUREMENT.C ======================================================CC SWEEP THROUGH THE TIME SPACE : LTIME SLICEC
DO 100 L=l,LTIMECC RECALCULATE GREEN'S FUNCTION AFTER LG SLICESC TO RESTORE PRECISION.
LL=MOD(L,LG)IF (LL.EQ.O) THEN
CALL GNPUD(GU,GD,L)END IF
CDO 1000 IPP=1,IPMAX
CC SWEEP WITHIN EACH TIME PARTITIONS. (FOR BSS, IPMAX=1)C
LIPP=(IPP-1)*LTIME+LCC UPDATE ISING SPIN AT LATTICE SITES : SIGMA
IF (IU .N E .l) THENDO 110 1=1 ,N
IIPP=(IPP-1)*N+IDNU= DUU*SIGMA(LIPP,I)+DUDDND=-DUU*SIGMA(LIPP, I)+DUDRU=1. 0D0+(1 . 0D0-GU(IIPP,IIPP))*DNURD=1. 0D0+(1 . 0D0-GD(IIPP,IIPP))*DNDRUD=RU*RDPRO=DABS(RUD)IF(DABS(VO).GT.1 .OD-2) PRO=PRO/(1 . ODO+PRO)
C
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C FLIP PROBABILITY: PROC RANDOM NUMBER : X
X=RAN(ISEED)C
IF(PRO.GT.X) THENCC PROPOSED SPIN FLIP SIGMA -> -SIGMA ACCEPTED.C
NUPDATE=NUPDATE+1 IF (RUD.LT.O.ODO) SIGN=-SIGN
SIGMA(LIPP,I)=-SIGMA(LIPP,I)RU=DNU/RURD=DND/RD
CC UPDATING THE GREEN’S FUNCTION ■C
DO 55 J l= l ,N P DJ1I=0. ODOIF (J l.E Q .IIP P ) DJ1I=1. ODO TRU1=-(DJ1I-GU( J l , IIPP ) )*RU TRD1=-(DJ1I-GD( J l , IIPP ) )*RD DO 55 J2=1,NPS U (J l , J2)=GU(J1, J2)+TRU1*GU(IIPP,J2)
55 SD(J1, J2)=GD(J1, J2)+TRD1*GD(IIPP»J2)C
DO 56 K1=1,NP DO 56 K2=l,NP GU(K1,K2)=SU(K1,K2)
56 GD(K1,K2)=SD(K1,K2)CC GREEN’S FUNCTION UPDATED
END IF 110 CONTINUE CC UPDATE ALL SIGMA’ S FOR THE CURRENT TIME SLICE.
END IFCC : UPDATE LINKS : TOTAL N2 LINKS C
IF(IU .N E .2) THEN DO 111 K=1,N2
I=LINKI(K)J=LINKJ(K)IIPP=(IPP-1)*N+IJIPP=(IPP-1)*N+J
CC : SPIN 2 , S2C
DNU= DVU*SP2(LIPP,K)+DVD DND=-DVU*SP2(LIPP,K)+DVD RU=1.0D0+(1 .0D0-GU(IIPP,IIPP))*DNU
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RD=1. 0D0+(1 .ODO-GD(JIPP,JIPP))*DNDRUD=RU*RDPRO=DABS(RUD)IF(DABS(VO).GT.1.0D-2) PRO=PRO/(1 .ODO+PRO)
CC FLIP PROBABILITY : PROC RANDOM NUMBER : XC
X=RAN(ISEED)IF(PRO.GT.X) THEN
C FLIP ACCEPTEDC
NUPDATE=NUPDATE+1 SP2(LIPP,K)=-SP2(LIPP,K)RU=DNU/RURD=DND/RD
CC UPDATE GREEN'S FUNCTIONSC
DO 551 J l= l ,N P DJ1I=0. ODOIF (J l.E Q .IIP P ) DJ1I=1.ODO DJ1J=0. ODOIF(Jl.E Q .JIPP) DJ1J=1.0D0 TRU1=-(DJ1I-GU(J1,IIPP))*RU TRD1=-(DJlJ-GDfJl, JIPP))*RD DO 551 J2=1,NPSUCJ1, J2)=G U (Jl, J2)+GU(IIPP,J2)*TRU1
551 SD(J1, J2)=GD(Jl,J2)+GD(JIPP,J2)*TRD1DO 561 K1=1,NP DO 561 K2=1,NP GU(K1,K2)=SU(K1,K2)
561 GD(K1,K2)=SD(K1,K2)END IF
CC : SPIN 3 , S3C
DNU= DVU*SP3(LIPP,K)+DVDDND=- DVU*SP3(LIPP, K) +DVDRU=1. ODO+(1 . ODO-GU(JIPP,JIPP))*DNDRD=1. ODO+(1 . ODO-GD(IIPP, IIPP))*DNURUD=RU*RDPRO=DABS(RUD)IF(DABS(VO).GT.1.0D-2) PRO=PRO/(1 . ODO+PRO)
CC FLIP PROBABILITY : PROC RANDOM NUMBER : XC
X=RAN(ISEED)IF(PRO.GT.X) THEN
C
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C SPIN FLIP ACCEPTEDC
NUPDATE=NUPDATE+1IF (RUD.LT.O.ODO) SIGN=-SIGNSP3(LIPP,K)=-SP3(LIPP,K)RU=DND/RURD=DNU/RD
CC UPDATE GREEN'S FUNCTIONC
DO 554 J1=1,NP DJ11=0.ODOIF (J l.E Q .IIP P ) DJ1I=1.ODO DJ1J=0. ODOIF(Jl.E Q .JIPP) DJ1J=1.ODO TRU1=-( DJ1J-GU( J l , JIPP))*RU TRD1=-(DJ1I-GD(Jl, IIPP))*RD DO 554 J2=1,NPS U (J l , J2)=G U (Jl, J2)+GU(JIPP,J2)*TRU1
554 SD (J1,J2)=G D (Jl, J2)+GD(IIPP,J2)*TRD1DO 564 Kl=l,NP DO 564 K2=l,NP GU(K1,K2)=SU(K1,K2)
564 GD(K1,K2)=SD(K1,K2)
END IFCC: SPIN 1, SI C
DNU= DVU*SP1(LIPP, K) +DVDDND=- DVU*SP1(LIPP,K)+DVDDETC1=1. 0D0+(1 . ODO-GUQIPP, IIPP))*DNUDETC2=1. ODO+(1 . ODO-GUCJIPP,JIPP))*DNDDETC3= GU(JIPP,IIPP) *DNUDETC4= GU(IIPP,JIPP) *DNDRU=DETC1*DETC2-DETC3*DETC4RUD=RUPRO=DABS(RUD)IF(DABS(V0).GT.1.0D-2) PRO=PRO/(1 . ODO+PRO)
CC FLIP PROBABILITY : PROC RANDOM NUMBER : XC
X=RAN(ISEED)IF(PRO.GT.X) THEN
CC SPIN FILP ACCEPTEDC
NUPDATE=NUPDATE+1 IF (RUD.LT.O.ODO) SIGN=-SIGN
SP1(LIPP,K)=-SP1(LIPP,K)
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RD=DND/RURU=DNU/RU
CC UPDATE GREEN'S FUNCTIONC
DO 552 J1=1,NP DJ11=0.ODOIF(J1.EQ .IIPP ) DJ1I=1.ODO DJ1J=0. ODOIF(J1.EQ.JIPP ) DJ1J=1.ODO TRU1=-(DJ1I-GU(Jl, IIPP) )*RU TRU2=-(DJ1J-GU(Jl, JIPP) )*RD TRUI=TRU1*DETC2+TRU2*DETC3 TRUJ=TRU1*DETC4+TRU2*DETC1 DO 552 J2= l,N P
552 S U (J l , J2)=GU(J1, J2)1 +TRUI*GU(IIPP,J2)+TRUJ*GU(JIPP,J2)
DO 562 Kl=l,NP DO 562 K2=l,NP
562 GU(K1,K2)=SU(K1,K2)END IF
CC: SPIN 4 , S4 C
DNU= DVU*SP4(LIPP,K)+DVDDND=-DVU*SP4( LIPP, K) +DVDDETC1=1. 0D0+(1 . ODO-GD(IIPP,IIPP))*DNUDETC2=1. ODO+(1 . ODO-GD(JIPP, JIPP))*DNDDETC3= GD(JIPP,IIPP) *DNUDETC4= GD(IIPP,JIPP) *DNDRU=DETC1*DETC2-DETC3*DETC4RUD=RUPRO=DABS(RUD)IF(DABS(VO).GT.1 . OD-2) PRO=PRO/(1 . ODO+PRO)
CC FLIP PROBABILITY : PROC RANDOM NUMBER : XC
X=RAN(ISEED)IF(PRO.GT.X) THEN
CC SPIN FLIP ACCEPTEDC
NUPDATE=NUPDATE+1 IF (RUD.LT.O.ODO) SIGN=-SIGN
SP4(LIPP,K)=-SP4(LIPP,K)RD=DND/RURU=DNU/RU
CC UPDATE GREEN'S FUNCTIONC
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DO 553 J l= l ,N P DJ1I=0. ODOIFCJ1.EQ.IIPP ) DJ1I=1. ODO DJ1J=0.ODOIF(J1.EQ.JIPP ) DJ1J=1.0D0 TRD1=-(DJ1I“GD(J1,IIPP) )*RU TRD2=-(DJ1J-GD(Jl, JIPP) )*RD TRDI=TRD1*DETC2+TRD2*DETC3 TRDJ=TRD1*DETC4+TRD2*DETC1 DO 553 J2=l,NP
553 SD(J1, J2)=G D (Jl, J2)1 +TRDI*GD(IIPP,J2)+TRDJ*GD(JIPP,J2)
DO 563 K1=1,NP DO 563 K2=1,NP
563 GD(K1,K2)=SD(R1,K2)END IF
111 CONTINUE CC ALL SPINS FOR THE CURRENT TIME SLICE UPDATED
END IFC1000 CONTINUE CC UPDATE GREEN'S FUNCTION TO NEXT TIME SLICE:C
DO 120 IP1=1,IPMAX DO 120 IP2=1,IPMAX
DO 121 J1=1,N DO 121 J2=1,N J1G=(IP1-1)*N+J1 J2G=(IP2-1)*N+J2 YU(J1,J2)=GU( JIG, J2G )
121 YD(J1,J2)=GD( JIG, J2G )CALL BUBDL(EU,ED,IP1,L)CALL FMMM(EU,YU,XU,N,N,N)CALL FMMM(ED,YD,XD,N,N,N)CALL INBUBDL(EU,ED,IP2,L)CALL FMMM(XU,EU,YU,N,N,N)CALL FMMM(XD,ED,YD,N,N,N)
DO 122 J1=1,NDO 122 J 2= l,NJ1G=(IP1-1)*N+J1J2G=(IP2-1)*N+J2GU( JIG, J2G) =YU(J1,J2)
122 GD( JIG, J2G) =YD(J1,J2)120 CONTINUECC FINISH ONE SWEEP THROUGH THE WHOLE TIME-SPACE LATTICE C100 CONTINUE C
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c =*======================================================C WHEN MS IS LESS THAN THE MC STEPS BETWEEN MEASUREMENTS (NSWEP) C UPDATE GREEN'S FUNCTION AFTER EACH SWEEPC
I F ( ( MT. LE. MMM). OR. (MS. LT. NSWEP)) THEN CALL GNPUD(GU,GD,1)
END IFCC ========================================================C *NSWEP* MONTE CARLO SWEEPS BETWEEN MEASUREMENTS DONE.64 CONTINUECC ========================================================C66 CONTINUE CC AFTER EACH MID MEASUREMENTS , REPORT THE RESULTS.C
MESMOD=MOD(MES,MID)IF(MESMOD. EQ. 0 . AND. MES. NE.0 ) THEN
CWRITE(6,*) 'MEASUREMENT MES',MES CALL DIV(SIG,ASIGN,XSIGN,MES,l.ODO)CALL DIV(TLN,ATLN,XTLN,MES,ASIGN)CALL DIV(TLNU,ATLNU,XTLNU,MES,ASIGN)CALL DIV(TLND, ATLND, XTLND, MES, ASIGN)CALL DIV(SPIN,ASPIN1.XSPINl,MES,ASIGN)CALL DIV(ASPIN,ASPIN2,XSPIN2,MES,ASIGN)CALL DIV(ZM,AZM,XZM,MES,ASIGN)
AZM=AZM/N XZM=XZM/N
CALL DIV(CZ,ACZ,XCZ,MES,ASIGN)ACZ=ACZ/N XCZ=XCZ/N
CALL DIV(SS,ASS,XSS,MES,ASIGN)ASS=ASS/N XSS=XSS/N
CALL DIV(CCO, ACCO, XCCO,MES, ASIGN)ACCO=ACCO/N XCCO=XCCO/N
CALL DIV(UDN,AUDN,XUDN,MES,ASIGN)AUDN=AUDN/N XUDN=XUDN/N
CALL DIV(SSI,ASS1 ,XSSI, MES,ASIGN)ASS1=ASS1/N*0. 5D0 XSS1=XSS1/N*0.5D0
CALL DIV(CC1,ACC1,XCC1,MES,ASIGN)ACC1=ACC1/N*0.SD0 XCC1=XCC1/N*0. 5D0
CALL DIV(SS2, ASS2.XSS2, MES, ASIGN)ASS2=ASS2/N*0.5D0 XSS2=XSS2/N*0. 5D0
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CALL DIV(CC2,ACC2,XCC2,MES,ASIGN) ACC2=ACC2/N*0.5D0 XCC2=XCC2/N*0. SDO
CALL DIV(SS3,ASS3,XSS3,MES,ASIGN) ASS3=ASS3/N XSS3=XSS3/N
CALL DIV(CC3, ACC3, XCC3, MES.ASIGN) ACC3=*ACC3/N XCC3=XCC3/N
CALL DIV(SS4,ASS4,XSS4,MES,ASIGN) ASS4=ASS4/N*0.5D0 XSS4=XSS4/N*0.5D0
CALL DIV(CC4,ACC4,XCC4,MES,ASIGN) ACC4=ACC4/N*0. 5D0 XCC4=XCC4/N*0.5DO
CALL DIV(SS5.ASS5.XSS5, MES.ASIGN) ASS5=ASS5/N*2.0D0 XSS5=XSS5/N*2. ODO
CALL DIV(CC5,ACC5,XCC5, MES,ASIGN) ACC5=ACC5/N*2. ODO XCC5=XCC5/N*2. ODO
CALL DIVCSUST, ASUST.XSUST, MES, ASIGN) ASUST=ASUST/N*0. 5DO XSUST=XSUST/N*0. 5D0
CALL DIV(SUS, ASUS, XSUS, MES, ASIGN) ASUST1=ASUS/N XSUST1=XSUS/N ASUS=ASUST1*BETA*0. 5D0 XSUS=XSUST1*BETA*0. 5D0
CALL DIV(SUSQ,ASUSQ,XSUSQ,MES,ASIGN) ASUSQ=ASUSQ/N XSUSQ=XSUSQ/N
CALL DIVCSUSQT, ASUSQT, XSUSQT, MES, ASIGN) ASUSQT=ASUSQT/N*0. 5DO XSUSQT=XSUSQT/N*0.5D0
CALL DIVC SUSCQ, ASUSCQ, XSUSCQ, MES.ASIGN) ASUSCQ=ASUSCQ/N XSUSCQ=XSUSCQ/N
CALL DIVC SUSCQT, ASUSCQT, XSUSCQT, MES, ASIGN) ASUSCQT=ASUSCQT/N XSUSCQT=XSUSCQT/N
CALL DIVCSUSC, ASUSC, XSUSC, MES, ASIGN) ASUSC=eASUSC-ATLN*ATLN)/N XSUSC=XSUSC/N
CALL DIVCSUSCT, ASUSCT, XSUSCT, MES.ASIGN) ASUSCT=ASUSCT/N-ATLN*ATLN/N*BETA XSUSCT-XSUSCT/N
CALL DIVe PPC, APPC, XPPC, MES, ASIGN) APPC=APPC/N XPPC=XPPC/N
CALL DIVePPCQ, APPCQ, XPPCQ,MES, ASIGN)
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APPCQ=APPCQ/N XPPCQ=XPPCQ/N
CALL DIV( PPS, APPS, XPPS, MES,ASIGN) APPS=APPS/N XPPS=XPPS/N
CALL DIV(PPSQ, APPSQ, XPPSQ, MES, ASIGN) APPSQ=APPSQ/N XPPSQ=XPPSQ/N
CALL DIV(ENRY, AENRY, XENRY, MES, ASIGN) WRITE(6,* ) ' ............................................................
WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITEC6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITEC6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITE(6 WRITEC6 WRITE(6 WRITEC6 WRITEC6
*) ' SIGN ' , ASIGN , ' + / - XSIGN*) ' TOTAL # ' , ATLN , ' + / - ' XTLN*) ' T# UP ' , ATLNU , ' + / - ' XTLNU*) ' T# DN , ATLND , ' + / - ' XTLND*) 1 SPIN 1 , ASPIN1, ' + / - ’ XSPIN1*) ' |SPIN| ' , ASPIN2, ' + / - ' XSPIN2*) ' < s * s > ' , ASS , ’ + / - ' XSS*) ' #U*D , AUDN , ' + / - ' XUDN*) ’ S-S 1 1 , ASS1 , ' + / - ' XSSI*) ' S-S 2 , ASS2 , ' + / - ' XSS2*) ' S-S 3 1 , ASS3 , ' + / - ' XSS3*) ' S-S 4 1 , ASS4 , ' + / “ ' XSS4*) 1 S-S 5 ' , ASS5 , ' + / - ’ XSS5*) 1 C-C 0 ' , ACCO , ' + / - ‘ XCCO*) ' C-C 1 , ACC1 , ' + / - ' XCC1*) ' C-C 2 , ACC2 , ' + / - ' XCC2*) ' C-C 3 1 , ACC3 , ' + / - ' XCC3*) ’ C-C 4 ‘ , ACC4 , ’ + / - ’ XCC4*) ' C-C 5 ' , ACCS , ' + / - * XCC5* ) ' <H> ' , AENRY , ' + / - ' XENRY*) ’ ZM , AZM , ’ + / - ' XZM*) CZ ’ , ACZ , ' + / - ’ XCZ* ) ' s u s , ASUS , ' + / - ' XSUS*) ' SUST ' , ASUST , ' + / - ' XSUST*) 1 SUSQ , ASUSQT, 1 + / - ' XSUSQT*) ' SQ(K=0) ’ , ASUST1, ' + / - ' XSUST1*) ' SQ(K=PI)1 , ASUSQ, ' + / - ’ XSUSQ*) ' CQ(K=0) ' , ASUSC, ’ + / “ ' XSUSC*) ' CQ(K=PI)' , ASUSCQ, ' + / - ' XSUSCQ* ) ' SUSCT ' , ASUSCT + / - ' XSUSCT*) ' SUSCQT ' , ASUSCQT,1 + / - ' XSUSCQT*) ' PPC ' , APPC , ’ + / - ’ XPPC* ) 1 PPCQ ' , APPCQ , ' + / - ’ XPPCQ*) ' PPS ' , APPS , ' + / - ' XPPS*) ' PPSQ ' , APPSQ , ' + / - ' XPPSQ
PRINT OUT THE TOTAL NUMBER OF UPDATES ACCEPTED. USED TO CALCULATE THE ACCEPTANCE RATIO.
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WRITE( 6 , * ) 'NUPDATE', NUPDATEC
IF (IW.NE.O) THEN C SAVE THE ISING SPIN CONFIGURATION, WRITE TO CHANNEL 3.C
OPEN(UNIT=3)WRITE(6,*) 'SIGMAS WRITTEN TO 3'WRITE(3,2000) ((SIGMA(K1,K2),K2=1,N),KI=1,LTIME*IPMAX) WRITEC3 ,2 0 0 0 ) ((SPlfKl,K2),K2=1,N2),K1=1,LTIME*IPMAX) WRITE(3,2000) ( (SP2(K1,K2),K2=1,N2),K1=1,LTIME*IPMAX) WRITEC3 ,2 0 0 0 ) ((SP3(K1,K2),K2=1,N2),K1=1,LTIME*IPMAX) WRITEC3 ,2 0 0 0 ) CCSP4CK1,K2),K2=I,N2),K1=1,LTIME*IPMAX) CLOSECUNIT=3)
END IF 2000 FORMATC10F8.2)CC INTERMEDIATE RESULTS AFTER MID MEASUREMENTS * DONE *
END IFC66 CONTINUE
ENDCC MONTE CARLO MAIN PROGRAM ENDSC
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c ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC SUBROUTINES CG Cc ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC .GNPUD. CALCULATES THE GREEN'S FUNCTION AT TIME SLICE L CC MAXIMUM ALLOWED TIME SLICES LA=40 CC INPUT TIME SLICES : LTIME MUST BE V< IA CC GREEN'S FUCTION SIZE: NP X NP MATRICES CC FOR BSS ALGORITHM, USE IPMAX=1 CC CC ....................................................................................................................................CC
SUBROUTINE GNPUD(GUNP,GDNP,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(LA=40, IPMAX=1J PARAMETER( ND=4, N=ND**2, NP=N*IPMAX)DIMENSION GUNP(NP,NP) , GDNP(NP,NP)DIMENSION GUINP(NP.NP), GDINP(NP,NP)DIMENSION GU(N,N) , EU(N,N) , SU(N,N)DIMENSION GD(N,N) , ED(N,N) , SD(N,N)COMMON /CLTIME/LTIME
CC INITIALIZE THE WORK AREA.
DO 555 1=1 ,NP DO 555 J=1,NP GUINP(I, J ) = 0 . ODO
555 GDINP(I,J ) = 0 . ODO CC....................................... ...........................................................
DO 100 IPP = 1 , IPMAX C CALCULATE B(L)
CALL BUBDL(EU, ED, IPP, L)CC CALCULATE ALL B'S ; AND B (L -1 ) . .B ( 1 ) . B(LTIME). . . B(L+1). B(L)
DO 5 I=L+1,LTIME+L-1 IS=I IPS=IPP
IF ( IPP.EQ.IPMAX .AND. I.GT.LTIME ) THEN IS=IS-LTIME IPS=1
ENDIFC
CALL BUBDL( SU, SD, IPS, IS )CALL FMMM(SU,EU,GU,N,N,N)CALL FMMM(SD,ED,GD,N,N,N)
CDO 7 J1=1,N DO 7 J2=1,N E U (J l , J2)=G U (Jl, J2)
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7 ED(J1, J2)=G D (Jl, J2 )C
5 CONTINUEC
IF(IPP.LT.IPMAX) THEN DO 8 1= 1 ,N DO 8 J=1,N
II=IPP*N+I JJ=(IPP-1)*N+J GUINP(II, JJ )= -G U (I , J )GDINPfII, JJ )= -G D (I , J )
8 CONTINUEC
ELSEC THIS IS THE CASE FOR BSS ALGORITHM, IPP=IPMAX=1
DO 9 1=1 ,N DO 9 J=1,N
JJ=(IPMAX-1)*N+J GUINP(I, JJ)=GU( I , J )GDINP(I, JJ)=GD(I, J )
9 CONTINUE ENDIF
C100 CONTINUE
FULL MATRIX BEFORE THE INVERSION: *M* OR *0*DET(M) OR DET(O) IN THE PARTION FUNCTIN.
DO 110 1=1 ,NPGUINPC1 , 1 ) =GUINP(I, I ) + l . ODO GDINP(I, I)=GDINP(I, I ) + 1 .ODO
10 CONTINUE
CALCULATE THE INVERSE, AND FIND G =(I+B .. . B ) - lTHE GREEN'S FUNCTION
CALL MATRINV(GUINP.GUNP)CALL MATRINV(GDINP.GDNP)
RETURNEND
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ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
GTOTAL CALCULATES THE TIME-DEPANDENT GREEN'S FUCNTIONS C CALLED BEFORE TAKING MEASUREMENT. C
CTOTAL TIME SLICE LTIME*IPMAX : C
CGUT0(L,I,J) = < CI(L)CJ+(1) > CGUTL(L,I,J) = < CJ+(L)CI(1) > CGUL(L,I, J ) = < CI(L)CJ+(L) > C
C■.............................................................................................. C
SUBROUTINE GTOTAL(GUNP, GDNP)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX) DIMENSION GUNP(NP.NP), GDNP(NP.NP)DIMENSION GUTO( LTT, N, N ) , GDTO( LTT, N,N)DIMENSION GUTL(LTT,N,N),GDTL(LTT,N,N)DIMENSION GUL(LTT,N,N), GDL(LTT,N,N)COMMON /CLTIME/LTIME COMMON /GB/GUTO, GDTO, GUTL, GDTL, GUL, GDL
CALCULATE GU AND GD FIRST
CALL GNPUD(GUNP, GDNP,1 )
CALCUALTE ALL G'S IN TWO STEPS:FISRT G'S FOR TIME SLICE \< HALF OF THE TOTAL SLICES SECOND G'S FOR TIME SLICE > HALF OF THE TOTAL SLICES
PURPOSE: REDUCE THE ROUND-OFF ERROR
DO 10 IP=1,IPMAXCALL GBTO(GUNP,GDNP,IP)
10 CALL GBTL(GUNP,GDNP, IP)RETURNEND
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cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
GTBO CALCULATES THE TIME-DEPANDENT GREEN'S FUCNTIONS CALLED BY GTOTALFOR TIME SLICE L \< HALF OF TOTAL TIME SLICES
GUT0(L,I, J ) = < CI(L)CJ+(1) >GUTL(L,I, J ) = < CJ+(L)CI(1) >GULfL,I, J ) = < CI(L)CJ+(L) >
INPUT : GREEN'S FUNCTIONS USED IN THE UPDATING PROCEDURES IP : PARTITION NUMBER; FOR BSS, IP=IPMAX=1
SUBROUTINE GBTO(GUNP,GDNP,IP)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX)
CDIMENSION GUNP(NP.NP) ,GDNP(NP,NP)DIMENSION GUTO( LTT, N, N) , GDTO( LTT, N,N)DIMENSION GUTL(LTT,N,N),GDTL(LTT,N,N)DIMENSION GUL(LTT,N,N), GDL(LTT,N,N)
CDIMENSION BU(N,N),BD(N,N)DIMENSION BUI(N,N),BDI(N,N)DIMENSION EU1(N,N),EU2(N,N)DIMENSION ED1(N,N),ED2(N,N)DIMENSION SU1(N,N),SU2(N,N)DIMENSION SD1(N,N),SD2(N,N)DIMENSION TU1(N,N),TU2(N,N)DIMENSION TD1(N,N),TD2(N,N)
CCOMMON /CLTIME/LTIMECOMMON /GB/GUTO, GDTO, GUTL, GDTL, GUL, GDL
CC INITIALIZE THE LOOP IP C
I F (IP .E Q .l) THEN C THIS IS ALSO THE CASE FOR BSS ALGORITHM: IP=IPAMX=I C
DO 125 M4=1,N DO 125 M5=1,N
GUT0(1,M4,M5)= GUNP(M4,M5)GDTO( 1 ,M4,M5)= GDNP(M4,M5) GUTL(1,M4,M5)=-GUNP(M4,M5) GDTL(1,M4,M5)=-GDNP(M4,M5)GUL(1,M4,M5) = GUNP(M4,M5)GDL(1,M4,M5) = GDNP(M4,M5)EU1(M4,MS) = GUNP(M4,M5)
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ED1(M4,M5) = GDNP(M4,M5) SU1(M4,M5) =-GUNP(M4,M5) SD1(M4,M5) =-GDNP(M4,M5) TU1(H4,M5) = GUNP(M4,M5)
125 TD1(M4,M5) = GDNP(M4,M5)C
DO 1255 M4=l,NGUTL( 1 , M4, M4)=-GUNP(M4, M4) + 1 .ODO GDTL(1 ,M4,M4)=-GDNP(M4,M4)+1.ODO SU1(M4,M4) =-GUNP(M4,M4)+l. ODO
1255 SD1(M4,M4) =-GDNP(M4,M4)+1.0D0C
ELSELIP=(IP-1)*LTIME+1 DO 126 M4=1,NDO 126 M5=1,N
IPM4=(IP-1)*N+M4 IPM5=(IP-1)*N+M5 GUTO(LIP,M4,M5)= GUNP(IPM4,M5) GDTO(LIP, M4, M5)= GDNP(IPM4,M5) GUTL(LIP,M4,M5)=-GUNP(M4,IPM5) GDTL(LIP,M4,M5)=-GDNP(M4,IPM5) GUL(LIP,M4,M5) = GUNP(IPM4,IPM5) GDL(LIP,M4,M5) = GDNP(IPH4,IPM5) EU1(M4,M5)= GUNP(IPM4,M5) ED1(M4,M5)= GDNP(IPM4,M5) SU1(M4,M5)=-GUNP(M4,IPM5) SD1(M4,M5)=-GDNP(M4,IPM5) TU1(M4,M5)= GUNP(IPM4,IPM5)
126 TD1(M4,M5)= GDNP(IPM4,IPM5)END IF
CC HALF OF THE TOTAL TIME SLICES (L2)
L2=LTIME/2+lC
DO 234 LK=2,L2 LK1=LK-1
* CALL BUBDIN(BU,BD,BUI,BDI,IP,LK1)CALL BUBDL(BU,BD,IP,LK1)CALL INBUBDL(BUI,BDI,IP,LK1)
CC * FOR GUTO, GDTOC
CALL FMMM(BU,EU1,EU2,N,N,N)CALL FMMM(BD,ED1,ED2,N,N,N)
CC * FOR GUTL, GDTLC
CALL FMMM(SU1,BUI,SU2,N,N,N)CALL FMMM(SD1,BDI,SD2,N,N,N)
C
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* FOR GUL, GDL
CALL FMMM(BU,TU1,TU2,N,N,N)CALL FMMM(BD,TD1,TD2,N,N,N)CALL FMMM(TU2,BUI,TU1,N,N,N) CALL FMMM(TD2,BDI,TD1,N,N,N)
IPLK=(IP-1)*LTIHE+LK
DO 127 M6=1,N DO 127 M7=1,N
GUT0(IPLK,M6,M7)=EU2(M6,M7) GDT0(IPLK,M6,M7)=ED2(M6,M7) EU1(M6,M7)=EU2(M6,M7) ED1(M6,M7)=ED2(M6,M7)GUTL(IPLK,M6,M7)=SU2(M6,M7) GDTLfIPLK,M6,M7)=SD2(M6,M7) SU1(M6,M7)=SU2(M6,M7) SD1(M6,H7)=SD2(M6,M7) GUL(IPLK,M6,M7)=TU1(M6,M7) GDL(IPLK,M6,M7)=TD1(M6,M7)
CONTINUE
CONTINUERETURNEND
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c ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC GTBL CALCULATES THE TIME-DEPANDENT GREEN'S FUCNTIONS CC CALLED BY GTOTAL CC FOR TIME SLICE L > HALF OF TOTAL TIME SLICES Cc cC GUTO(L,I,J) = < CI(L)CJ+(1) > cC GUTL(L,I,J) = < CJ+(L)CI(1) > CC GUL(L,I,J) = < CI(L)CJ+(L) > CC CC ..................................................... C
SUBROUTINE GBTL(GUNP,GDNP,IP)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX)
CDIMENSION GUNP(NP,NP) ,GDNP(NP,NP)DIMENSION GU(N,N) ,GD(N,N)DIMENSION GUT0(LTT,N,N),GDTO(LTT,N,N)DIMENSION GUTL(LTT,N,N),GDTL(LTT,N,N)DIMENSION GUL(LTT,N,N), GDL(LTT,N,N)
CDIMENSION BU(N,N),BD(N,N)DIMENSION BUI(N,N),BDI(N,N)DIMENSION EU1(N,N),EU2(N,N)DIMENSION ED1(N,N),ED2(N,N)DIMENSION SU1(N,N),SU2(N,N)DIMENSION SD1(N,N),SD2(N,N)DIMENSION TU1(N,N),TU2(N,N)DIMENSION TD1(N,N),TD2(N,N)
CCOMMON /CLTIME/LTIMECOMMON /GB/GUTO,GDTO,GUTL,GDTL,GUL,GDL
CIF(IP.EQ.IPMAX) THEN
C THIS IS ALSO THE CASE FOR BSS ALGORITHM; IP=IPMAX=1 C
LIP=(IPMAX"1)*LTIME DO 125 M4=1,N DO 125 M5=1,N
EU1(M4,M5)=-GUNP(M4,M5)ED1CM4,M5)=-GDNP(M4,M5)SU1(M4,M5)= GUNP(M4,M5)SD1(M4,M5)= GDNPCM4.M5)TU1(M4,M5)= GUNP(M4,M5)
125 TD1(M4,M5)= GDNP(M4,M5)C
DO 1255 M4=1,NEU1(M4,M4)=1.0D0+EU1(M4,M4)
1255 ED1(M4,M4)=1.0DO+ED1(M4,M4)
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ELSEDO 126 M4=1,N DO 126 M5=1,N
IPM4=IP*N+M4 IPM5=IP*N+M5EU1(M4,M5)= GUNP(IPM4,M5) ED1(M4,M5)= GDNP(IPM4,M5) SU1(M4,M5)=“GUNP(M4,IPM5) SD1(M4,M5)=-GDNP(M4,IPM5) TU1(M4,M5)= GUNP(IPM4,IPM5)
126 TD1(M4,M5)= GDNP(IPM4,IPM5)END IF
HALF OF THE TOTAL TIME SLICES (L2) L2=LTIME/2 L21=L2+2
DO 334 LP=LTIME,L21,-1CALL BUBDL(BU, BD, IP , LP)CALL INBUBDL(BUI,BDI, IP,LP)
* GUTO.GDTO **
CALL FMMM(BUI,EU1,EU2,N,N,N)CALL FMMM(BDI, EDI, ED2, N, N, N)
* GUTL.GDTL **
CALL FMMM(SU1,BU,SU2,N,N,N)CALL FMMM( SD1 , BD, SD2, N, N, N )
* GUL,GDL **
CALL FMMM(BUI,TU1,TU2,N,N,N)CALL FMMM(BDI,TD1,TD2,N,N,N)CALL FMMM(TU2,BU,TU1,N,N,N)CALL FMMM(TD2,BD,TD1,N,N,N)
DO 226 M1=1,N DO 226 M2=1,N
IPLP=(IP-1)*LTIME+LP GUL(IPLP,M1,M2)=TU1(M1,M2) GDL(IPLP,M1,M2)=TD1(M1,M2) GUT0(IPLP,M1,M2)=EU2(M1,M2) GDT0(IPLP,M1,M2)=ED2(M1,M2) EU1(M1,M2)*EU2(M1,M2) ED1(M1,M2)=ED2(M1,M2)GUTL(IPLP.Ml,M2)=SU2(M1,M2) GDTL(IPLP,M1,M2)=SD2(M1,M2) SU1(M1,M2)=SU2(M1,M2)SD1(M1,M2)=SD2(M1,M2)
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CONTINUE
CONTINUERETURNEND
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c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC EXPERT CALCULATES THE MATRCIES EXP(K) AND EXP(-K)CC INPUT: HOPPING MATRIX K,C SPECIFIES THE GEOMETRY OF THE SYSTEM.CC RESULTS: EXPEK, AND ERIN CC .............................................................- ...........................................................
SUBROUTINE EXPEKT(EKO)PARAMETER( ND=4, N=ND**2)DIMENSION E 1(N ,N ),E 2(N ,N ),E0(N ,N )DIMENSION EXPEK(N,N),EKIN(N,N),EK1(N,N),EK0(N,N)
CCOMMON /CDELTAT/DELTAT COMMON /CEXPEK/EXPEK COMMON /CEKIN/EKIN
CDO 10 1 = 1 ,N DO 10 J=1,N
EO(I,J)=O.ODO IF (I .E Q .J ) E 0 (I ,J )= 1 .0 D 0
10 EK1(I,J)=-DELTAT*EKO(I,J)C
DO 20 1 = 1 ,N DO 20 J=1,N
EXPEK( I , J ) = E 0 ( I , J)+E K 1(I, J )20 EKIN(I,J) =EO(I, J ) -E K 1 (I ,J )C
FACT= l.ODO FACI=-1.0D0CALL FMMM(EK1,E0,E1,N,N,N)
CDO 30 M=2,100
FACT= FACT*M FACI=-FACI*M HH=1.0D0/FACT IF(HH.LE.1.0D-99) GOTO 70 CALL FMMM(EK1,E1,E2,N,N,N)
CDO 40 1=1,N DO 40 J=1,N
EXPEK(I, J)=EX PEK (I,J)+ l. 0D0/FACT*E2(I, J )40 EKIN(I, J ) =EKINCI,J)+1.0D0/FACI*E2(I,J)30 CALL FMMM(E2,E0,E1,N,N,N)C70 CONTINUE
RETURN END
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ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
THIS SUBROUTINE SPECIFIES THE GEOMETRY OF THE SYSTEM.TO PERFORM CALCULATIONS FOR A DIFFERENT SYSTEM, WE NEED CHANGE THIS SUBROUTINE.
EK1 THROUGH EK5 ARE THE FIRST THROUGH FIFTH NEAREST NEIGHBORS.EQQ SPECIFIES THE TWO SUB-LATTICES
SUBROUTINE HER( EK1 , EK2, EK3, EK4, EK5, EQQ)IMPLICIT REAL*8(A-H,0-Z)PARAMETER( ND-4, N=ND**2)DIMENSION EKl(N.N),EK2(N,N),EK3(N,N),EK4(N,N),ER5(N,N),EQQ(N,N) DIMENSION IJX(N.N)COMMON /CIJX/IJX
CC ( I 1 , J 1 ) j COORDINATE OF THE FIRST LATTICE SITE C N1 : LABEL NUMBER OF THE FIRST LATTICE SITE C ( 1 2 , J 2 ) : COORDINATE OF THE SECOND LATTICE SITE C N2 : LABEL NUMBER OF THE SECOND LATTICE SITE
DO 1 I I =1,ND DO 1 J1 =I,ND DO 1 12 = 1 ,ND DO 1 J2 = 1 ,ND
N1=(I1-1)*ND+J1 N2=(I2-1)*ND+J2 EK1(N1,N2)=0. ODO EK2(N1,N2)=0. ODO EK3(N1,N2)=0. ODO EK4(N1,N2)=0.ODO EK5(N1,N2)=0.ODO EQQ(N1,N2)=1. ODO
CC FIND THE DISTANCE (L ) BETWEEN THE TWO SITESC USE THE PERIODIC BOUNDARY CONDITIONSC
I= IA B S (I1 -I2 )IF(I.EQ .C ND -l)) 1=1 J=IABS(J1-J2)IF(J.EQ.CND-l)) J=1 L=I*I+J*J LL=MOD((I+J),2 )
CIF (L .E Q .l) EK1(N1,N2)=1.ODO IF(L.EQ.2) EK2(N1,N2)=1. ODO IF(L.EQ.4) EK3(N1,N2)=1.0D0 IF(L.EQ.5) EK4(N1,N2)=1.ODO IF(L.EQ.S) EK5(N1,N2)=1.0D0 IF(LL.EQ.l) EQQ(N1,N2)=-1 . ODO 11=12-11+1
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jI F ( I I .L T . l ) 11= II+ND JJ=J2-J1+1I F (J J .L T . l ) JJ= JJ+ND NT=(II-1)*ND + JJ IJX(N1,NT)=N2
C1 CONTINUE
RETURN END
ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc.LINKS. INITIALIZES THE TWO SITES ( I , J ) CONNECTING LINK K
I=LINKI(K); J=LINKJ(K)
RESULTS: COMMON /CLINK/LINKI.LINKJ
SUBROUTINE LINKS(EKl)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(ND=4, N=ND**2, N2=N*2) DIMENSION LINKICN2),LINKJ(N2),EK1(N,N) COMMON /CLINK/LINKI,LINKJ
CK=0DO 10 1= 2 ,N DO 10 J = 1 ,I -1IF( E K 1(I ,J ) .G T .0 .5D 0) THEN
K=K+1 LINKI(K)=I LINKJ(K)=J
END IF 10 CONTINUE
RETURN END
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c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cc cC SUBROUTINE HEVL CALCULATES EXP{V(L)} CC WHERE V IS THE POTENTIALS DUE TO ISING SPINS CC CC L: TIME SLICE CC IP : PARTITION NUMBER ; FOR BSS IP=IPMAX=1 CC CC RESULTS: COMMON /CHEV/EVU(L), EVD(L) CC CC ...............................................................................................................................................c
SUBROUTINE HEVL(IP,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX)
CDIMENSION SIGMA(LTT.N)DIMENSION SP1(LTT,N2),SP2(LTT,N2)DIMENSION SP3(LTT,N2),SP4(LTT,N2)DIMENSION EVU(LTT.N) , EVD(LTT.N)DIMENSION LINKI(N2) , LINKJ(N2)
CC : COMMON INPUT DATA :C
COMMON /CLTIME/LTIMECOMMON /CDL/DLU,DLVCOMMON /CUI/UO.VOCOMMON /CEI/UMCOMMON /CDELTAT/DELTATCOMMON /CLINK/LINKI.LINKJCOMMON /CSPINS/SIGMA, SP1, SP2, SP3, SP4
CC : OUTPUT DATA /COMMON OUT/ :C
COMMON /CHEV/EVU.EVDC
LL=(IP-1)*LTIME+LCC FOR SPIN SIGMAC
DO 20 1=1 ,NEVU(LL, I )= DLU*SIGMA(LL, I)+DELTAT*(UM-0 . 5DO*UO)
20 EVD(LL, I ) =-DLU*SIGMA(LL, I ) +DELTAT*(UM-0 . 5DO*UO)CC FOR SPINS S I , S2, S3 , AND S4C
DO 30 K=1,N2 I=LINKI(K)J=LINKJ(K)EVU(LL, I ) =EVU( LL, I ) +DLV*(SP1(LL,K)+SP2(LL,K))-DELTAT*VO EVD(LL, I)=EVD(LL, I ) +DLV*(SP3(LL, K) +SP4(LL, K)) -DELTAT*VO
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EVU(LL,J)=EVU(LL,J)-DLV*(SP1(LL,K)+SP3(LL,K))-DELTAT*VO 30 EVD(LL,J)=EVD(LL,J)-DLV*(SP2(LL,K)+SP4(LL,K))-DELTAT*VOC
DO 40 1=1,NEVU(LL,I)=DEXP(EVU(LL,I))
40 EVDfLL,I)=DEXP(EVD(LL, I ) )RETURNEND
CCCCc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cc cC *INHEVL* CALCULATES EXP{-V(L)} CC WHERE V IS THE POTENTIALS DUE TO ISING SPINS CC CC L: TIME SLICE CC IP: PARTITION NUMBER ; FOR BSS IP=IPMAX=1 CC CC RESULTS: COMMON /CHEVIN/EVU(L), EVD(L) CC CC ...................................................................................................................................... C
SUBROUTINE INHEVL(IP.L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX)
CDIMENSION SIGMA(LTT.N)DIMENSION SP1(LTT,N2),SP2(LTT,N2)DIMENSION SP3(LTT,N2),SP4(LTT,N2)DIMENSION EVU(LTT.N) , EVD(LTT,N)DIMENSION LINKICN2) , LINKJCN2)
C :C : INPUT DATA :C :
COMMON /CLTIME/LTIMECOMMON /CDL/DLU, DLVCOMMON /CUI/UO.VOCOMMON /CEI/UMCOMMON /CDELTAT/DELTATCOMMON /CLINK/LINKI,LINKJCOMMON /CSPINS/SIGMA, SP1, SP2, SP3, SP4
C :C : OUTPUT DATA /COMMON OUT/ :C :
COMMON /CHEVIN/EVU.EVDC
LL=(IP-1)*LTIME+LCC FOR SPIN SIGMA
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cDO 2 0 1 = 1 , N
EVU(LL, I )= DLU*SIGMA(LL, I)+DELTAT*(UM-0. 5DO*UO)20 EVDfLL, I ) =-DLU*SIGMA( LL, I ) +DELTAT*(UM-0 . 5DO*UO)CC FOR SPIN S I , S2, S3, AND S4 C
DO 30 K=1,N2 I=LINKICK)J=LINKJ(K)EVU( LL, I ) =EVU(LL, I ) +DLV*( SP1 (LL, K)+S P2( LL, K) ) - DELTAT*VO EVD( LL, I ) =EVD(LL, I ) +DLV*( SP3( LL, K) +SP4( LL, K) ) -DELTAT*VO EVU( LL, J ) =EVU( LL, J ) - DLV*(SP1 (LL, K) +SP3( LL, K) ) -DELTAT*VO
30 EVD(LL,J)=EVD(LL,J)-DLV*(SP2(LL,K)+SP4(LL,K))-DELTAT*V0C
DO 40 1=1 ,NE VU ( LL, I ) =DEXP ( - E VU ( LL, I ) )
40 EVD(LL,I)=DEXP(-EVD(LL,I))RETURNEND
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c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC *BUBDL* CALCULATES (BU.BD) AT TIME SLICE : (IP-1)*LTIME+L C C WHERE IP : PARTITION NUMBER, FOR BSS IP=IPMAX=1 CC L : TIME SLICE CC CC .................................................................................................................................................. c
SUBROUTINE BUBDL(BU,BD,IP,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX) DIMENSION BU(N,N),BD(N,N)DIMENSION EEK(N,N)DIMENSION EVU(LTT,N),EVD(LTT,N)
CCOMMON /CLTIME/LTIME COMMON /CEXPEK/EEK COMMON /CHEV/EVU,EVD
CC FIRST CALCULATE EXP{V(L)J C
CALL HEVL(IP,L)CC THEN CALCULATE BL{I,J}C
LL=(IP-1)*LTIME+L DO 10 1= 1 ,N DO 10 J=1,N
BU(I, J)=EEK(I, J)*EVU(LL, J)10 BD(I, J)=EEK(I, J)*EVD(LL,J)
RETURNEND
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c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC INBUBDL CALCULATES (BU-l.BD-l) AT TIME SLICE: (IP-1)*LTIME+L C C WHERE IP : PARTITION NUMBER, FOR BSS IP=IPMAX=I CC L : TIME SLICE CC Cc .............................................................C
SUBROUTINE INBUBDLfBU.BD,IP,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX) DIMENSION BU(N,N),BD(N,N)DIMENSION EEK(N,N)DIMENSION EVU(LTT,N),EVD(LTT,N)
CCOMMON /CLTIME/LTIME COMMON /CEKIN/EEK COMMON /CHEVIN/EVU.EVD
CC FIRST CALCULATE EXP{-V(L)}C
CALL INHEVL(IP.L)CC THEN CALCULATE BL(-l)fI,J]C
LL=(IP-1)*LTIME+L DO 10 1=1,N DO 10 J=1,N
BU(I,J)=EEK(I,J)*EVU(LL,I)10 BD(I,J)=EEK(I,J)*EVD(LL,I)
RETURNEND
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C GCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCcC EXAM IS USED TO CHECK THE PRECISION OF GREEN'S FUNCTIONS C GUI, GDI: GREEN’S FUNCTIONS AT TIME L AFTER UPDATING STEPS.CC NEW GREEN’S FUNCTION GU, GD ARE CALCULATED FROM THE SCRACHC PRECISION IS CHECKED.C ..................................................................................................................................................
SUBROUTINE EXAMfGUl,GD1,L)IMPLICIT REAL*8(A*H,0-Z)PARAMETER( LA=40, ND=4, N=ND**2, IPMAX=1, NP=N*IPMAX)DIMENSION GUl(NP.NP),GD1(NP,NP)DIMENSION GU(NP.NP), GD(NP.NP)DIMENSION EU(NP.NP), ED(NP,NP)
CC INPUT GREEN’S FUNCTIONS AT TIME LC
WRITE( 6 , * ) ’ L=',L WRITE(6,* ) 'GU AFTER UPDATE*WRITE(6,*) ( (G U 1(II , J J ) , JJ=1,N P ),1 1 = 1 ,NP)WRITEC6,*) 'GD AFTER UPDATE'WRITE(6,*) ( (G D 1(II, J J ) , JJ=1 ,N P ),1 1 = 1 ,NP)
CC CALCULATED GREEN'S FUNCTIONS AT TIME LC
CALL GNPUD(GU,GD,L)WRITEC6,*) 'GU FROM GNPUD'WRITEC6,*) ( (G U (II , J J ) , JJ=1,N P ), II=1,NP)WRITE(6,*) 'GD FROM GNPUD'WRITEC6,*) ( (G D (II , J J ) , JJ=1 ,N P ),11= 1 ,NP)
CC CHECK ERRORSC
DO 787 11=1,NP DO 787 JJ=1,NP
E U (II , JJ)=G U 1(II , J J ) -G U (I I , JJ)787 E D (II , JJ)=G D 1(II, J J ) -G D (I I ,JJ )C
WRITE(6,*) 'DIFF GU 'WRITEC6,* ) ( (E U (II , J J ) , JJ= 1 ,N P ),1 1 = 1 ,NP)WRITE( 6 , * ) 'DIFF GD 'WRITEC6,*) C (E D (I I ,J J ) ,J J = 1 ,N P ) ,I I = 1 ,N P )
CC CALCULATE DETERMINANTS:C
CALL DETM(GU,DETGU)CALL DETM(GD,DETGD)WRITE(6,*)'DETGU=',DETGU,' DETGD=' ,DETGDRETURNEND
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c ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC *DIV* CALCULATES THE AVERAGE AND STANDARD DIVIATION CC A : INPUT ARRAY CC D : ACTUAL INPUT NUMBERS; DIMENSION OF A CC B : RETURNS THE AVERAGE OF A CC C : STANDARD DIVIATION OF A, CC SIGN: INPUT AVERAGE SIGN OF THE DERMINANT(M). CC CC ...........................................................................................................................................C
SUBROUTINE DIV(A,B,C,M,SIGN)IMPLICIT REAL*8 (A-H.O-Z)DIMENSION A (10000)
CB=0. ODO DO 10 1= 1 ,M
10 B=B+A(I)B=B/FLOAT(M)B=B/SIGN
CC=0. ODO DO 20 1= 1 ,M
20 C=C+(A(I)-B)**2C=C/FLOAT(M-1)C=DSQRT(C/FLOAT(M))C=C/SIGN
CRETURNEND
CCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc cC INVERSE A MATRIX A - -> TO AINV CC INTORMAKE THERINVERSIONTSUBROUTINE MORE PORTABLE. CC CC ................................................................................................................................................C
SUBROUTINE MATRINV(A,AINV)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(ND=4, N=ND**2, IPMAX=1, NP=IPMAX*N) DIMENSION A(NP,NP), AINV(NP,NP)DIMENSION WV(NP+1)CALL PFINV(NP,A,WV,AINV,IERR)RETURNEND
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c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cC *DETM* CALCULATES THE DETERMINANT OF AN ORDINARY MATRIX B.CC USES THE CROUT REDUCTION METHODC NP IS ACTUAL DIMENSION OF MATRIXC C(NP.NP) ARE WORK SPACE.CC ........................................................ - .- ....................... ................ ...........
SUBROUTINE DETM(B,DET)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(ND=4, N=ND**2, IPMAX=1, NP=N*IPMAX )DIMENSION B(NP,NP),C(NP,NP)
CDO 10 1=1 ,NP DO 10 J=1,NP
10 C(I,J)=O.ODOC
DO 50 1=1,NP DO 50 J=1,NP
Q=B(I,J)DO 20 K=1,NP
20 Q=Q-C(I,K)*C(K,J)C
IF (I .L T .J ) THEN C (I ,J )= Q /C (I ,I )
ELSEC ( I , J)=Q
ENDIFC50 CONTINUEC
DET=1. ODO DO 60 1=1,NP
60 DET=DET*C(I,I)RETURN END
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ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c
*RESULTS* PERFORMS MONTE CARLO MEASUREMENTS. CMEASUREMENT NUMBER : MES CSIGN OF DETMINANTS FOR THIS MEASUREMENT: SIGN CALL MEASUREMENT COMMONED OUT. C
C............................................................................................................................................. C
SUBROUTINE RESULTS(MES,SIGN,GU,GD)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX)
DIMENSION GU(NP,NP), GD(NP,NP)DIMENSION EK1(N,N), EK2(N,N),EK3(N,N),EK4(N,N),EK5(N,N),EQQ(N,N) DIMENSION LINKI(N2), LINKJ(N2), IJX(N,N)DIMENSION SIG(LB), TLN(LB), TLND(LB), TLNU(LB)DIMENSION SS(LB), UDN(LB)DIMENSION SPIN(LB),ASPIN(LB),ENRY(LB)DIMENSION ZM(LB), CZ(LB)DIMENSION S S l(L B ), SS2(LB), SS3(LB), SS4(LB), SS5(LB)DIMENSION CCO(LB), CCl(LB), CC2(LB), CC3(LB), CC4(LB),CC5(LB)DIMENSION SUS(LB), SUSQ(LB), SUST(LB), SUSQT(LB)DIMENSION SUSC(LB),SUSCQ(LB),SUSCT(LB).SUSCQT(LB)DIMENSION PPC(LB), PPCQ(LB), PPSQ(LB), PPS(LB)DIMENSION PTC(LB), PTCQ(LB), PTSQ(LB), PTS(LB)
COMMON /CUI/UO,VO COMMON /CLINK/LINKI,LINKJ COMMON /CIJX/IJXCOMMON /CEKS/EK1,EK2,EK3,EK4, EK5, EQQCOMMON /CNS/SIG,TLN, TLNU, TLND, UDN, SPIN, ASPIN, ZM, CZ, ENRYCOMMON /C S S /S S ,S S 1 ,S S 2 , SS3, SS4, SS5COMMON /CCC/CCO, CC1 ,CC2, CC3, CC4, CC5COMMON /CSS1/SUS, SUSQ, SUST, SUSQTCOMMON /CSS2/SUSC, SUSCQ, SUSCT, SUSCQTCOMMON /CSS3/PPC, PPCQ, PPSQ, PPSCOMMON /CSST/PTC, PTCQ, PTSQ, PTS
T=1.0DOSIG(MES) =SIGN TLN(MES) = 0 .ODO TLNU(MES) = 0 .ODO TLND(MES) = 0 .ODO SS(MES) = 0 .ODO CCO(MES) * 0 .ODO UDN(MES) =0.0D0 SPIN(MES) = 0 .ODO ASPIN(MES)=0. ODO ZM(MES) = 0 .ODO CZ(MES) = 0 .ODO
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ENRY(MES) = 0 .ODOCC SUM OVER THE SITES C
DO 651 1 = 1 ,NZM(MES) = ZM(MES) + ( G D (I ,I ) -G U (I ,I ) ) * EQQ(1,I)CZ(MES) = CZ(MES) + ( 2 . ODO -GD(I, I ) -G U (I , I ) ) * EQQ(l.I)SPIN(MES) = SPIN(MES)+ GD(I, I ) -G U (I , I )TLN(MES) = TLN(MES) + 2 .0 D 0 -G U (I ,I ) -G D (I ,I )TLNU(MES) = TLNU(MES)+ 1 . ODO-GU(I, I )TLND(MES) = TLND(MES)+ 1 . ODO~GD(I, I )SS(MES) = SS(MES) + GU(I, I)+G D (I, I ) - 2 . QDO*GU(I,I)*GD(I, I )CCO(MES) = CCO(MES) + 4.0D0-3.OD0*( G U (I ,I )+ G D (I ,I ) )
1 + 2 . ODO*GU(I, I)*GD(1 ,1 )651 UDN(MES) = UDN(MES) + ( 1 . ODO-GU(I, I ) ) * ( 1 . ODO-GD(I, I ) )C
ASPIN(MES)= DABS(SPIN(MES))ZM(MES) = DABS(ZMCMES))CZ(MES) = DABS(CZ(MES))
CC SUSCEPTIBILITIES AND STRUCTURE FACTORS C
SUS(MES) = O.ODO SUST(MES) = O.ODO SUSQ(MES) = O.ODO SUSQT(MES)= O.ODO SUSC(MES) = O.ODO SUSCT(MES)= O.ODO SUSCQ(MES)= O.ODO SUSCQT(MES)=0. ODO PPC(MES) = O.ODO PPCQ(MES) = O.ODO PPS(MES) = O.ODO PPSQ(MES) = O.ODO PTC(MES) = O.ODO PTCQ(MES) = O.ODO PTS(MES) = O.ODO PTSQ(MES) = O.ODO
CCC
DO 85 1=2 ,N DO 85 J = 1 ,I -1
TES + ( 1 .ODO-GU(I,I)) * ( 1 . 0 -G U (J ,J )) -G U (J,I)*G U (I, J ) + ( 1 . ODO-GD(I, 1 ) ) * ( 1 . 0-GD(J, J ) ) -G D (J,I)*G D (I, J) - ( 1 . ODO-GU(I, I ) ) * ( 1 . ODO-GD(J, J ) )
111 - ( 1 . ODO-GU(J, J ) ) * ( 1 . ODO-GD(I, I ) )
111
TEC + (1 .0 D 0 -G U (I ,I ) )* (1 .0 -G U (J ,J ) ) -G U (J ,I )* G U (I ,J ) + ( 1 .ODO-GD(I, I ) ) * ( 1 . 0*G D (J,J)) -G D (J,I)*G D (I,J) + ( 1 . ODO-GU(I, I ) ) * ( 1 . ODO-GD(J,J))+ (1 .0 D 0 -G U (J ,J ) )* (1 .0 D 0 -G D (I ,I ) )
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SUS(MES) = SUS(MES) + TESSUSQ(MES) = SUSQ(MES) + TES*EQQ(I,J)SUSC(MES) = SUSC(MES) + TECSUSCQ(MES)= SUSCQ(MES)+ TEC*EQQ(I,J)PP = G U (I,J)*G D (I,J) + G U (J ,I)*G D (J ,I)PPC(MES) = PPC(MES) + PPPPCQ(MES) = PPCQ(MES) + PP* EQQ(I,J)PTIJ = 0 .ODO
CDO 855 K -l,N
IX=IJX(I,K)JX=IJX(J,K)PTIJ=PTIJ + GU(I, J ) * GU(IX,JX) - GU(I.JX) * GU(IX,J)
1 + G U (J,I)* GU(JX.IX) - GU(J,IX) * GU(JX.I)855 CONTINUE C
PTC(MES) = PTC(MES) + PTIJ/N PTCQ(MES) = PTCQ(MES) + PTIJ/N * EQQ(I.J)
85 CONTINUEC
SUS(MES) = 2 .ODO*SUS(MES)SUSQ(MES) = 2 . 0D0*SUSQ( MES)SUSC(MES) = 2.0D0*SUSC(MES)SUSCQ(MES)= 2 . 0D0*SUSCQ(MES)
CDO 86 1=1 ,N
PPC(MES) = PPC(MES) + GU(I, I)*GD(1 ,1 )PPCQ(MES) = PPCQ(MES)+ G U (I ,I )* G D (I ,I )PTIJ=0. ODO
CDO 865 K=1,N IX=IJX(I,K)
865 PTIJ = PTIJ + GU(I, I)*GU(IX,IX) - GU(I,IX)*GU(IX,I) C
PTC(MES) = PTC(MES) + PTIJ/N PTCQ(MES)= PTCQ(MES)+ PTIJ/NSUSC(MES)= SUSC(MES)+ 4 .0 D 0 -3 .0 D 0 * (G U (I ,I )+ G D (I ,I ) )
1 + 2 . 0D0*GU( 1 , 1 )*GD(1 ,1 )86 SUSCQ(MES)=SUSCQ(MES)+4.ODO-3. 0D0*(GU(I, I)+GD(1 ,1 ) )
1 + 2 .0 D 0 * G U ( I ,I )^ D ( I , I )C
SUS(MES) = SUS(MES) +SS(MES)SUSQ(MES)= SUSQ(MES)+SS(MES)
CALCUALTE SUCEPTIBILITIES
CALL SUSLL(EQQ, SST, SSQ, CCT, CCQ, PPS1 , PPS2, PTS1 , PTS2) SUST(MES) = SST SUSQT(MES) = SSQ SUSCT(MES) = CCT SUSCQT(MES)= CCQ
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PPS(MES) = PPS1 PPSQ(MES) = PPS2 PTS(HES) = PTS1 PTSQ(MES) = PTS2
CC TOTAL ENERGY C
DO 856 1 = 1 ,N DO 856 J=1,N
856 ENRY(MES) = ENRY(MES) - T * (G U (I,J)+G D (I,J) ) * EK1(I,J)C
DO 8561 I = 1 ,NENRY(MES) = ENRY(MES) + UO* (1 .0 D 0 -G U (I ,I ) ) * (1 .0 D 0 -G D (I ,I ) )
8561 CONTINUE C
DO 8563 K = 1 ,N2 I=LINKI(K)J=LINKJ(K)VNINJ = (1 .0 D 0 -G U (I ,I ) )* (1 .0 -G U (J ,J ) ) -G U (J ,I )* G U (I ,J )
1 + ( 1 .0D0-GD(I, I ) ) * ( 1 . 0 -G D (J ,J ))-G D (J ,I)*G D (I , J )1 + ( 1 .ODO-GU(I, I ) ) * ( 1 .0D 0-G D (J,J))1 + ( 1 . ODO-GU(J, J ) ) * ( 1 . ODO-GD(1 , 1 ) )ENRY( MES) =ENRY(MES) + VO* VNINJ
8563 CONTINUE CC CORRELATION FUCTIONS C
SS1(MES)=0. ODO SS2(MES)=0. ODO SS3(MES)=0. ODO SS4(MES)=0. ODO SS5(MES)=0. ODO CC1(MES)=Q.ODO CC2(MES)=0. ODO CC3(MES)=0, ODO CC4(MES)=0.ODO CC5(MES)=0.0D0 DO 891 1=2 ,N DO 891 J = 1 ,I -1
TERMS = ( 1 . ODO-GU(I, I ) ) * ( 1 . ODO-GU(J,J)) -G U (J,I)*G U (I, J )1 + ( 1 . 0D0-GD(I, I ) ) * ( 1 . ODO-GD(J, J ) ) -G D (J ,I )* G D (I ,J )1 - ( l .O D 0 -G U (I ,I ) )* ( l .O D 0 -G D (J ,J ) )1 - ( 1 . ODO-GUCJ, J ) ) * ( 1 . ODO-GDC1 , 1 ) )
TERMC = ( 1 . ODO-GU(I, I ) ) * ( 1 . ODO-GU(J,J)) -GU(J, I)*G U(I, J )1 + ( 1 . 0D0-GD(I, 1 ) ) * ( 1 . ODO-GDCJ,J))-GD(J, I)*G D(I, J )1 + ( 1 .ODO-GU(I, I ) )* (1 .0 D 0 -G D (J ,J ) )1 + (1 .0 D 0 -G U (J ,J ) )* (1 .0 D 0 -G D (I ,I ) )
SSl(MES) = SS1 (MES) +TERMS*EK1 ( I , J )SS2(MES) = SS2( MES) +TERMS*EK2( I , J )SS3(MES) = SS3(MES)+TERMS*EK3(I, J)SS4(MES) = SS4(MES)+TERMS*EK4(I, J)
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SS5(MES) = SS5( MES) +TERMS*EK5( I , J ) CCl(MES) = CC1 (MES) +TERMC*EK1 ( I , J ) CC2CMES) = CC2(MES)+TERMC*EK2(I, J ) CC3(MES) = CC3(MES)+TERMC*EK3(I, J ) CC4(MES) = CC4(MES)+TERMC*EK4(I,J ) CCS(MES) = CC5(MES)+TERMC*EK5(I,J)
891 CONTINUE
FIX-UP THE SIGN PROBLEM
TLN(MES) = TLN(MES) * SIGNTLNU(MES) = TLNU(MES) * SIGNTLND(MES) = TLND(MES) * SIGNSS(MES) SS(MES) * SIGNUDN(MES) = UDN(MES) a SIGNSPIN(MES) = SPIN(MES) a SIGNSUS(MES) = SUS(MES) a SIGNSUST(MES) = SUST(MES) * SIGNSUSQ(MES) = SUSQ(MES) a SIGNSUSQT(MES)= SUSQT(MES) a SIGNSUSC(MES) = SUSC(MES) * SIGNSUSCT(MES)= SUSCT(MES) * SIGNSUSCQ(MES)= SUSCQ(MES) * SIGNSUSCQT(MES)= SUSCQT(MES)* SIGNPPC(MES) = PPC(MES) * SIGNPPCQ(MES) = PPCQ(MES) a SIGNPPS(MES) = PPS(MES) * SIGNPPSQ(MES) = PPSQ(MES) * SIGNENRY(MES) = ENRY(MES) * SIGNSSl(MES) = SSI(MES) ★ SIGNSS2(MES) = SS2(MES) * SIGNSS3(MES) = SS3(MES) * SIGNSS4(MES) = SS4(MES) A SIGNSS5(MES) = SSS(MES) * SIGNCCO(MES) = CCO(MES) * SIGNCCl(MES) = CCl(MES) A SIGNCC2(MES) = CC2(MES) * SIGNCC3(MES) = CC3(MES) A SIGNCC4(MES) = CC4(MES) * SIGNCCS(MES) = CCS(MES) A SIGNRETURNEND
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c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc cC SUSLL CALCULATES THE SUSCEPTIBILTIESCC ..............................................................................................................................
CSUBROUTINE SUSLL(EQQ, SUST, SUSQT, SUSCT, SUSCQT, PPS, PPSQ, PTS, PTSQ) IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N*IPMAX)DIMENSION GUTO(LTT»N,N),GDTO(LTT,N,N),EQQ(N,N)DIMENSION GUTL(LTT,N,N),GDTL(LTT,N,N)DIMENSION GUL(LTT,N,N),GDL(LTT,N,N)DIMENSION IJX(N,N)
CCOMMON /CLTIME/LTIME COMMON /CDELTAT/DELTAT COMMON /GB/GUTO,GDT0,GUTL, GDTL,GUL, GDL COMMON /CIJX/IJX
CSUST = 0 . ODO SUSQT = O.ODO SUSCT = O.ODO SUSCQT= O.ODO PPS = O.ODO PPSQ = 0 . ODO PTS = 0 . ODO PTSQ = 0 . ODO
CC TOTAL TIME SLICES = LTIME * IPMAXC
DO 10 L=1,LTIME*IPMAX DO 10 1 = 1 ,N DO 10 J=1,N
PPS = PPS + GUTO(L, I , J ) * GDTO(L,I, J )PPSQ = PPSQ + GUTO(L,I, J ) * GDTO(L,I,J) a EQQ(I,J)TERM1 = ( 1 . ODQ-GUL(L, I , I ) ) * ( 1 .0D0-G UL(1,J,J))
1 + GUTLCL,J,I) * GUTO(L,I,J)TERM2 = (1 .0 D 0-G D L (L ,I ,I )) a ( 1 . ODO-GDL(1 ,J,J)D
1 + GDTL(L.J.I) a GDTO(L,I,J)TERM3 = ( 1 . ODO-GUL(L, 1 , 1 ) ) a ( 1 ,0D0-G DL(1.J,J))TERM4 = ( 1 . 0D0-GDL(L, 1 , 1 ) ) a ( l .ODO-GUL(l,J,J))SUST = SUST+ TERM1+TERM2- TERM3 - TERM4SUSQT = SUSQT+(TERM1+TERM2-TERM3-TERM4)AEQQ(I,J)SUSCT = SUSCT+ TERM1+TERM2+TERM3+TERM4SUSCQT= SUSCQT+(TERM1+TERM2+TERM3+TERM4)AEQQ( I , J)PTIJ = O.ODO
CDO 20 K=1,N IX=IJX(I,K)JX=IJX(J,K)
a n
n ca
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20 PTIJ = PTIJ+ GUT0(L,I, J)*GUT0(L, IX, JX)-GUT0(L,I,JX)*GUT0(L, IX,J )C
PTS = PTS + PTIJ/NPTSQ = PTSQ+ PTIJ/N *EQQ(I,J)
10 CONTINUEC
PPS = PPS * DELTAT PPSQ = PPSQ * DELTAT PTS = PTS * DELTAT PTSQ = PTSQ * DELTAT SUST = SUST * DELTAT SUSQT= SUSQT* DELTAT SUSCT- SUSCT* DELTAT SUSCQT=SUSCQT*DELTAT RETURN END
CC ================================— =======================C//LKED.APLMOD DD DSN=PHHANG.FPS.C0M(HBN4P1),DISP=SHR»SPACE=
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//ADN4P1 JOB ( 1 1 0 3 , 6 6 9 4 5 ,1 5 ,4 ) , 'YZ’ ,MSGCLASS=S,REGION=4096K / / * NOTIFY=PHHANG/*JOBPARM SHIFT=D /*AFTER NONE/ /* S 1 EXEC FPSCL, FTNPARH='LIST, LIN, SUBCHK' , VOPT=0/ / S I EXEC FPSCL,FTNPARM='LIST,LIN,XOFF(ALL)’ ,VOPT=3//FORT.SYSIN DD *C ......................................................................................................................................CC SYMMETRIC PERIODIC ANDERSON MODEL (U,V) IN 2-DIMENSIONS.c F-ELECTRON ELEVEL: EF = - U/2cP
CHEMICAL POTENCIAL: UM = 0
c SYSTEM SIZE: N = ND * NDcp
TOTAL ORBITALS: N2 = N * 2L*c OTHER PARAMETERS:c IPMAX - > PARTITION NUMBER (HIRSCH'S ALGORITHM)c (FOR BSS ALGORITHM, USE IPMAX=1)c LTIME - > INPUT TIME SLICES (FOR EACH PARTITION;c (TOTAL TIME SLICES IN BSS ALGORITHM)cp
IPMAX * LTIME - > MAXIMUM TIME SLICES ALLOWEDLic NP = N2 * IPMAX - > GREEN FUCTION MATRIX SIZEc FOR BSS ALGORITHM NP=N2c MATRICES USED:c EXPEK(I.J), EKIN -> EXP(K), EXP(-K) USED TO CALCULATE B'Sc K: HOPPING MATRIXc EV U(L,I), EVD - > EXP{VU(L)} , EXP{VD(L)jc V : POTENTIAL DUE TO ISING SPINS.c EVUIN(L,I), EVDIN - > EXP(-VU(L)} , EXP{-VD(L))c ( I : FROM 1 TO N )c E K I(I , J) - > ITH NEAREST NEIGHBORSc E ( I , J ) = l . 0 IF < I ,J >c EQQ(IjJ) - > 1, -1 FOR TWO SUBLATTICEc EQQ(I,J)=1 . 0 IF I , J ARE INc THE SAME SUBLATTICEcp
SIGMA(L,I) AND - > SPIN VARIABLES:Li
c GU(NP,NP), GD(NP,NP) - > SUBSET OF GREEN'S FUNCTIONSc (USED IN THE UPDATING PROCEDURES)c TIME DEPENDENT GREEN ' S FUNCTIONS:c G U T 0(L ,I ,J ) , GDTO - > < CI(L) * CJ+(1) >c GUTL , GDTL - > < CJ+(L) * C I(1) >cp
GUL , GDL - > < CI(L) * CJ+(L) >Li
c MEASURED QUANTITIES:c SIG(MES) - > SIGNc TLN, TLNU, TLND - > TOTAL# , TOTAL F # UP, TOTAL F # DOWN.c UDN - > < NI(UP) * NI(DN) > FOR F ELECTRONS.c ENRY - > TOTAL ENERY
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cc SAO. . . SA5 -> (F+D) - (F+D) SPIN CORRELATIONSc SFFO. . . SFF5 - > F-F SPIN CORRELATIONSc SFDO.. .SFD5 - > F-D SPIN CORRELATIONSc CAO. . . CA5 - > (F+D) - (F+D) CHARGE CORRELATIONSc CFFO. . .CFF5 - > F-F CHARGE CORRELATIONScn
CFDO. . .CFD5 - > F-D CHARGE CORRELATIONSuc SQFO, SQFQ - > F-SPIN STRUCTURE- FACTOR: K=0, K=PIc SQAO, SQAQ - > F+D SPIN STRUCTURE FACTOR: K=0, K=PIc SXFO, SXFQ -> F-SPIN SUSCEPTIBILITIES: K=0, K=PIc SXXO, SXXQ -> F-SPIN SUSCEPTIBILITIES: K=0, K=PIcr»
SXAO, SXAQ - > ALL SPIN SUSCEPTIBILITIES: K=0, K=PI\»r
c CQFO, CQFQ - > F-CHARGE STRUCTURE FACTOR: K=0, K=PIc CQAO, CQAQ - > F+D CHARGE TRUCTURE FACTOR: K=0, K=PIc CXFO, CXFQ - > F-CHARGE USCEPTIBILITIES: K=0, K=PIc CXXO, CXXQ -> F-CHARGE SUSCEPTIBILITIES: K=0, K=PIcn ■■ .
CXAO, CXAQ -> ALL CHARGE SUSCEPTIBILITIES: K=0, K=PI
cIMPLICIT REAL*8(A-H, O - Z )
PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)
nPARAMETER (LB=5000, ND=4, :N=ND**2, N2=N*2, NP=N2*IPMAX)
DIMENSION EK0(N2,N2),EK1(N,N),EK2(N,N), EK3(N,N)DIMENSION EK4(N,N), EK5(N,N),EQQ(N,N)D I M E N S I O N S I G M A ( L T T , N )
C
D I M E N S I O N E X P E K ( N 2 , N 2 ) , E K I N ( N 2 , N 2 )
D I M E N S I O N E V U ( L T T . N ) , E V D ( L T T . N )
D I M E N S I O N E V U I N ( L T T . N ) , E V D I N ( L T T . N )C
D I M E N S I O N G U ( N P , N P ) , S U ( N P , N P ) , X U ( N 2 , N 2 ) , E U ( N 2 , N 2 ) , Y U ( N 2 , N 2 )
D I M E N S I O N G D ( N P . N P ) , S D ( N P . N P ) , X D ( N 2 , N 2 ) , E D ( N 2 , N 2 ) , Y D ( N 2 , N 2 ) D I M E N S I O N G U 1 C N P . N P ) , G D 1 ( N P , N P )
C
D I M E N S I O N S I G ( L B ) , T L N ( L B ) , T L N U ( L B ) , T L N D ( L B ) , U D N ( L B )
D I M E N S I O N S P I N ( L B ) , E N R Y ( L B )C
D I M E N S I O N S A O ( L B ) , S A 1 ( L B ) , S A 2 ( L B ) , S A 3 ( L B ) , S A 4 ( L B ) , S A 5 ( L B )
D I M E N S I O N S F F O ( L B ) , S F F 1 ( L B ) , S F F 2 ( L B ) , S F F 3 ( L B ) , S F F 4 ( L B ) , S F F 5 ( L B )
D I M E N S I O N S F D O ( L B ) , S F D 1 ( L B ) , S F D 2 ( L B ) , S F D 3 ( L B ) , S F D 4 ( L B ) , S F D 5 ( L B )C
D I M E N S I O N C A O ( L B ) , C A 1 ( L B ) , C A 2 ( L B ) , C A 3 ( L B ) , C A 4 ( L B ) , C A 5 ( L B )
D I M E N S I O N C F F O ( L B ) , C F F 1 ( L B ) , C F F 2 ( L B ) , C F F 3 ( L B ) , C F F 4 ( L B ) , C F F 5 ( L B )
D I M E N S I O N C F D O ( L B ) , C F D 1 ( L B ) , C F D 2 ( L B ) , C F D 3 ( L B ) , C F D 4 ( L B ) , C F D 5 ( L B )C
D I M E N S I O N S Q F O ( L B ) , S Q F Q ( L B )
D I M E N S I O N S Q A O ( L B ) , S Q A Q ( L B )
D I M E N S I O N S X F O ( L B ) , S X F Q ( L B )
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D I M E N S I O N S X X O ( L B ) , S X X Q ( L B )
D I M E N S I O N S X A O ( L B ) , S X A Q ( L B )
CD I M E N S I O N C Q F O ( L B ) , C Q F Q ( L B )
D I M E N S I O N C Q A O ( L B ) , C Q A Q ( L B )
D I M E N S I O N C X F O ( L B ) , C X F Q ( L B )
D I M E N S I O N C X X O ( L B ) , C X X Q ( L B )
D I M E N S I O N C X A O ( L B ) , C X A Q ( L B )
DIMENSION GUT0(LTT,N2,N2), GDT0(LTT,N2,N2)DIMENSION GUTL(LTT,N2,N2), GDTL(LTT,N2,N2)DIMENSION GUL(LTT,N2,N2), GDL(LTT,N2,N2)
CCOMMON /CDELTAT/DELTATCOMMON /CLTIME/LTIMECOMMON /CDL/DLUCOMMON /CUI/UO,VO,EFCOMMON /CEI/UMCOMMON /CEXPEK/EXPEKCOMMON /CEKIN/EKINCOMMON /CSPINS/SIGMACOMMON /CHEV/EVU.EVDCOMMON /CHEVIN/EVUIN,EVDINCOMMON /GB/GUTO,GDT0,GUTL, GDTL,GUL, GDLCOMMON /CIJX/IJX
CCOMMON / C E K S / E K O , E K 1 , E K 2 , E K 3 , E K 4 , E K 5 , EQ Q
COMMON / C N S / S I G , T L N , T L N U , T L N D , U D N , S P I N , E N R Y COMMON / C S A / S A O , S A 1 , S A 2 , S A 3 , S A 4 , S A 5
COMMON / C S F F / S F F O , S F F 1 , S F F 2 , S F F 3 , S F F 4 , S F F 5
COMMON / C S F D / S F D O , S F D 1 , S F D 2 , S F D 3 , S F D 4 , S F D 5
COMMON / C C A / C A O , C A 1 , C A 2 , C A 3 , C A 4 , C A 5
COMMON / C C F F / C F F O , C F F 1 , C F F 2 , C F F 3 , C F F 4 , C F F 5
COMMON / C C F D / C F D O , C F D 1 , C F D 2 , C F D 3 , C F D 4 , C F D 5 COMMON / C S Q / S Q F O , S Q F Q , S Q A O , SQ A Q
COMMON / C S X / S X F O , S X F Q , S X A O , S X A Q , S X X O , SX X Q
COMMON / C C Q / C Q F O , C Q F Q , C Q A 0 , CQAQ
COMMON / C C X / C X F O , C X F Q , C X A O , C X A Q , C X X O , CXXQC
OPEN(UNIT=l)C ..........................................................................................................................................C READ INPUTS:C T : HOPPING CONSTANT, USUALLY T=1C UO : INPUT PARAMETER UC VO : VC EF : F-ORBITAL LEVE.C UM : CHEMICAL POTENTIAL.C BETA : 1 OVER TEMPERATUREC LG : AFTER UPDATING GREEN'S FUNCTION FOR LG TIME SLICESC RECALCULATE G'S FROM SCRACH.
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ccc L T I M E :
c N M E M S:
c N S W E P :
c MMM :
c I R :
c IW :
ccc M ID
L» “
c
m i1112C
C - - -
C
C
C O U R S E : R O U N D - O F F E R R O R S .
U S U A L L Y : U S E L G = 4
T I M E S L I C E S . F O R B S S A L G O R I T H M , T H I S I S T H E T O T A L S L I C E S .
NUM BER O F MO NTE C A R L O M E A S U R E M E N T S
NUM BER O F MC S T E P S B E T W E E N M E A S U R E M E N T SMMM X N S W E P I S T H E W A R M -U P S W E E P S
I F N O T E Q U A L T O 0 , S I G M A R E A D FROM C H A N N E L 2
I F N O T E Q U A L T O 0 , S IG M A W R I T T E N T O C H A N N E L 3
S E T I R A ND IW T O 0 I F T H E C A L C U A L T I O N I S O N E S H O T .
: A F T E R E V E R Y M I D M E A S U R E M N E T S , C A L C U L A T E R E S U L T S .
R E A D ( 1 , 1 1 1 1 ) U O . V O
R E A D ( 1 , 1 1 1 1 ) B E T A
R E A D ( 1 , 1 1 1 2 ) L T I M E , L G
R E A D f 1 , 1 1 1 2 ) N M E M S ,M M M ,N S W E P R E A D ( 1 , 1 1 1 2 ) I R . I W R E A D C 1 , 1 1 1 2 ) I S E E D
R E A D ( 1 , 1 1 1 2 ) M I D
C L O S E ( U N I T = l )
F O R M A T ( F l l . 6 )
F O R M A T ( I 6 )
CC
c
cccc
I N I T I A L I Z E T H E P R O G R A M :D L U = L A M D A ( U )
N U P D A T E = 0
N W ARMUP=M MM*NSW EPU M = O .O D O
E F = - 0 . 5 * U O
D E L T A T = B E T A / ( L T I M E * I P M A X )
W R I T E C 6 , * ) ' S I T E S = ’ , N , ’ P M A X = ' , I P M A X
W R I T E ( 6 , * ) ' U = ' , U O , ' V = ' , V O
W R I T E ( 6 , * ) ' U M = ' , U M , 1 E F = * , E F
W R I T E ( 6 , * ) ' B E T A = ' , B E T A , ' L T I M E = ' , L T I M E , ’ L G = ' , L G
W R I T E ( 6 , * ) ’ M O N TE C A R L O M E S U R E M E N T S : ' .N M EM S
W R I T E ( 6 , * ) ' M O N TE C A R L O S T E P S ( M C S ) . N S W E P
W R I T E ( 6 , * ) ' WARMUP S T E P S : ' .NW AR M UP
W R I T E ( 6 , * ) ' I R , I W \ I R , I W
D L U : L A M D A ( U )
D L U = D T A N H ( D E L T A T * U O * 0 . 2 5 D 0 )
D L U = D L O G ( ( 1 . 0 D 0 + D S Q R T ( D L U ) ) * * 2 / ( 1 . O D O - D L U ) ) D U U = D S I N H ( - 2 . 0 D 0 * D L U )
D U D = D C O S H ( - 2 . O D O * D L U ) - 1 . ODO
S P E C I F Y T H E I N I T I A L I S I N G S P I N C O N F I G U R A T I O N S :
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DO 50 I=1,LTIME*IPMAXC
DO 511 K=1,N LX=MOD(K,2)IF(LX.EQ.O) THEN
SIGMA(IjK)=-1 . ODO ELSE
SIGMA(I,K)= l.ODO END IF
511 CONTINUE 50 CONTINUECC ..............................................................................................................................................................C IF IR IS NOT 0 , ISING SPINS ARE READ FROM CHANNEL 2.
IF (IR.NE.O) THEN OPEN (UNIT=2)WRITE(6,*) 'SIGMAS READ FROM 2'READ(2,2000) ( (SIGMA(K1,K2),K2=1,N) ,K1=1,LTIME*IPMAX) CL0SE(UNIT=2)
END IFCC = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
C GEOMETRY: EK1_5 DEFINE THE SYSTEM WE ARE STUDYINGC
CALL HEK( EKO, EK1 , EK2, EK 3 , EK4, EK5, EQQ)CC CALCULATE EXP(K) AND EXP(-K)C
CALL EXPEKT(EKO)CC CALCUATE THE INITIAL GREEN'S FUNCTIONS FOR MC UPDATING C
CALL GNPUD(GU,GD,1)CC INITIAL DETERMINANTS OF THE GREEN'S FUNCTIONS C
CALL DETM(GU,DETGU)CALL DETM(GD,DETGD)WRITE(6,*)'DETGU=', DETGU„' DETGD=' , DETGD SIGN=DETGU/DABS(DETGU)SIGN=DETGD/DABS(DETGD)*SIGN WRITE(6,*) ' SIGN=' , SIGN
CC =======— ==*=======================================!===C MONTE CARLO STEPS START:C
MES=0DO 66 MT=1,MMM+NMEMS
CC TAKE MEASUREMENT AFTER (MMM X NSWEP) WARM-UP SWEEPS OF MC STEPS. C ** RESULTS ** CALCULATES THE AVERAGES OF THE QUANTITIES.
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cI F ( M T . G T . M M M ) TH EN
C A L L G T O T A L ( G U , G D )
M E S = M E S + 1
CALL RESULTS(MES,SIGN,GU,GD)E N D I F
C
DO 6 4 M S = 1 , N S W E P
C
C MC S T E P S : N S W E P S W E E P S B E T W E E N E A C H M E A R S U R E M E N T .C = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
C
C S W E E P T H R O U G H T H E T I M E S P A C E : L T I M E S L I C E
C
DO 1 0 0 L = l , L T I M EC
C R E C A L C U L A T E G R E E N ' S F U N C T I O N A F T E R L G S L I C E S
C T O R E S T O R E P R E C I S I O N .
L L = M O D ( L , L G )I F ( L L . E Q . O ) T H E N
C A L L G N P U D ( G U , G D , L )
E N D I FC
D O 1 0 0 0 I P P = 1 , I P M A XC
C S W E E P W I T H I N E A C H T I M E P A T I T I O N S . F O R B S S , I P M A X = 1C
L I P P = ( I P P - 1 ) * L T I M E + LC
C U P D A T E I S I N G S P I N A T S I T E : S IG M A DO 1 1 0 1 = 1 , N
C
I I P P = ( I P P - 1 ) * N 2 + I
D N U = D U U * S I G M A ( L I P P , I ) + D U D
D N D = - D U U * S I G M A ( L I P P , I ) + D U DR U = 1 . O D O + ( 1 . O D O - G U ( I I P P . I I P P ) ) * D N U
R D = 1 . 0 D 0 + ( 1 . O D O - G D ( I I P P , I I P P ) ) * D N D
R U D = R U * R D
P R O = D A B S ( R U D )C
C F L I P P R O B A B I L I T Y : P R O
C RANDOM N U M B E R : X
X = R A N ( I S E E D )C
I F ( P R O . G T . X ) T H E NC
C P R O P O S E D S P I N F L I P S I G M A - > - S I G M A A C C E P T E D .C
N U P D A T E = N U P D A T E + 1
I F ( R U D . L T . O . O D O ) S I G N = - S I G N
S I G M A ( L I P P , I ) = - S I G M A ( L I P P , I )
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RU=DNU/RURD=DND/RD
CC UPDATE THE GREEN'S FUNCTIONC
D O 5 5 J 1 = 1 , N P
D J 1 I = O . O D O
IF (J l.E Q .IIP P ) DJ1I=1. 0D0 TRU1=-(DJ1I-GU( J l , IIPP ) )*RU TRD1=-( DJ1I-GD( J l , IIPP ) )*RD DO 55 J2=l,NPSU(J1,J2)=GU(J1,J2)+TRU1*GU(IIPP,J2)
55 S D (J l , J2)=G D (Jl, J2)+TRD1*GD(IIPP,J2)DO 56 Kl=l,NPDO 56 K2=l,NP GU(K1,K2)=SU(K1,K2)
56 GD(K1,K2)=SD(K1,K2)CC GREEN'S FUNCTION UPDATED
END IF 110 CONTINUEC UPDATE ALL SIGMA'S FOR THE CURRENT TIME SLICE.C1000 CONTINUE CC UPDATE GREEN'S FUNCTION TO NEXT TIME SLICES:C
DO 120 IP1=1,IPMAXDO 120 IP2=1,IPMAX
DO 121 J l= l ,N 2 DO 121 J2=l,N2 J1G=(IP1-1)*N2+J1 J2G=(IP2-1)*N2+J2 YU(J1,J2)=GU( JIG, J2G )
121 Y D (J l , J2)=GD( JIG, J2G )CALL BUBDL(EU,ED,IP1,L)CALL FMMM(EU,YU,XU,N2,N2,N2)CALL FMMM(ED,YD,XD,N2,N2,N2)CALL INBUBDL(EU,ED,IP2,L)CALL FMMM(XU,EU,YU,N2,N2,N2)CALL FMMM(XD,ED,YD,N2,N2,N2)
DO 122 J1=1,N2DO 122 J2= l,N 2J1G=(IPI-1)*N2+J1J2G=(IP2-1)*N2+J2GU( JIG, J2G) =YU(Jl, J2)
122 GD( JIG, J2G) =YD(J1,J2)120 CONTINUECC FINISH ONE SWEEP THROUGH THE WHOLE TIME-SPACE LATTICE C
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1 0 0 C O N T I N U E
CC = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
C WHEN MS I S L E S S T H E N T H E MC S T E P S B E T W E E N M E A S U R E M N E T S ( N S W E P )
C U P D A T E G R E E N ' S F U N C T I O N A F T E R E A C H S W E E P
C
I F ( ( M T . L E . MMM) . O R . ( M S . L T . N S W E P ) ) T H E N
C A L L G N P U D ( G U , G D , 1 )E N D I F
CC = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
C * N S W E P * MO NTE C A R L O S W E E P S B E T W E E N M E A S U R E M E N T S D O N E .
6 4 C O N T I N l f eCC = = = = = = = = = = = = = = = = = = = = = = = = = = : = = = = = = = = = = = = = = = = = = = = = = = = : = = = = = =
C
C A F T E R E A C H M I D M E A S U R E M E N T S , R E P O R T T H E R E S U L T S .
C
M E S M O D = M O D ( M E S ,M I D )
I F ( M E S M O D . E Q . 0 . A N D . M E S . N E . 0 ) T H E NC
W R I T E ( 6 , * ) 'M E A S U R E M N E T M E S ' . M E S
C A L L D I V ( S I G , A S I G N , X S I G N , M E S , 1 . 0 D 0 )C A L L D I V ( T L N , A T L N , X T L N , M E S , A S I G N )
C A L L D I V ( T L N U , A T L N U , X T L N U , M E S , A S I G N )
C A L L D I V ( T L N D , A T L N D , X T L N D , M E S , A S I G N )
C A L L D I V ( U D N , A U D N , X U D N , M E S , A S I G N )
A U D N = A U D N /N
X U D N = X U D N /N C A L L D I V ( S P I N , A S P I N . X S P I N , M E S , A S I G N )C A L L D I V ( S A O , A S A O , X S A O , M E S , A S I G N )
A S A O = A S A O / N
X S A 0 = X S A 0 / N
C A L L D I V ( S A 1 , A S A 1 , X S A 1 , M E S , A S I G N )
A S A 1 = A S A 1 / ( N * 4 )
X S A 1 = X S A 1 / ( N * 4 )
C A L L D I V ( S A 2 , A S A 2 , X S A 2 , M E S , A S I G N )
A S A 2 = A S A 2 / ( N * 4 )
X S A 2 = X S A 2 / ( N * 4 )
C A L L D I V ( S A 3 , A S A 3 , X S A 3 , M E S , A S I G N )
A S A 3 = A S A 3 / N 2
X S A 3 = X S A 3 / N 2 C A L L D I V ( S A 4 , A S A 4 , X S A 4 , M E S , A S I G N )
A S A 4 = A S A 4 / ( N * 4 )
X S A 4 = X S A 4 / ( N * 4 )
C A L L D I V ( S A 5 , A S A 5 , X S A 5 , M E S , A S I G N )
A S A 5 = A S A 5 / N
X S A 5 = X S A 5 / N
C A L L D I V ( S F F O , A S F F O , X S F F O , M E S , A S I G N )
A S F F O = A S F F Q / N
X S F F O = X S F F O / N
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CALL D IV (SFF1 ,A SFF1,X SFF1,M ES,A SIG N ) ASFF1=A SFF1/(N *4)XSFF1=XSFF1 / ( N*4)
CALL D IV (S F F 2 , A SFF2,X SF F2, MES, ASIGN) A SFF2=A SFF2/(N *4) X SFF2=X SFF2/(N *4)
CALL D IV (S F F 3 .A S F F 3 ,X S F F 3 , MES, ASIGN) ASFF3=ASFF3/N2 XSFF3=XSFF3/N2
CALL D IV ( S F F 4 , A SFF4, X SFF4, MES, ASIGN) A SFF4=A SFF4/(N *4) X SFF4=X SFF4/(N *4)
CALL DIVC SFF5 , A SFF5, X SFF5, MES, ASIGN) ASFF5=ASFF5/N XSFF5=XSFF5/N
CALL D IV ( SFDO, ASFDO, XSFDO, MES, ASIGN) ASFDO=ASFDO/N XSFDO=XSFDO/N
CALL DIVCSFD1.ASFD1.XSFD1,MES,ASIGN) ASFD1=ASFD1/(N*4) X SFD l=X SFD l/eN *4)
CALL D IV eSFD 2, ASFD2, XSFD2, MES, ASIGN) ASFD2=ASFD2/eN*4) XSFD2=XSFD2/eN*4)
CALL D IV eSFD 3, ASFD3,XSFD3, MES, ASIGN) ASFD3=ASFD3/N2 XSFD3=XSFD3/N2
CALL DIVe SFD4, ASFD4, DSFD4, MES, ASIGN) ASFD4=ASFD4/eN*4) XSFD4=XSFD4/eN*4)
CALL DIVe SFD 5, ASFD5, XSFD5, MES, ASIGN) ASFD5=ASFD5/N XSFD5=XSFD5/N
CALL DIVeCAO, ACAO,XCAO, MES, ASIGN) ACAO=ACAO/N XCAO=XCAO/N
CALL DIVCCA1 , ACA1, XCA1 , MES, ASIGN) ACA1=ACA1/CN*4)XCAl=XCAl/eN*4)
CALL DIVeCA2,ACA2,XCA2,MES,ASIGN) ACA2=ACA2/eN*4)XCA2=XCA2/eN*4)
CALL DIVecA3,ACA3,XCA3,MES,ASIGN) ACA3=ACA3/N2 XCA3=XCA3/N2
CALL DIV e CA4, ACA4, XCA4,MES, ASIGN) ACA4=ACA4/eN*4)XCA4=XCA4/(N*4)
CALL DIVCCA5,ACA5,XCA5,MES,ASIGN) ACA5=ACA5/N XCA5=XCA5/N
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CALL D IV (C F F O , ACFFO, XCFFO, MES, A S IG N ) ACFFO=ACFFO/N XCFFO=XCFFO/N
CALL D I V ( C F F 1 ,A C F F I ,X C F F 1 ,M E S ,A S I G N ) ACFF1=ACFF1 / ( N * 4 ) X C F F 1 = X C F F 1 / (N * 4 )
CALL D I V C C F F 2 .A C F F 2 .X C F F 2 , M E S .A S IG N ) A C F F 2 = A C F F 2 / (N * 4 ) X C F F 2 = X C F F 2 / (N * 4 )
CALL D I V ( C F F 3 ,A C F F 3 ,X C F F 3 ,M E S , A S IG N ) A C F F 3=A C F F 3/N 2 X C F F 3= X C F F 3 /N 2
CALL D IV ( C F F 4 , A C F F 4 , X C F F 4 , MES, A S IG N ) A C F F 4 = A C F F 4 / (N * 4 ) X C F F 4 = X C F F 4 / (N * 4 )
CALL D I V ( C F F 5 , A C F F 5 , X C F F 5 , MES, A S IG N ) A C FF5=A C FF5/N X C F F 5=X C F F5/N
CALL D IV (C FD O ,A C FD O ,X C FD O , MES, A SIG N ) ACFDO=ACFDO/N XCFDO=XCFDO/N
CALL D I V C C F D l .A C F D l .X C F D l , M E S ,A S IG N ) ACFD1=ACFD1 / ( N * 4 )XCFD1=XCFD1 / ( N * 4 )
CALL D IV C C F D 2, A C F D 2, X C F D 2, MES, A SIG N ) A C F D 2= A C F D 2/C N *4) X C F D 2= X C F D 2/C N *4)
CALL D IV C C F D 3 ,A C F D 3 .X C F D 3 , M E S ,A S IG N ) A CFD3=AC FD 3/N 2 XCFD3=XC FD 3/N 2
CALL D IV C C F D 4, A C F D 4, D C F D 4, MES, A S IG N ) A C F D 4= A C F D 4/C N *4) X C F D 4= X C F D 4/C N *4)
CALL D IVC CFD S, A C F D 5, X C F D 5, MES, A S IG N ) ACFD5=ACFD5/N XCFDS =XCFD5/ N
CALL DIV C SQFO, ASQFO, XSQFO, MES, A S IG N ) ASQFO=ASQFO/N XSQFO=XSQFO/N
CALL DIVC SQFQ, ASQFQ, XSQFQ, MES, A SIG N ) ASQFQ=ASQFQ/N XSQFQ=XSQFQ/N
CALL DIVC SQ A O ,A SQ A O ,X SQ A O ,M ES,A SIG N ) ASQAO=ASQAO/N2 X SQ A 0=X SQ A 0/N 2
CALL DIVCSQAQ, ASQAQ, XSQAQ,MES, A SIG N ) ASQAQ=ASQAQ/N2 XSQAQ=XSQAQ/N2
CALL D IVC SXFO, ASXFO, XSXFO, MES, A S IG N ) ASXFO=ASXFO/N
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XSXFO=XSXFO/N CALL DIVCSXFQ, ASXFQ, XSXFQ , MES, A SIG N )
ASXFQ=ASXFQ/N XSXFQ=XSXFQ/ N
CALL D IV (S X A O , ASXAO,XSXAO, MES, A SIG N ) A SXA 0=ASX A0/N2 X SXA 0=XSX A0/N2
CALL DIVCSXAQ, ASXAQ, XSXAQ, MES, A SIG N ) ASXAQ=ASXAQ/N2 XSXAQ=XSXAQ/N2
CALL DIV C SXXO, ASXXO, XSXXO, MES, A S IG N ) ASXXO=ASXXQ/N XSXXO=XSXXO/N
CALL DIVC SXXQ, ASXXQ, XSXXQ, MES, A SIG N ) ASXXQ=ASXXQ/N XSXXQ=XSXXQ/N
CALL DIVCCQFO, ACQFO, XCQFO, MES, A SIG N ) ACQFO=CACQFO- CATLNU+ATLND)**2 ) / N XCQFO=XCQFO/N
CALL DIVCCQFQ, ACQFQ, XCQFQ, MES, A SIG N ) ACQFQ=ACQFQ/N XCQFQ=XCQFQ/N
CALL DIVC CQAO, ACQAO, XCQAO, ME S , A S IG N ) ACQAO=CACQAO - A T L N * * 2 ) / N 2 XCQAO=XCQAO/N2
CALL DIVCCQAQ, ACQAQ, XCQAQ, MES, A SIG N ) ACQAQ=ACQAQ/N2 XCQAQ=XCQAQ/N2
CALL DIVCCXFO, ACXFO, XCXFO, M E S.A SIG N ) ACXF0=CACXF0-BETA*CATLNU+ATLND)**2 ) / N XCXFO=XCXFO/N
CALL DIVCCXFQ,ACXFQ,XCXFQ, M E S,A SIG N ) ACXFQ=ACXFQ/N XCXFQ=XCXFQ/N
CALL DIV C CXAO, ACXAO, XCXAO, MES, A SIG N ) ACXAO=CACXAO- BETA* A T L N **2 ) / N 2 XCXAO=XCXAO/ N2
CALL DIVCCXAQ, ACXAQ, XCXAQ, M E S.A SIG N ) ACXAQ=ACXAQ/N2 XCXAQ=XCXAQ/N2
CALL DIVCCXXO, ACXXO, XCXXO, M E S,A SIG N ) ACXXO=CACXXO - CATLNU+ATLND)*ATLN ) / N XCXXO=XCXXO/N
CALL DIVCCXXQ, ACXXQ, XCXXQ, M E S .A S IG N ) ACXXQ=ACXXQ/N XCXXQ=XCXXQ/N
CALL DIVCENRY, AENRY, XENRY, MES, A SIG N ) W R IT E C 6 ,* ) * ...........................................................................................
K R I T E C 6 ,* ) 1 SIGN ' , A S IG N , ' XASIGNW R IT E C 6 ,* ) ' TOTAL# ’ , ATLN , 1 XTLN
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WRITE( 6 * ) ' # U P - F ' , ATLNU, ' XTLNUWRITE( 6 * ) ' #DM -F 1 , ATLND, ’ XTLNDWRITE( 6 * ) ’ UP*DM-F ' , AUDN , ’ XUDNWRITE( 6 * ) ' S P IN i , A S P I N , ' X S P INW R IT E (6 * ) ' ENERGY ' , AENRY, ' XENRYWRITE( 6 * ) ' S Z - S Z F F O ' , A S F F O , ' XSFFOWRITE( 6 * ) ’ S Z -S Z F F l ' , A S F F 1 , ' X S FF 1WRITEC6 * ) 'S Z - S Z F F 2 ' , A S F F 2 , ' X SFF2W R IT E (6 * ) 'S Z - S Z F F 3 1 , A S F F 3 , ’ ■ X SFF 3W R IT E (6 * ) 'S Z - S Z F F 4 ' , A S F F 4 , ' X S F F 4WRITE( 6 * ) ' S Z - S Z F F 5 ' , A S F F S , ' X SFF5W R IT E (6 * ) ' S Z - S Z FDO' , ASFDO, ' XSFDOW R IT E (6 * ) ’ S Z -S Z F D 1 ' , A S F D 1 , ’ XSFD1WRITE( 6 * ) ’ S Z -S Z F D 2 1 , A S F D 2 , ’ XSFD2WRITE( 6 * ) ' S Z - S Z F D 3 ' , A S F D 3 , ' XSFD3WRITE( 6 * ) ' S Z - S Z F D 4 ’ , A S F D 4 , ' X SFD 4WRITE( 6 * ) ' S Z - S Z F D S ’ , A S F D 5 , ’ XSFD5WRITE( 6 * ) ' S Z - S Z AAO ’ , ASAO , ' XSAOW R IT E (6 * ) ' S Z - S Z A A 1 ’ , ASA1 , ' XSA1WRITEC6 * ) ' S Z - S Z A A 2 ' , ASA2 , ' XSA2WRITEC6 * ) 'S Z - S Z A A 3 ' , A SA3 , ' XSA3WRITE( 6 * ) 'S Z - S Z AA4* , A SA4 , ’ XSA4WRITE( 6 * ) ' S Z - S Z A A 5 ' , ASA5 , ' XSA5WRITEC6 * ) ' N -N FFO' , ACFFO, ' XCFFOWRITE( 6 * ) ' N -N F F l ' , A C F F 1 , ’ XCFF1WRITE( 6 * ) 1 N-N F F 2 ' , A C F F 2 , ' + / - * , XCFF2WRITEC6 * ) ’ N -N F F 3 ' , A C F F 3 , ' XCFF3WRITE( 6 * ) ’ N -N F F 4 ' , A G F F 4 , ’ XCFF4W R IT E (6 * ) ' N -N F F 5 ' , A C F F 5 , ' XCFF5WRITEC6 * ) ' N -N FDO' , ACFDO, ’ XCFDOWRITEC6 * ) ' N -N F D 1 ' , A C F D 1, ' XCFD1WRITEC6 * ) ' N -N F D 2 ' , A C F D 2, ' XCFD2WRITE( 6 * ) ' N -N F D 3 ' , A C F D 3, ' XCFD3W R IT E (6 * ) ' N -N F D 4 ' , A C F D 4, ' XCFD4WRITEC6 * ) ' N -N F D 5 ' , A C F D 5, ‘ XCFD5WRITEC6 * ) ' N -N AAO' , ACAO , ’ XCAOW R IT E (6 * ) ' N -N AA1' , ACA1 , ' XCA1WRITEC6 * ) ' N -N A A2' , ACA2 , ' XCA2WRITEC6 * ) ' N -N A A 3' , ACA3 , ' XCA3WRITE( 6 * ) ' N -N A A4' , ACA4 , ’ + / - ' » XCA4WRITE( 6 * ) ' N -N A A 5 ' , ACA5 , XSA5WRITE( 6 * ) ' S Q F (K = 0 ) ' , ASQFO, ’ XSQFOW R IT E (6 * ) ' S Q F ( K = P I ) ' , ASQFQ, XSQFQWRITE( 6 * ) ’ S Q A (K = 0 ) ' , ASQAO, ’ XSQAOWRITE( 6 * ) ' S Q A ( K = P I ) * , ASQAQ, ’ XSQAQWRITE( 6 * ) ' X F S ( K = 0 ) ' , A SXFO, ' XSXFOW R IT E (6 * ) ' X F S ( K = P I ) ' , ASXFQ, ' XSXFQW R IT E (6 * ) ’ X A S (K = 0 ) ' , ASXAO, ’ XSXAOW R IT E (6 * ) ’ X A S ( K = P I ) ' , ASXAQ, ’ XSXAQWRITE( 6 * ) ’ T * X F ( K = 0 ) ' , ASXXO, ' XSXXOW R IT E (6 * ) ' T * X F ( P I ) ’ , ASXXQ, ' + / - ’ » XSXXQ
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W R I T E ( 6 ,* ) W R I T E ( 6 , * ) W R IT E C 6 ,* ) W R I T E ( 6 , * ) W R I T E ( 6 , * } W R I T E ( 6 ,* ) W R I T E ( 6 , * ) W R IT E C 6 ,* ) W R I T E ( 6 , * ) W R I T E ( 6 , * ) W R IT E C 6 ,* )
’ C Q F ( K = 0 ) '
' C Q F ( K = P I ) '
' C Q A ( K = 0 ) '
' C Q A ( K = P I ) '
' X F C ( K = 0 ) ’
' X F C ( K = P I ) '
* X A C ( K = 0 ) '
’ X A C ( K = P I ) '
’ T * X C ( K = 0 ) ' ' T * X C ( P I ) ’
ACQFO, ACQFQ, ACQAO, ACQAQ, ACXFO, ACXFQ, ACXAO, ACXAQ, ACXXO, ACXXQ,
+ / -+ / -+ / "+ / -+ / -+ / “+ / -+ / -+ / "+ / -
XCQFOXCQFQXCQAOXCQAQXCXFOXCXFQXCXAOXCXAQXCXXOXCXXQ
CCCC
Cc
2000CC
c66
CCcccccccccccccccc
PRINT OUT THE TOTAL NUMBER OF UPDATES ACCEPTED. USED TO CALCULATE THE ACCEPTANCE RATIO.
W R I T E ( 6 , * ) 'N U PD A T E ', NUPDATE
I F ( I W .N E . O ) THENSAVE THE IS IN G S P IN CONFIGURATIONS, WRITE TO CHANNEL 3 .
0 P E N (U N I T = 3 )W R I T E ( 6 , * ) ’ SIGMAS WRITTEN TO 3 ’W R I T E ( 3 , 2 0 0 0 ) ( ( S I G M A ( K 1 ,K 2 ) , K 2 = I , N ) , K 1 = 1 , L T I M E * I P M A X ) CLOSE( U N IT = 3 )
END IFF 0 R M A T C 1 0 F 8 .2 )
INTERMEDIATE RESULTS AFTER MID MEASUREMENTS * DONE *END IF
CONTINUEEND
MONTE CARLO MAIN PROGRAM ENDS
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
SUBROUTINES CC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc
GNPUD CALCULATES THE GREEN'S FUNCTION AT TIME SL IC E L MAXIMUM ALLOWED TIME SL IC E S L A =40 INPUT TIME SL IC E S : LTIME MUST BE \ < LA
GREEN'S FUCTION S I Z E : NP X NP MATRICESFOR BSS ALGORITHM, USE IPMAX=1
CCCCCC
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SUBROUTINE GNPUD(GUNP, GDNP, L )IM P L IC IT R E A L * 8 ( A - H , 0 - Z )PARAMETER(LA=40, IPM AX=1)PARAMETER(ND=4, N = 2 * N D * * 2 , NP=N*IPMAX)DIMENSION G U N P (N P .N P ) , G D N P (N P .N P )DIMENSION G U I N P ( N P , N P ) , G D I N P (N P .N P )DIMENSION G U ( N ,N ) , E U ( N , N ) , S U ( N ,N )DIMENSION G D ( N ,N ) , E D ( N , N ) , S D ( N . N )COMMON /C LTIM E/LTIM E
CC I N I T I A L I Z E THE WORK AREA.
DO 5 5 5 1 = 1 , NP DO 5 5 5 J = 1 , N P G U I N P ( I , J ) = 0 . 0D0
5 5 5 G D I N P ( I , J ) = 0 . 0D0CC ...........................................................................................................................
DO 1 0 0 I P P = 1 , IPMAX C CALCULATE B ( L )
CALL B U B D L ( E U ,E D ,I P P ,L )CC CALCULATE ALL B ' S ; AND B ( L - l ) . . B ( l ) . B ( L T I M E ) . . . B ( L + 1 ) . B ( L )
DO 5 I = L + 1 ,L T I M E + L - 1 I S = I I P S = I P P
I F ( IP P .E Q .IP M A X .A ND . I .G T .L T IM E ) THEN I S = I S -L T I M E I P S = 1
END I FC
CALL B U B D L ( S U ,S D , I P S , I S )CALL F M M M (S U ,E U ,G U ,N ,N ,N )CALL F M M M (S D ,E D ,G D ,N ,N ,N )
CDO 7 J 1 = 1 , N DO 7 J 2 = 1 , N E U ( J 1 , J 2 ) = G U ( J 1 , J 2 )
7 E D ( J 1 , J 2 ) = G D ( J 1 , J 2 )C
5 CONTINUEC
I F ( I P P . L T . I P M A X ) THEN DO 8 1 = 1 , N DO 8 J = 1 , N
I I = I P P * N + I J J = ( I P P - 1 ) * N + J G U I N P ( I I , J J ) = - G U ( I , J )G D I N P ( I I , J J ) = - G D ( I , J )
8 CONTINUE
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cELSE
C T H IS I S THE CASE FOR BSS ALGORITHM, IPP=IPM AX =1DO 9 1 = 1 , N DO 9 J = 1 , N
J J = ( I P M A X - 1 ) * N + J G U I N P ( I , J J ) = G U ( I , J )G D I N P ( I , J J ) = G D ( I , J )
9 CONTINUEEND IF
C1 0 0 CONTINUE CC FULL MATRIX BEFORE THE INVERSION: *M* OR * 0 *C DET(M) OR D E T (O ) IN THE PARTION FUNCTIN.C
DO 1 1 0 1 = 1 ,N PG U I N P ( I , I ) = G U I N P ( I , I ) + l . ODO G D IN P C I , I ) = G D I N P ( I , I ) + l . ODO
1 1 0 CONTINUE CC CALCULATE THE IN VER SE, AND F IN D G = ( I + B . . . B ) - 1 C THE GREEN'S FUNCTION
CALL MATRINV(GUINP,GUNP)CALL MATRINV(GDINP.GDNP)
RETURNEND
CCcc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cc cc GTOTAL CALCULATES THE TIME-DEPANDENT GREEN'S FUCNTIONS CC CALLED BEFORE TAKING MEASUREMENT. CC CC TOTAL TIME SL IC E LTIME*IPMAX : CC CC G U T O ( L , I , J ) = < C I ( L ) C J + ( 1 ) > CC G U T L ( L , I , J ) = < C J + ( L ) C I ( 1 ) > CC G U L ( L , I , J ) = < C I ( L ) C J + ( L ) > CC Cc ................................................................................................... - ....................................................................... Cc
SUBROUTINE GTOTAL(GUNP,GDNP)IM P L IC IT R E A L * 8 ( A - H , 0 - Z )PARAMETER ( L A = 4 0 , IPM AX=1, LTT=LA*IPMAX) PARAMETER ( L B = 5 0 0 0 , N D = 4 , N = 2 * N D * * 2 , NP=N*IPMAX) DIMENSION G U N P ( N P .N P ) , G D N P (N P .N P )DIMENSION G U T O ( L T T .N .N ) ,G D T 0 ( L T T ,N ,N )DIMENSION G U T L ( L T T ,N ,N ) ,G D T L ( L T T ,N ,N )DIMENSION G U L ( L T T , N , N ) , G D L (L T T ,N ,N )
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COMMON /CLTIME/LTIMECOMMON /GB/GUTO,GDTO,GUTL,GDTL,GUL,GDL
CC CALCULATE GU AND GD FIRSTC
CALL GNPUD(GUNP,GDNP,1)CC CALCUALTE ALL G'S IN TWO STEPS:C FISRT G'S FOR TIME SLICE \< HALF OF THE TOTAL SLICESC SECOND G'S FOR TIME SLICE > HALF OF THE TOTAL SLICESC PURPOSE: REDUCE THE ROUND-OFF ERROR C
DO 10 IP=1,IPMAXCALL GBTO(GUNP,GDNP,IP)
10 CALL GBTL(GUNP,GDNP,IP)RETURNEND
CCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc cC GTBO CALCULATES THE TIME-DEPANDENT GREEN’S FUCNTIONS CC CALLED BY GTOTOAL CC FOR TIME SLICE L \< HALF OF TOTAL TIME SLICES CC CC GUTO(L,I, J ) = < CI(L)CJ+(1) > CC GUTL(L,I,J) = < CJ+(L)CI(1) > CC GUL(L, I , J ) = < CI(L)CJ+(L) > CC CC INPUT : GREEN'S FUNCTIONS USED IN THE UPDATING PROCEDURES CC IP : PARTITION NUMBER; FOR BSS, IP=IPMAX=1 CC CC ............................................................................................................- ............................ cc
SUBROUTINE GBTO(GUNP,GDNP,IP)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX) PARAMETER (LB=SOOO, ND=4, N=2*ND**2, NP=N*IPMAX)
CDIMENSION GUNP(NP.NP) ,GDNP(NP,NP)DIMENSION GUTO( LTT, N, N) , GOTO( LTT, N, N)DIMENSION GUTL( LTT, N, N) , GDTL( LTT, N, N)DIMENSION GUL(LTT,N,N), GDL(LTT,N,N)
CDIMENSION BU(N,N),BD(N,N)DIMENSION BUI(N,N),BDI(N,N)DIMENSION EU1(N,N),EU2(N,N)DIMENSION ED1(N,N),ED2(N,N)DIMENSION SU1(N,N),SU2(N,N)DIMENSION SD1(N,N),SD2(N,N)
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DIMENSION TU1(N,N),TU2(N,N)DIMENSION TD1(N,N),TD2(N,N)
CCOMMON /CLTIME/LTIMECOMMON /GB/GUTO, GDTO,GUTL, GDTL, GUL, GDL
CC INITIALIZE THE LOOP IP C
IF (IP .E Q .l) THEN C THIS IS ALSO THE CASE FOR BSS ALGORITHM: IP=IPAMX=1 C
DO 125 M4=1 ,N DO 125 M5=1,N
GUT0(1 ,M4,M5)= GUNP(M4,M5)GDT0(1,M4,M5)= GDNP(M4,M5)GUTL(1 ,M4,M5)=-GUNP(M4,M5)GDTL(1 ,M4,M5)=-GDNP(M4,M5)GUL(1 ,M4,M5) = GUNP(M4,M5)GDL(1,M4,M5J ■ GDNP(M4,M5)EU1(M4,M5) = GUNP(M4,M5)ED1(M4,M5) ■ GDNP(M4,M5)SU1(M4,M5) =-GUNP(M4,M5)SD1(M4,M5) =-GDNP(M4,M5)TU1(M4,M5) = GUNP(M4,M5)
125 TD1(M4,M5) = GDNP(M4,M5)C
DO 1255 M4=1,NGUTL(1 ,M4,M4)=-GUNP(M4,M4)+1. ODO GDTL(1 ,M4,M4)=-GDNP(M4,M4)+1. ODO SU1(M4,M4) =-GUNP(M4,M4)+1.0D0
1255 SD1(M4,M4) =-GDNP(M4,M4)+1. ODOC
ELSELIP=(IP-1)*LTIME+1 DO 126 M4=1,NDO 126 M5=1,N
IPM4=(IP-1)*N+M4 IPM5=(IP-1)*N+M5 GUTO(LIP,M4,M5)= GUNP(IPM4.M5)GDTO(LIP,M4,M5)= GDNP(IPM4.M5) GUTL(LIP,M4,M5)=-GUNP(M4,IPM5)GDTL(LIP,M4,M5)=-GDNP(M4, IPM5) GUL(LIP,M4,M5) = GUNP(IPM4,IPM5) GDL(LIP,M4,M5) = GDNP(IPM4, IPM5) EU1(M4,M5)= GUNP(IPM4,M5)ED1(M4,M5)= GDNP(IPM4,M5) SU1(M4,M5)=-GUNP(M4,IPM5) SD1(M4,M5)=-GDNP(M4>IPM5)TU1(M4,M5)= GUNP(IPM4,IPM5)
126 TD1(M4,M5)= GDNP(IPM4,IPM5)END IF
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nCC H A L F O F T H E T O T A L T I M E S L I C E S ( L 2 )
L 2 = L T I M E / 2 + l
CDO 2 3 4 L K = 2 , L 2
L K 1 = L K - 1
* C A L L B U B D I N ( B U , B D , B U I , B D I , I P , L K 1 )C A L L B U B D L ( B U , B D , I P , L K 1 )
C A L L I N B U B D L C B U I , B D I , I P , L K 1 )
CC * F O R G U T O , G D T O
C
C A L L F M M M ( B U , E U 1 , E U 2 , N , N , N )
C A L L F M M M ( B D , E D 1 , E D 2 , N , N , N )
C
C * F O R G U T L , G D T L
CC A L L F M M M ( S U 1 , B U I , S U 2 , N , N , N )
C A L L F M M M ( S D 1 , B D I , S D 2 , N , N , N )
CC * F O R G U L , G D L
C
C A L L F M M M ( B U , T U 1 , T U 2 , N , N , N )
C A L L F M M M ( B D , T D 1 , T D 2 , N , N , N )C A L L F M M M ( T U 2 , B U I , T U 1 , N , N , N )
C A L L F M M M ( T D 2 , B D I , T D I , N , N , N )C
I P L K = ( I P - 1 ) * L T I M E + L KC
DO 1 2 7 M 6 = 1 , N
DO 1 2 7 M 7 = 1 , N
G U T 0 ( I P L K , M 6 , M 7 ) = E U 2 ( M 6 , M 7 )
G D T 0 ( I P L K , M 6 , M 7 ) = E D 2 ( M 6 , M 7 )
E U 1 ( M 6 , M 7 ) = E U 2 ( M 6 , M 7 )
E D 1 ( M 6 , M 7 ) = E D 2 ( M 6 , M 7 )
G U T L ( I P L K , M 6 , M 7 ) = S U 2 ( M 6 , M 7 )
G D T L ( I P L K , M 6 , M 7 ) = S D 2 ( M 6 , M 7 )
S U 1 ( M 6 , M 7 ) = S U 2 ( M 6 , M 7 )
S D 1 ( M 6 , M 7 ) = S D 2 ( M 6 , M 7 )
G U L ( I P L K , M 6 , M 7 ) = T U 1 ( M 6 , M 7 )
G D L ( I P L K , M 6 , M 7 ) = T D 1 ( M 6 , M 7 ) 1 2 7 C O N T I N U E
C
2 3 4 C O N T I N U E
R E T U R N
E N D
C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C
c
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GTBL CALCULATES THE TIME-DEPANDENT GREEN'S FUCNTIONS C CALLED BY GTOTOAL; CFOR TIME SLICE L > HALF OF TOTAL TIME SLICES C
CGUT0(L,I, J ) = < CI(L)CJ+(1) > CGUTL(L,I, J ) = < CJ+(L)CI(1) > CGUL(L,I, J ) = < CI(L)CJ+(L) > C
CC
SUBROUTINE GBTL(GUNP,GDNP,IP)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=2*ND**2, NP=N*IPMAX)
CDIMENSION GUNP(NP.NP) ,GDNP(NP,NP)DIMENSION GU(N,N) ,GD(N,N)DIMENSION RUTO( LTT, N, N) , GDTO( LTT,N, N)DIMENSION GUTL(LTT,N,N),GDTL(LTT,N,N)DIMENSION GUL(LTT,N,N), GDL(LTT,N,N)
CDIMENSION BU(N,N),BD(N,N)DIMENSION BUI(N,N),BDI(N,N)DIMENSION EUI(N,N),EU2(N»N)DIMENSION EDl(N.N),ED2(N,N)DIMENSION SU1(N,N),SU2(N,N)DIMENSION SD1(N,N),SD2(N,N)DIMENSION TU1(N,N),TU2(N,N)DIMENSION TD1(N,N),TD2(N,N)
CCOMMON /CLTIME/LTIMECOMMON /G B/GUTO, GDTO, GUTL, GDTL,GUL, GDL
CIF(IP.EQ.IPMAX) THEN
C THIS IS ALSO THE CASE FOR BSS ALGORITHM; IP=IPMAX=1 C
LIP=(IPMAX-1)*LTIME DO 125 M4=1,N DO 125 M5=1,N
EU1(M4,M5)=-GUNP(M4,M5)ED1(M4,M5)=-GDNP(M4,M5)SU1(M4,M5)= GUNP(M4,M5)SD1(M4,M5)= GDNP(M4,M5)TU1(M4,M5)= GUNP(M4,M5)
125 TD1(M4,M5)= GDNP(M4,M5)C
DO 1255 M4=1,NEU1(M4,M4)=1.0D0+EU1(M4,M4)
1255 ED1(M4,M4)=1.0D0+ED1(M4,M4)C
ELSE
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DO 1 2 6 M 4 = l , N DO 1 2 6 M 5 = 1 ,N
IPM 4=IP*N +M 4 IPM 5=IP*N +M 5E U 1 ( M 4 ,M 5 ) = G U N P (IP M 4 ,M 5 ) E D 1 ( M 4 ,M 5 ) = G D N P (IP M 4 ,M 5 )S U 1 ( M 4, M5) = - GUNP( M 4 , IP M 5 ) S D 1 ( M 4 ,M 5 ) = - G D N P ( M 4 ,I P M 5 ) T U 1 ( M 4 ,M 5 )= G U N P (IP M 4 , IP M 5 )
1 2 6 T D 1 ( H 4 , M 5 ) = G D N P ( I P M 4 ,I P M 5 )END IF
CC HALF OF THE TOTAL TIME S L IC E S ( L 2 )
L 2 = L T IM E /2 L 2 1 = L 2 + 2
CDO 3 3 4 L P = L T I M E ,L 2 1 , - 1
* CALL B U B D I N ( B U , B D , B U I , B D I , I P , L P )CALL BUBDL( B U , B D , I P , L P )CALL I N B U B D L (B U I ,B D I , I P . L P )
CC * GUTO,GDTO **'C
CALL F M M M (B U I ,E U 1 ,E U 2 ,N ,N ,N )CALL F M M M (B D I ,E D 1 ,E D 2 ,N ,N ,N )
CC * GUTL,GDTL * *C
CALL FMMMfSUl, B U , S U 2 , N , N , N )CALL F M M M (S D 1 ,B D ,S D 2 ,N ,N ,N )
CC * GUL, GDL * *C
CALL F M M M (B U I ,T U 1 ,T U 2 ,N ,N ,N )CALL F M M M (B D I ,T D 1 ,T D 2 ,N ,N ,N )CALL F M M M (T U 2 ,B U ,T U 1 ,N ,N ,N )CALL F M M M (T D 2 ,B D ,T D 1 ,N ,N ,N )
CDO 2 2 6 M 1 = 1 ,N DO 2 2 6 M 2 = 1 ,N
I P L P = ( I P - 1 )*L T IM E +L P G U L ( I P L P ,M 1 ,M 2 ) = T U 1 ( M 1 ,M 2 ) G D L ( I P L P ,M 1 ,M 2 ) = T D 1 ( M 1 ,M 2 ) G U T 0 ( I P L P ,M 1 ,M 2 ) = E U 2 ( M 1 ,M 2 ) G D T 0 ( I P L P ,M 1 JM 2 )= E D 2 (M 1 ,M 2 ) E U 1 ( M 1 ,M 2 )= E U 2 ( M 1 ,M 2 )E D 1 ( M 1 ,M 2 )= E D 2 ( M 1 ,M 2 )G U T L f I P L P ,M 1 ,M 2 ) = S U 2 ( M 1 ,M 2 ) G D T L ( I P L P ,M 1 ,M 2 )= S D 2 (M 1 ,M 2 ) S U 1 ( M 1 ,M 2 ) = S U 2 ( M 1 ,M 2 ) S D 1 ( M 1 ,M 2 ) = S D 2 ( M 1 ,M 2 )
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226 CONTINUEC334 CONTINUE
RETURN END
CCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc CC SUBROUTINE HEVL CALCULATE EXP{V(L)J CC WHERE V IS THE POTENTIALS DUE TO ISING SPINS CC ON F-ORBITALS. CC L: TIME SLICE CC IP: PARTITION NUMBER ; FOR BSS IP=IPMAX=1 CC CC RESULTS: COMMON /CHEV/EVU(L), EVD(L) CC CC ......................................................................................................................................C
SUBROUTINE HEVL(IP.L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N2*IPMAX)
CDIMENSION SIGMA(LTT.N)DIMENSION EVU(LTT,N),EVD(LTT,N)
CC : COMMON INPUT DATA :C
COMMON /CLTIME/LTIME COMMON /CDL/DLU COMMON /CUI/UO,VO,EF COMMON /CEI/UM COMMON / CDELTAT/DELTAT COMMON /CSPINS/SIGMA
CC : OUTPUT DATA /COMMON OUT/ :C
COMMON /CHEV/EVU.EVDC
LL=(IP“ 1)*LTIME+LCC FOR SPIN SIGMA C
DO 20 1=1,NEVU(LL,I)= DLU*SIGMA(LL,I)+DELTAT*(UM-0.5DO*UO-EF)
20 EVD(LL,I)=-DLU*SIGMA(LL,I)+DELTAT*(UM-0. 5DO*UO-EF)DO 40 1= 1 ,N
EVU(LL,I)=DEXP(EVU(LL,I))40 EVD(LL,I)=DEXP(EVD(LL,I))
RETURN
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ENDCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCGCCCGCGCCCCCCCCCCCCCCCCCCCCCCc cC *INHEVL* CALCULATES EXP[-V(L)} CC WHERE V IS THE POTENTIALS DUE TO ISING SPINS- CC CC L: TIME SLICE CC IP: PARTITION NUMBER ; FOR BSS IP=IPMAX=1 CC CC RESULTS: COMMON /CHEVIN/EVU(L), EVD(L) CC CC ...................................................................................................................................... C
SUBROUTINE INHEVL(IP,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N2*IPMAX)
CDIMENSION SIGMA(LTT.N)DIMENSION EVU(LTTjN),EVD(LTT,N)
C :C : INPUT DATA :C :
COMMON /CLTIME/LTIME COMMON /CDL/DLU COMMON /CUI/UO,VO,EF COMMON /CEI/UM COMMON /CDELTAT/DELTAT COMMON /CSPINS/SIGMA
C :C : OUTPUT DATA /COMMON OUT/ :C :
COMMON /CHEVIN/EVU.EVDC
LL=(IP-I)*LTIME+LCC FOR SPIN SIGMA C
DO 20 1=1 ,NEVU( LL, I )= DLU*SIGMA(LL, I ) +DELTAT*( UM-0 . 5DO*UO-EF)
20 EVDfLL,I)=-DLU*SIGMA(LL, I)+DELTAT*(UM-0. 5DO*UO-EF)C
DO 40 1= 1 ,NEVU(LL,I)=DEXP(-EVU(LL,I))
40 EVD(LL,I)=DEXP(-EVD(LL,I))RETURNEND
C
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c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cC *BUBDL* CALCULATES (BU,BD) AT TIME SLICE :(IP-1)*LTIME+L C WHERE IP : PARTITION NUMBER, FOR BSS IP=IPMAX=IC L : TIME SLICECC ............................................-...........
SUBROUTINE BUBDL(BU,BD,IP,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N2*IPMAX) DIMENSION BU(N2,N2),BD(N2,N2)DIMENSION EEK(N2,N2)DIMENSION EVU(LTT,N),EVD(LTT,N)
CCOMMON /CLTIME/LTIME COMMON /CEXPEK/EEK COMMON /CHEV/EVU.EVD
CC FIRST CALCULATE EXP{V(L)}C
CALL HEVL(IP.L)CC THEN CALCULATE BL{I,J}C
LL=(IP-1)*LTIME+L DO 10 J=1,N DO 10 1=1,N2
BU(I,J)=EEK(I,J)*EVU(LL,J)10 BD(I,J)=EEK(I,J)*EVD(LL,J)
DO 20 J=N+1,N2 DO 20 1=1,N2
BU(I,J)=EEK(I, J)20 BD(I,J)=EEK(I,J)
RETURN END
C C CC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC cc INBUBDL CALCULATES (BU-1,BD-1) AT TIME SLICE: (IP-1)*LTIME+L C C WHERE IP : PARTITION NUMBER, FOR BSS IP=IPMAX=1 CC L : TIME SLICE CC CC .......................................................... C
SUBROUTINE INBUBDL(BU,BD,IP,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)
n o
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PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N2*IPMAX) DIMENSION BU(N2,N2),BD(N2,N2)DIMENSION EEK(N2,N2)DIMENSION EVU(LTT,N),EVD(LTT,N)
CCOMMON /CLTIME/LTIME COMMON /CEKIN/EEK COMMON /CHEVIN/EVU.EVD
CC FIRST CALCULATE EXP{-V(L)}C
CALL INHEVL(IP.L)CC THEN CALCULATE BL(-1){I,J)C
LL=(IP-1)*LTIME+L DO 10 1=1,NDO 10 J=1,N2
BU(I,J)=EER(I,J)*EVU(LL,I)10 BD(I ,J) =EEK( I, J)*EVD(LL, I )
DO 20 I=N+1,N2DO 20 J=1,N2
BU(I,J)=EEK(I,J)20 BD(I,J)=EEK(I,J)
RETURNEND
CCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc cC EXPERT CALCULATES THE MATRIES EXP(K) AND EXP(-K) CC CC INPUT: HOPPING MATRIX K, CC SPECIFIES THE GEOMETRY OF THE SYSTEM. CC CC RESULTS: EXPEK, AND ERIN CC Cc ..................................................... C
SUBROUTINE EXPERT(ERO)PARAMETER(ND=4,N=2*ND**2)DIMENSION E0(N,N),E1(N,N),E2(N,N)DIMENSION EXPER(N,N),ERIN(N,N),ER1(N,N),ER0(N,N)
CCOMMON /CDELTAT/DELTAT COMMON /CEXPER/EXPEK COMMON /CERIN/EKIN
CDO 10 1=1,N DO 10 J=1,N
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E 0 ( I , J)=Q. ODO IF (I .E Q .J ) E 0 ( I , J J = 1 . ODO
10 EK1(I,J)=-DELTAT*EK0(I,J)C
DO 20 1=1 ,NDO 20 J=1,N
EXPEKfl, J)=E O (I, J)+EK 1(I, J )20 EKIN(I,J) =EO(I, J ) -E K 1 (I , J )C
FACT= l.ODO FACI=-1.ODOCALL FMMM(EK1,E0,E1,N,N,N)
CDO 30 M=2,100
FACT= FACT*M FACI=-FACI*M HH=1. ODO/FACT IF(HH.LE.1 . OD-99) GOTO 70 CALL FMMM(EK1,E1,E2,N,N,N)
CDO 40 1=1 ,N DO 40 J=1,N
EXPEK(I, J)=EXPEK(I, J ) + l . 0D0/FACT*E2(I, J )40 EKIN(I, J ) =EKIN(I,J)+l.OD0/FACI*E2(I,J)30 CALL FMMM(E2,E0,E1,N,N,N)C70 CONTINUEC WRITE(6,*)'EXPEK'C WRITE(6,*) ( (EXPEK(I, J ) , J = 1 ,N ) , I=1,N)
RETURNEND
CCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC cC THIS ** HER ** SPECIFIES THE GEOMETRY OF THE SYSTEM.C TO PERFORM CALCULATIONS FOR A DIFFERENT, WE NEED FIRST C CHANGE THIS PART.CC EK1 THROUGH EK5 ARE THE FIRST THROUGH FIFTH NEAREST NEIGHBORS.C EQQ SPECIFIES THE TWO SUB-LATTICES CC ...........................................................................................................................................
SUBROUTINE HEKfEKO,EK1,EK2,EK3,EK4,EK5,EQQ)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(ND=4, N=ND**2, N2=N*2)DIMENSION EK0(N2,N2),EK1(N,N),EK2(N,N)DIMENSION EK3(N,N),EK4(N,N),EK5(N,N),EQQ(N,N)DIMENSION IJX(N.N)COMMON /CIJX/IJX
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COMMON /CUI/UO.VO.EFCC ( I 1 ,J 1 ) : COORDINATE OF THE FIRST LATTICE SITE C N1 : LABEL NUMBER OF THE FIRST LATTICE SITE C ( 1 2 , J2 ) : COORDINATE OF THE SECOND LATTICE SITE C N2 : LABEL NUMBER OF THE SECOND LATTICE SITE
DO 1 I I =1,ND DO I J l = 1 ,ND DO 1 12 =1,ND DO 1 J2 =1,ND
NI=(I1-1)*ND+J1 NJ=(I2-1)*ND+J2 EK1(NI,NJ)=0.ODO EK2(NI,NJ)=0. ODO EK3(NI,NJ)=0. ODO EK4(NI,NJ)=0. ODO EK5(NI,NJ)=0.ODO EQQCNI,NJ)=1.0D0
FIND THE DISTANCE BETWEEN THE TWO SITES USE THE PERIODIC BOUNDARY CONDITIONS
I=IABS(I1-I2)IF(I.E Q .(N D -1)) 1=1 J=IABS(J1-J2)IF(J.EQ .(N D-1)) J=1 L=I*I+J*J LL=MOD(( I+J) , 2)
CIF(L .EQ .l) EK1(NI,NJ)=1. ODO IF(L.EQ.2 ) EK2(NI,NJ)=1.ODO IF(L.EQ.4) EK3(NI,NJ)=1. ODO IF(L.EQ.S) EK4(NI,NJ)=1.0D0 IF(L.EQ.S) EK5(NI,NJ)=1. ODO IF(LL.EQ.l) EQQCNI,NJ)=-l.ODO 11= 12 - 11+1I F ( I I .L T . l ) 11= II+ND JJ=J2-J1+1I F (J J .L T .l ) JJ= JJ+ND NT=(II-1)*ND + JJ IJX(NI,NT)=N2
1 CONTINUEC
DO 33 1=1 ,N DO 33 J=1,N
33 EKO(I,J)=O.ODO DO 34 I=N+1,N2DO 34 J=N+1,N2
EK0(I,J)=EK1(I-N,J-N)IF (I .E Q .J ) EK0(I,J)=-UM
34 CONTINUE
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DO 35 1=1 ,N DO 35 J=N+1,N2
35 EKO(I,J)=VO*EK1(I,J-N)DO 36 I=N+1,N2DO 36 J=1,N
36 EK0(I, J)=VO*EKl(I-N,J)RETURNEND
CCc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cC EXAM IS USED TO CHECK THE PRECISION OF GREEN'S FUNCTIONS C GUI, GDI: GREEN'S FUNCTIONS AT TIME L AFTER UPDATING STEPS.CC NEW GREEN'S FUNCTION GU, GD ARE CALCULATED FOR THE SCRACHC PRECISION IS CHECKED.CC ............... - .....................- ..........................- ..................................... ......... ............................
SUBROUTINE EXAM(GU1,GD1,L)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(LA=40,ND=4,N=2*ND**2, IPMAX=1, NP=N*IPMAX) DIMENSION GUl(NP.NP),GD1(NP,NP)DIMENSION GU(NP.NP), GD(NP.NP)DIMENSION EU(NP,NP), ED(NP.NP)
CC INPUT GREEN'S FUNCTIONS AT TIME LC
WRITE(6,*) ' L=',L WRITE(6,*) 'GU AFTER UPDATE'WRITE(6,* ) ( (G U 1(II, J J ) , J J= 1 ,N P ) ,1 1 = 1 ,NP)WRITE(6,*) 'GD AFTER UPDATE1 WRITEC6,*) ( (G D 1 (II ,J J ) ,J J = 1 ,N P ) ,I I= 1 ,N P )
CC CALCULATED GREEN’S FUNCTIONS AT TIME LC
CALL GNPUD(GU,GD,L)WRITEC6,*) 'GU FROM GNPUD1WRITE(6,*) ( ( G U ( I I ,J J ) ,JJ= 1 ,N P ) ,1 1 = 1 ,NP)WRITE(6,*) 'GD FROM GNPUD'WRITEC6,* ) ( (GDCII, J J ) ,J J= 1 ,N P ) ,1 1 = 1 ,NP)
CC CHECK ERRORS C
DO 787 11=1,NP DO 787 JJ=1,NP
E U (II , JJ )= G U 1 (II ,J J )-G U (II ,J J )787 E D (II ,JJ )= G D 1(II ,J J ) -G D (I I ,J J )
WRITE(6,*) 'DIFF GU 'WRITEC6,*) ( ( E U ( I I ,J J ) , JJ=1 ,N P ), II=1,NP)
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WRITEC6 ,* ) 'DIFF GD 'WRITE(6,*) ( (E D (I I , J J ) , JJ= 1 ,N P ), II=1,NP)
CC CALCULATE DETERMINANTS:C
CALL DETMCGU,DETGU)CALL DETMCGD,DETGD)WRITE( 6 , * ) 'DETGU=', DETGU,1 DETGD=‘ , DETGDRETURNEND
CCCc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cc cc *DIV* CALCULATES THE AVERAGE AND STANDARD DIVIATION CC A : INPUT ARRAY CC D : ACTUAL INPUT NUMBERS; DIMENSION OF A CC B : RETURNS THE AVERAGE OF A CC C : STANDARD DIVIATION OF A. CC SIGN: INPUT AVERAGE SIGN OF THE DERMINANT(M). CC CC .................... - ..........................- ......................... - .............. C
SUBROUTINE DIV(A,B,C,M,SIGN)IMPLICIT REAL*8 (A-H,0-Z)DIMENSION A (10000)
CB=0. ODO DO 10 1= 1 ,M
10 B=B+A(I)B=B/FLOAT(M)B=B/SIGN
CC=0.ODO DO 20 1 = 1 ,M
20 C=C+(A(I)-B)**2C=C/FLOAT(M-l)C=DSQRT(C/FLOAT(M))C=C/SIGN
CRETURNEND
CCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCc cc *DETM* CALCULATES THE DETERMINANT OF AN ORDINARY MATRIX B. C C CC USES THE CROUT REDUCTION METHOD CC NP IS ACTUAL DIMENSION OF MATRIX C
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C C(NP,NP) ARE WORK SPACE. CC CC ...................................................................................................... - ................................... C
SUBROUTINE DETM(B.DET)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(ND=4, N=ND**2, IPMAX=1, N2=N*2, NP=N2*IPMAX ) DIMENSION B(NP,NP),C(NP,NP)
CDO 10 1= 1 ,NP DO 10 J=1,NP
10 C ( I , J ) = 0 . ODO C
DO 50 1=1,NPDO 50 J=1,NP
Q=B(I,J)DO 20 K=1,NP
20 Q=Q-C(I,K)*C(K,J)C
I F (I .L T .J ) THEN C (I ,J )= Q /C (I ,I )
ELSEG(I,J)=Q
ENDIFC50 CONTINUEC
DET=l.ODO DO 60 1=1 ,NP
60 DET=DET*C(I,I)RETURNEND
Cc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c cc cC INVERSE A MATRIX A - -> TO AINV CC TO MAKE THE INVERSION SUBROUTINE MORE PORTABLE. CC CC ............................................................................................................................................... C
SUBROUTINE MATRINV(A.AINV)IMPLICIT REAL*8(A-H,0-Z)PARAMETER(ND=4, N=ND**2, N2=N*2, IPMAX=1, NP=N2*IPMAX) DIMENSION A(NP,NP), AINV(NP.NP)DIMENSION WV(NP+1)CALL PFINV(NP, A, WV, AINV, IERR)RETURNEND
CCC
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CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCGGCGCCCCCCCGCCCCCCCCCCCCC
SUBROUTINE RESULTS(MES,SIGN,GU,GD)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=5000, ND=4, N=ND**2, N2=N*2, NP=N2*IPMAX)
CDIMENSION EE0(N,N)DIMENSION GU(NP,NP), GD(NP,NP)DIMENSION EK0(N2,N2),EK1(N,N),EK2(N,N),EK3(N,N)DIMENSION EK4(N,N), EK5(N,N),EQQ(N,N)
CDIMENSION SIG(LB),TLN(LB),TLNU(LB),TLND(LB),UDN(LB)DIMENSION SPIN(LB), ENRY(LB)
CDIMENSION SAO(LB),SA1(LB),SA2(LB),SA3(LB),SA4(LB),SA5(LB) DIMENSION SFFO(LB),SFF1(LB),SFF2(LB),SFF3(LB),SFF4(LB),SFF5(LB) DIMENSION SFDO( LB) , SFD1(LB) , SFD2( LB) , SFD3( LB) , SFD4(LB) , SFD5( LB)
CDIMENSION CAO(LB),CAl(LB),CA2(LB),CA3(LB),CA4(LB),CA5(LB) DIMENSION CFFO(LB),CFF1(LB),CFF2(LB),CFF3(LB),CFF4(LB),CFF5(LB) DIMENSION CFDO( LB) , CFD1( LB) , CFD2( LB) , CFD3(LB) , CFD4( LB) , CFD5( LB)
CDIMENSION SQFO(LB), SQFQ(LB)DIMENSION SQAO(LB), SQAQ(LB)DIMENSION SXFO(LB), SXFQ(LB)DIMENSION SXXO(LB), SXXQ(LB)DIMENSION SXAO(LB), SXAQ(LB)
CDIMENSION CQFO(LB), CQFQ(LB)DIMENSION CQAO(LB), CQAQ(LB)DIMENSION CXFO(LB), CXFQ(LB)DIMENSION CXXO(LB), CXXQ(LB)DIMENSION CXAO(LB), CXAQ(LB)
CDIMENSION GUT0(LTT,N2,N2), GDT0(LTT,N2,N2)DIMENSION GUTL(LTT,N2,N2), GDTL(LTT,N2,N2)DIMENSION GUL(LTT,N2,N2), GDL(LTT,N2,N2)
CCOMMON /CDELTAT/DELTAT COMMON /CLTIME/LTIME COMMON /CUI/UO,VO,EF COMMON /CEI/UMCOMMON /GB/GUTO, GDTO, GUTL, GDTL, GUL,GDL COMMON /CIJX/IJX
^RESULTS* PERFORMS MONTE CARLO MEASUREMENTS. MEASUREMENT NUMBER : MESSIGN OF DETMINANTS FOR THIS MEASUREMENT: SIGN ALL MEASUREMENT COMMONED OUT.
CCCcccc
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COMMON /CEKS/EKO,EK1,EK2,EK3, EK4, EK5, EQQ COMMON / CNS/SIG, TLN, TLNU, TLND, UDN, SPIN, ENRY COMMON /CSA/SAO, SA1 , SA2, SA3, SA4, SA5 COMMON /CSFF/SFFO, SFF1, SFF2, SFF3, SFF4, SFF5 COMMON /CSFD/SFDO, SFD1, SFD2, SFD3, SFD4, SFD5 COMMON /CCA/CAO,CA1,CA2,CA3,CA4,CA5 COMMON /CCFF/CFFO, CFF1, CFF2, CFF3, CFF4, CFF5 COMMON /CCFD/CFDO, CFD1, CFD2, CFD3, CFD4, CFD5 • COMMON / CSQ/SQFO, SQFQ, SQAO, SQAQ COMMON /CSX/SXFO, SXFQ, SXAO, SXAQ, SXXO, SXXQ COMMON /CCQ/CQFO, CQFQ, CQAO, CQAQ COMMON /CCX/CXFO, CXFQ,CXAO,CXAQ,CXX0,CXXQ
CDO 10 1= 1 ,N DO 10 J=1,N
10 EEO(I,J)=O.ODO DO 11 1= 1 ,N
11 E E 0(I ,I )= 1 .0D 0C
SIG(MES) =SIGN SPIN(MES) = 0 .ODO TLN(MES) =0.0D0 TLNU(MES) = 0 .ODO TLND(MES) = 0 .ODO UDN(MES) = 0 .ODO ENRY(MES) =0.0D0 SAO(MES) = 0 .ODO SAl(MES) = 0 .ODO SA2CMES) =O.ODO SA3(MES) = 0 .ODO SA4(MES) = 0 .ODO SA5(MES) = 0 .ODO SFFO(MES) = 0 .ODO SFFl(MES) = 0 .ODO SFF2(MES) =0.0D0 SFF3(MES) = 0 .ODO SFF4CMES) = 0 .ODO SFF5(MES) =O.ODO SFDO(MES) =0.0D0 SFDl(MES) = 0 .ODO SFD2(MES) = 0 .ODO SFD3(MES) =O.ODO SFD4CMES) = 0 ,ODO SFD5CMES) = 0 .ODO CAO(MES) = 0 .ODO CAl(MES) = 0 .ODO CA2(MES) =0.0D0 CA3(MES) = 0 .ODO CA4(MES) = 0 .ODO CAS(MES) = 0 .ODO
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CFFO(MES) = 0 .ODOCFFl(MES) =0. ODOCFF2(MES) = 0 .ODOCFF3(MES) = 0 .ODOCFF4(MES) = 0 .ODOCFF5(MES) = 0 .ODOCFDO(MES) =O.ODOCFDl(MES) = 0 .ODOCFD2( MES) =0. ODOCFD3(MES) = 0 .ODOCFD4(MES) =0.0D0CFD5(MES) =O.ODOCFD5(MES) = 0 .ODOSQFO(MES) = 0 .ODOSQFQ(MES) =0.0D0SQAO(MES) = 0 .ODOSQAQ(MES) = 0 .ODOSXFO(MES) = 0 .ODOSXFQ(MES) =0.0D0SXAO(MES) = 0 .ODOSXAQ(MES) = 0 .ODOSXXO(MES) = 0 .ODOSXXQ(MES) =O.ODOCQFO(MES) = 0 .ODOCQFQ(MES) =0.0D0CQAO(MES) = 0 .ODOCQAQ(MES) = 0 .ODOCXFO(MES) = 0 .ODOCXFQ(MES) = 0 .ODOCXAO(MES) = 0 .ODOCXAQ(MES) =O.ODOCXXO(MES) = 0 .ODOCXXQ(MES) = 0 .ODO
cC SUM OVER THE SITES C
DO 651 1=1 ,N2SPIN(MES) = SPIN(MES)+ G D (I ,I ) -G U (I ,I )
651 TLN(MES) = TLN(MES) + 2 .0 D 0 -G U (I ,I ) -G D (I ,I )DO 652 1=1 ,NTLNU(MES) = TLNU(MES)+ l.ODO-GU(I.I)TLND(MES) = TLND(MES)+ 1 . 0D0-GD(I, I )
652 UDN(MES) = UDN(MES) + ( l .O D 0 -G U (I ,I ) )* ( l .O D O -G D (I ,I ))ENRY(MES) = UO*UDN(MES) +EF*( TLNU(MES)+TLND(MES))
CG CORRELATION FUCTIONS AND STRUCTURE FACTORS C
DO 891 1=1 ,N DO 891 J=1,N
DIJ=0. ODOIF (I .E Q .J ) DIJ=1.0D0
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IN=I+NJN=J+N
TIJS = ( l .O D 0-G U (I ,I ))* ( l .O D O -G U (J ,J ))+ (D IJ-G U (J ,I ))*G U (I ,J )+ (1 .0 D 0 -G D (I ,I ) )* (1 .0 D 0 -G D (J ,J ))+ (D IJ -G D (J ,I ) )* G D (I ,J ) - ( 1 . ODO-GU(I, I ) ) * ( 1 . ODO-GDCJ,J))“( 1 . ODO-GUCJ,J)) * ( 1 .ODO-GD(I, I ) )
TINJS = (1 .0D 0-G U (IN ,IN ))*(1 .0D 0-G U (J,J))-G U (J,IN )*G U (IN ,J)+ ( 1 .0D0-G D(IN ,IN))*(1 .0D0-G D(J,J))-G D(J,IN)*G D (IN ,J) -Cl.ODO-GU(IN, IN )) * ( 1 . ODO-GDCJ, J ) )- ( 1 . ODO-GUCJ,J)) * ( 1 . ODO-GDCIN, IN ))
TIJNS = C1.0D0-GUCI,I))*C1-0D0-GUCJN,JN))-GUCJN,I)*GUCI,JN)+ C1 . ODO-GDCI, I ) )*C1 •ODO-GDCJN, JN )) -GDCJN»I)*G D (I, JN) -Cl.ODO-GUCI, I ) )* C 1 •ODO-GDCJN, JN )) - C i . o d o - g u c j n , j n ) ) * C 1 . o d o - g d c i , i ) )
TINJNS= C1 •ODO-GUCIN, IN))*C1 . ODO-GUCJN, JN ))+CDIJ-GUCJN.IN) )*GUCIN,JN) + CDIJ-GDCJN.IN) )*GDCIN,JN) + C1 •ODO-GDCIN, IN))*C1 •ODO-GDCJN, JN )) -C1.0D0-GUCIN,IN))*C1.0D0-GDCJN,JN))-C1 . ODO-GUCJN,JN))*C1.ODO-GDCIN,IN))
TIJC = C1.0D0-GUCI,I))*C1.0D0-GUCJ,J))+CDIJ-GUCJ,I))*GUCI,J)+ C1 •ODO-GDCI, I ))* C 1■ODO-GDCJ,J))+CDIJ-GDCJ, I ))* G D (I , J) +C1 -ODO-GUCI, I ) ) * c 1 •ODO-GDCJ, J ) )+C1.ODO-GUCJ,J))*C1.ODO-GDCI,I))
TINJC = Cl-ODO-GUCIN,IN))*C1.0DO-GUCJ,J))-GUCJ,IN)*GUCIN,J)+ CI•ODO-GDCIN,IN))*c1 •ODO-GDCJ.J))-GDCJ, IN)*GDCIN, J)+C1 . ODO-GUCIN, I N ) )* c 1 •ODO-GDCJ, J ) )+C1 . ODO-GUCJ,J ) ) * ( 1 • ODO-GDCIN, IN ))
TUNC = C1 • ODO-GUC I , I ) )*C 1 .ODO-GUC JN,JN))-GUCJN, I)*GUC I , JN)+ C1 . ODO-GDCI,I))*C1-ODO-GDCJN,JN))-GDCJN,I)*GDCI,JN)+C1 .ODO-GUCI, I ) ) * ( 1 . ODO-GDCJN, JN ))+C1 . ODO-GUCJN, JN))*C1 . ODO-GDC1 ,1 ) )
TINJNC= C1 • ODO-GUCIN, IN))*C1 . ODO-GUCJN, JN))+CDIJ-GUCJN.IN) )*GU(IN,JN) + CDIJ-GDfJN, IN) )*GDCIN,JN) + C1 . ODO-GDCIN, IN) )* C I•ODO-GDCJN, JN) )+C1 . ODO-GUCIN, I N ) )* c 1 . ODO-GDCJN, JN))+C1 . ODO-GUCJN, J N ))* c 1 . ODO-GDCIN, IN ))
SAOCMES) = CTIJS+TINJS+TIJNS+TINJNS) •k EEOC I , J ) + SAOCMES)SA1CMES) = CTIJS+TINJS+TIJNS+TINJNS) k EK1CI.J) + SA1CMES)SA2CMES) * C TIJS+TINJS+TIJNS+TINJNS) * EK2CI, J) + SA2CMES)SA3CMES) = ( TIJS+TINJS+TIJNS+TINJNS) * EK3CI.J) + SA3CMES)SA4CMES) = CTIJS+TINJS+TIJNS+TINJNS) k EK4CI.J) + SA4CMES)SA5CMES) = C TIJS+TINJS+TIJNS+TINJNS) k EK5CI.J) + SA5CMES)SFFOCMES)= TIJS k EEOCI.J) + SFFOCMES)SFF1CMES)= TIJS k EK1(I, J ) + SFF1CMES)SFF2CMES)= TIJS k EK2CI,J) + SFF2CMES)SFF3CMES)= TIJS k EK3CI.J) + SFF3CMES)SFF4CMES)- TIJS k EK4CI.J) + SFF4CMES)SFF5CMES)* TIJS k EK5CI.J) + SFF5CMES)SFDOCMES)= TUNS k EEOCI.J) + SFDO(MES)SFD1CMES)= TUNS k EK1CI,J) + SFDl(MES)SFD2CMES)= TUNS k EK2CI.J) + SFD2CMES)
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SFD3(MES)= TUNS * EK3( I J + SFD3CMES)SFD4(MES)= TUNS * EK4(I J + SFD4CMES)SFD5(MES)= TUNS * EK5(I J + SFD5CMES)CAO(MES) = ( TIJC+TINJC+TIJNC+TINJNC) * EEO(I J + CAOCMES)CAl(MES) = ( TIJC+TINJC+TIJNC+TINJNC) * EK1(I J + CAl(HES)CA2CMES) = c TIJC+TINJC+TIJNC+TINJNC) * EK2( I J + CA2CMES)CA3(MES) = ( TIJC+TINJC+TIJNC+TINJNC) * EK3(I J + CA3CMES)CA4(MES) = ( TIJC+TINJC+TIJNC+TINJNC) * EK4(I J + CA4CMES)CA5(MES) = c TIJC+TINJC+TIJNC+TINJNC) * EK5( I J + CAS(MES)CFFO(MES)= TIJC * EEO( I J + CFFO(MES)CFF1(MES)= TIJC * EK1(I J + CFFl(MES)CFF2(MES)= TIJC * EK2(I J + CFF2(MES)CFF3(MES)= TIJC * EK3(I J + CFF3CMES)CFF4(MES)= TIJC * EK4( I J + CFF4CMES)CFF5(MES)= TIJC * EK5(I J + CFFS(MES)CFDO(MES)= TUNC * EEO( I J + CFDO(MES)CFD1(MES)= TIJNC * EK1CI J + CFDl(MES)CFD2(MES)= TUNC * EK2CI J + CFD2CMES)CFD3(MES)= TUNC •te EK3CI J + CFD3CMES)CFD4(MES)= TUNC ★ EK4CI J + CFD4CMES)CFD5(MES)= TUNC * EK5CI J + CFD5CMES)
SQFO(MES)SQFQ(MES)SQAO(MES)SQAQ(MES)CQFO(MES)CQFQ(MES)CQAO(MES)CQAQ(HES)
SXXO(MES) SXXQ(MES) CXXO(MES) CXXQ(MES)
891 CONTINUE
SQFO(MES)+ TIJS SQFQ(MES)+ TIJS * SQAO(MES)+ TIJS + SQAQ(MES)+(TIJS + CQFO(HES)+ TIJC CQFQ(MES)+ TIJC * CQAO(MES)+ TIJC + CQAQ(MES)+(TIJC +
EQQ(I.J)TINJS + TIJNS + TINJS + TIJNS +
EQQ(I.J)TINJC + TUNC + TINJC + TUNC +
TINJNSTINJNS)*EQQ(I, J)
TINJNCTINJNC)*EQQ(I,J)
SXXO(MES)+ TIJS+ TUNS SXXQ(MES)+(TIJS+ TUNS) * EQQ(I,J) CXXO(MES)+ TIJC+ TUNC CXXQ(MES)+(TIJC+ TUNC) * EQQ(I.J)
CALCUALTE SUCEPTIBILITIES
CALL SUSLL(EQQ.MES)
TOTAL ENERGY
DO 856 1=1,N2 DO 856 J=1,N2
56 ENRY(MES) = ENRY(MES) - (GU(I,J)+GD(I,J) ) * EKO(I.J)
FIX-UP THE SIGN PROBLEM
RETURNEND
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ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccSUSLL CALCULATES THE SUSCEPTIBILTIES
SUBROUTINE SUSLL(EQQ,MES)IMPLICIT REAL*8(A-H,0-Z)PARAMETER (LA=40, IPMAX=1, LTT=LA*IPMAX)PARAMETER (LB=SOOO, ND=4, N=ND**2, N2=N*2, NP=N2*IPMAX) DIMENSION GUTO(LTT,N2,N2),GDTO(LTT,N2,N2), EQQ(N,N)DIMENSION GUTL(LTT, N2, N2) , GDTL( LTT, N2»N2)DIMENSION GUL(LTT,N2,N2) ,GDL(LTT,N2,N2)
CDIMENSION SXFO(LB), SXFQ(LB)DIMENSION SXXO(LB), SXXQ(LB)DIMENSION SXAO(LB), SXAQ(LB)DIMENSION CXFO(LB), CXFQ(LB)DIMENSION CXXO(LB), CXXQ(LB)DIMENSION CXAO(LB), CXAQ(LB)
CCOMMON /CLTIME/LTIME COMMON /CDELTAT/DELTAT COMMON /GB/GUTO, GOTO, GUTL, GDTL, GUL,GDL COMMON /CSX/SXFO, SXFQ, SXAO, SXAQ, SXXO, SXXQ COMMON /CCX/CXFO,CXFQ, CXAO, CXAQ, CXXO, CXXQ
CSXFO(MES) = 0 .ODO SXFQ(MES) = 0 .ODO SXAO(MES) = 0 ,ODO SXAQ(MES) =O.ODO CXFO(MES) = 0 .ODO CXFQ(MES) = 0 .ODO CXAO(MES) = 0 .ODO CXAQ(MES) = 0 .ODO
CC TOTAL TIME SLICES = LTIME * IPMAX C
DO 10 L=1,LTIME*IPMAXDO 10 I~1,NDO 10 J=1,NIN=I+NJN=J+NTIJ1 = ( 1 . ODO-GUL(L, I , I ) ) * ( l .O D 0-G U L (l ,J ,J ))
1 + GUTL(L,J,I) * GUTO(L,I,J)1 + (1 .0 D 0-G D L (L ,I ,I )) * ( 1 .0D 0-G D L(1,J ,J))1 + GDTL(L,J,I) * GDTO(L,I, J )TIJ2 = ( 1 . ODO-GUL(L, I , I ) ) * ( 1.0D0“G D L (1,J ,J))
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1 + ( 1 . ODO-GDL(Tj , I , I ) ) * ( 1 . ODO-GUL( 1 , J , J ) )TINJ1 = (1.0D0-GUL(L,IN,IN)) * ( 1 .0D 0-G U L(1,J ,J))
1 + GUTL(L,J,IN) * GUTQ(L,IN,J)1 + (1.0D0-GDL(L,IN,IN)) * ( 1 .0D 0-G D L (I ,J ,J ))1 + GDTL(L.J.IN) * GDTO(L,IN,J)TINJ2 = ( 1 . ODO-GUL(L, IN, IN )) * ( 1 . ODO-GDL(1 ,J , J ) )
1 + (l.OD0-GDL(L,IN,IN)) * ( 1 . ODO-GUL(I, J , J ) )TIJN1 = ( 1 . ODO-GUL(L, 1 , 1 ) ) * ( l.ODO-GUL(l.JN.JN))
1 + GUTL(L,JN,I) * GUTO(L,I,JN)I + ( 1 . ODO-GDL(L,I, 1 ) ) * ( 1 .ODO-GDL(I,JN,JN))1 + GDTL(L,JN,I) * GDTO(L,I, JN)
TIJN2 = ( 1 . ODO-GUL(L,I, 1 ) ) * ( L.ODO-GDL(l,JN,JN))1 + (1 .0D 0-G D L (L ,I ,I ) ) * c 1 . ODO-GUL(1 ,JN ,JN ))
TINJN1 = ( 1 . ODO-GUL(L,IN,IN)) * ( 1.0D0-GUL(1,JN,JN)) 1 + GUTL(L,JN,IN) * GUTO(L,IN,JN)1 + ( 1 . ODO-GDL(L,IN,IN)) * ( l.OD0-GDL(l,JN,JN))1 + GDTL(L.JN.IN) * GDTO(L,IN,JN)TINJN2 = (1.0DO-GUL(L,IN,IN)) * ( l,ODO-GDL(l,JN,JN))
1 + ( 1 . ODO-GDL(L,IN,IN)) * ( l.OD0-GUL(l,JN,JN))C
SXFO(MES) = SXFO(MES) + TIJ1-TIJ2SXFQ(MES) = SXFQ(MES) + (T IJ1-T IJ2 )*EQQ(I,J)SXAO(MES) = SXAO(MES) + TIJ1-TIJ2 + TINJ1-TINJ2
1 + TIJN1-TIJN2 + TINJN1-TINJN2SXAQ(MES) = SXAQ(MES) + (TIJ1-T IJ2 + TINJ1-TINJ2
1 + TIJN1-TIJN2 + TINJN1-TINJN2) * EQQ(I,J)CXFO(MES) = CXFO(MES) + TIJI+TIJ2CXFQ(MES) = CXFQ(MES) + (TIJ1+TIJ2 )*EQQ(I1J )CXAO(MES) = CXAO(MES) + TIJ1+TIJ2 + TINJ1+TINJ2
1 + TIJN1+TIJN2 + TINJN1+TINJN2CXAQ(MES) = SXAQ(MES) + (TIJ1+TIJ2 + TINJ1+TINJ2
1 + TIJN1+TIJN2 + TINJN1+TINJN2) * EQQ(I.J)CONTINUE
SXFO(MES) -DELTAT* SXFO(MES)SXFQ(MES) =DELTAT* SXFQ(MES)SXAO(MES) =DELTAT* SXAO(MES)SXAQ(MES) =DELTAT* SXAQ(MES)CXFO(MES) =DELTAT* CXFO(MES)CXFQ(MES) =DELTAT* CXFQ(MES)CXAO(MES) =DELTAT* CXAO(MES)CXAQ(MES) =DELTAT* CXAQ(MES)RETURNEND
CC ========================================================C//LKED.APLMOD DD DSN-PHHANG.FPS.C0M(ADN4P1),DISP=SHR,SPACE=
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VITA.
Yi Zhang was bom on December 23, 1963 in Hangzhou, China.
He received a degree of Bachelor of Science in Physics from the
University of Science and Technology of China in July 1984. In
August 1984, he came to the United State for post-graduate studies
through the CUSFEA program. He joined the group of Professor
Callaway in 1985, and received a Master of Science in Physics from
Louisiana State University, Baton Rouge, Louisiana in August 1988.
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Candidate:
Major Field:
Title of Dissertation:
DOCTORAL EXAMINATION AND DISSERTATION REPORT
Yi Zhang
Physics
Quantum Monte Carlo S im ulations o f Hubbard and Anderson Models
Approved:
r Prolessor and Chairman
Dean of the Graduate^Khool
EXAMINING COMMITTEE;
jKL~ J c.
Date of Examination:
17 January 1990