University of Iowa Iowa Research Online eses and Dissertations Summer 2011 Fisher's zeros in laice gauge theory Daping Du University of Iowa Copyright 2011 Daping Du is dissertation is available at Iowa Research Online: hps://ir.uiowa.edu/etd/1217 Follow this and additional works at: hps://ir.uiowa.edu/etd Part of the Physics Commons Recommended Citation Du, Daping. "Fisher's zeros in laice gauge theory." PhD (Doctor of Philosophy) thesis, University of Iowa, 2011. hps://doi.org/10.17077/etd.bfnqfycu
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University of IowaIowa Research Online
Theses and Dissertations
Summer 2011
Fisher's zeros in lattice gauge theoryDaping DuUniversity of Iowa
Copyright 2011 Daping Du
This dissertation is available at Iowa Research Online: https://ir.uiowa.edu/etd/1217
Follow this and additional works at: https://ir.uiowa.edu/etd
Part of the Physics Commons
Recommended CitationDu, Daping. "Fisher's zeros in lattice gauge theory." PhD (Doctor of Philosophy) thesis, University of Iowa, 2011.https://doi.org/10.17077/etd.bfnqfycu
3.1 The results of the zeros with the increasing ”volumes”. . . . . . . . 67
3.2 Both of the actual zeros and the f ′′(x) zeros are shown with dif-ferent ranges of fitting for the volume 44. The semi-major of thecorresponding ellipses are also given ( the focus length is half therange). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3 Both of the actual zeros and the f ′′(x) zeros are shown with differentranges of fitting for the volume 64. . . . . . . . . . . . . . . . . . . . 86
3.4 The actual zeros of various orders of approximation for the twovolumes 44 and 64 discussed in the text. . . . . . . . . . . . . . . . . 87
3.5 The zeros with the increasing ”volumes” using the density of statesf4 and f6 of the 44 and 64 lattices. . . . . . . . . . . . . . . . . . . . . 88
3.6 βS, s1 and s2 defined in the text for L = 4, 6 and 8. . . . . . . . . . . 91
3.7 Real part of the first three zeros for L = 4, 6 and 8. . . . . . . . . . . 93
3.8 Imaginary part of the first three zeros for L = 4, 6 and 8. . . . . . . 94
3.9 The lowest three zeros in the three volumes 44,64 and 84. Column 1-4 are, the real parts of the zeros, the estimate error σs from differentseeds of Monte Carlo runs and the error σc due to the orders ofChebyshev interpolation( we used three different orders 40,44 and50 for all three volumes). Same for the imaginary part. . . . . . . . 95
viii
LIST OF FIGURES
2.1 Weak and strong coupling expansions of the average plaquetteP for SU(2) at various orders in the weak and strong couplingexpansion compared to the numerical values. . . . . . . . . . . . . 19
2.2 Numerical value of f (x) compared to the strong coupling expansionat successive orders for the SU(2) lattice gauge fields. . . . . . . . . 23
2.3 Numerical value of f (x) compared to the strong coupling expansionat successive orders for the U(1) lattice gauge fields. . . . . . . . . 23
2.4 The expansion coefficients h2m and g2m in the log scale are plottedversus the order 2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Logarithm of the absolute difference between the numerical dataand the strong coupling expansion of P at successive orders forthe SU(2) model (error diagram). The numerical data are takenfrom the volume 64. The error on numerical P are obtained from50 bootstraps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 The error diagram of P at successive orders for the U(1) Model.The numerical data are from the volume 44. The error is estimatedthrough 20 seeds of data. . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 The error diagram of f (x) at successive orders for the SU(2) Model.The numerical data are from the volume 64. The error on numericalf are obtained from 50 bootstraps. . . . . . . . . . . . . . . . . . . . 26
2.8 The error diagram of f (x) at successive orders for the U(1) Model.The numerical data are from the volume 84. The error is estimatedthrough 20 seeds of data. . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 The difference of f (x) for two different volumes 44 and 64 is plottedas a function of x = S/Np. We see that in the weak coupling region(small x), the volume effect is apparent and cannot be ignored inthe expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.10 The successive orders of f (x) expansion from U(1) weak couplingexpansion are plotted in contrast with the numerical f (x) frommulti-canonical simulations. . . . . . . . . . . . . . . . . . . . . . . 32
2.11 The error diagram of f (x) at successive orders of weak couplingexpansion for the U(1) model. . . . . . . . . . . . . . . . . . . . . . 32
ix
2.12 The error diagram of P at successive orders of weak coupling ex-pansion for the U(1) model. . . . . . . . . . . . . . . . . . . . . . . . 33
2.13 Numerical value of f (x) compared to the weak coupling expan-sion at successive orders. The coefficients are calculated from thevolume 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.14 The error diagram of f at successive orders for the SU(2) model. . 35
2.15 The error diagram of P at successive orders for the SU(2) model(above).The error for the numerical P is calculated using 50 bootstraps ofthe data. The graph below is the case without the contribution ofthe zero mode in b1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.16 The log-log plot of the autocorrelation vs t at β = 2.20 for a 44 SU(2)lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.17 The integrated autocorrelation time is plotted verse β for four dif-ferent volume V = 44, 64, 84, 104. The peak of the distribution isfound at around β = 2.20(for 44), 2.34 (for 64) and 2.40 (for 84). Thepeak is not obvious for the volume 104. . . . . . . . . . . . . . . . . 40
2.18 The overlapping of neighboring data sets. . . . . . . . . . . . . . . 41
2.19 A comparison between the trapezoidal and the generalized Simp-son’s rules that are used to obtained the initial values of F (βα).The data is from the 44 lattice and skipping every 5 datasets. ∆χ2
is the χ-square difference between successive iterations ( definedin the text ), the slope of which indicates the convergent rate of theiteration. It is obvious that when the datasets are sparse, the gener-alized Simpson’s rule gives better initial values and improves theconvergence significantly. . . . . . . . . . . . . . . . . . . . . . . . . 47
2.20 This plot is a comparison of reweighting with various x bin number:200(circles), 500(squares) and 1000(diamonds). They all result inthe same convergence down to the noise level of the data. . . . . . 48
2.21 Two slightly different sets of initials are fed to the reweighting ofa SU(2) data on a 44 lattice and reach the same convergence. Theplot is ∆ f (x) at iteration 10000 (middle two curves) and 200(outertwo curves). Here ∆ f means the difference of f (x) to the stabilizedvalue ( at iteration 15000). . . . . . . . . . . . . . . . . . . . . . . . . 49
x
2.22 An example of the iteration process. All iterations are comparedwith the convergent values f (x)con. Two different initials are com-pared here. For the first one we started with initial values f (x) = 0,for all x. The convergence is slow (upper five curves). For thesecond we used the integration method to get the initials. Themethod is efficient for the iterations (lower two curves). . . . . . . 50
2.23 Convergent iterations may not always lead to satisfactory results.Reweightings with different numbers of β’s are compared. Theresulting f (x)’s are then subtracted from the result with 284 β’s(normalized at x = 1). The solid, dashed, double-dashed curvesare using 142,57 and 29 β’s respectively. Although all lead to con-vergence, the reweighting with fewer datasets show much biggerundesired fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.24 The convergence is sensitive to the number of datasets. The reweight-ing using the same data set while including different numbers ofdatasets at volume 64. The plot is showing the chi2 varies withiterations. The upper one which is using 246 β’s shows obviousdivergence at iteration around 6000, while the lower one with 449datasets indicates an strong sign of convergence. Similar difficultyappears with higher volumes such as 84 and 104. . . . . . . . . . . 52
2.25 The overlapping plots of the previous example. The boxes in eachgraph show two-σ spreading (centered at the average) of each datapoint. The vertical increments have no meaning but to illustratethe neighboring overlapping in a clearer manner. The left graphcorresponds to the diverging reweighting and showing in sufficientoverlapping among the data. . . . . . . . . . . . . . . . . . . . . . . 52
2.26 The errors of the density of states due to the different seeds. . . . . 53
2.27 The collapse of the difference of f (x) of three different volumes,∆ fV4,V8 and ∆ fV6,V8 with the overall constant subtracted (describedin the text). The differences are then divided by their correspondinginverse volume difference. . . . . . . . . . . . . . . . . . . . . . . . 55
2.28 The difference of the differences ∆ fV4,V8 and ∆ fV6,V8 . The flat partcorresponds to the overall constant 0.035. . . . . . . . . . . . . . . . 55
3.1 The plot shows the contours of both real and imaginary part of thepartition function. The intersections of these curves are the zerosof the partition function. . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 The probability distribution of total action in the Gaussian toymodel described in the text. . . . . . . . . . . . . . . . . . . . . . . . 59
xi
3.3 Logarithm of the autocorrelation versus the time series distancefor the original set of 1,600,000 values. . . . . . . . . . . . . . . . . . 60
3.4 Zeros of the real (circles) and imaginary (crosses) part for 40,000Gaussian configurations. The solid lines are the circle of confidenceand the hyperbolas of the normal distribution. . . . . . . . . . . . . 61
3.5 Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the first example. The small dotsare the accurate values for the real (gray) and imaginary (black)parts. The exclusion region boundary for d = 0.12 is representedby boxes (red). The solid line is the circle of confidence of theGaussian approximation. . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the second example. The smalldots are the accurate values for the real (gray) and imaginary(black) parts. The exclusion region boundary for d = 0.15 is rep-resented by boxes. The solid line is the circle of confidence of theGaussian approximation. . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7 The left are the imaginary parts of the zeros in the toy modeldescribed in the text in different ”volumes” plotted against thef ′′(x) = 0 zeros which corresponds to the zeros in infinite ”volume”limit. The right is the log-log plot of the difference of the zeros tothe f ′′ = 0 zero vs the volume. We found that Im(βL − β∞) ∝ L−2.6457. 68
3.8 The logarithm of the absolute coefficients of the Chebyshev ap-proximation discussed in the text versus the order number n. . . . 71
3.9 The ellipse of convergence of a Chebyshev series ( described in thetext) and the roots of f ′(x) = βwith the values given in the text. Thetwo saddle points which are obviously inside the ellipse merge tothe root of f ′′(x) = 0 (red cross) as the value of β approaches to thevalue corresponding to the root of f ′′(x) = 0. . . . . . . . . . . . . . 71
3.10 The ellipse of convergence of a Chebyshev series and the rootsof f ′′(x) = 0(empty circles). The empty squares are the roots off ′(x) = βwhere β is given by f ′(rootsof f ′′ = 0) ( the cross ). . . . . . 72
3.11 The second moment of the SU(2) model at volume 44. The numer-ical result is plotted against various orders of Chebyshev approxi-mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.12 The third moment. Similar to Fig. (3.11). . . . . . . . . . . . . . . . 73
3.14 The plot is showing the correspondence of a loop in the β-plane tothe loop in the x-plane. . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.15 An actual loop in which two zeros are presented. The path isdetoured to get a better integration. . . . . . . . . . . . . . . . . . . 79
3.16 The contour plot shows the zeros of a SU(2) gauge model on a 44
lattice. Blue and green lines are the zero curves of the imaginaryand real part of the partition function from the Monte Carlo sim-ulation at β = 2.18. The crosses and circles are the counterpartsfrom a quasi-Gaussian approximation which overlaps the data inthe bottom part. Both show isolated zeros of the partition func-tion. However they are all lying outside the radius of confidence,or more strictly, they are all above the level of confidence ( thesquares). Therefore these zeros are artificial. The Monte Carlodata of a neighboring β doesn’t indicate the consistent locations ofthese zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.17 Distribution of eros of the real part of the partition function in thecomplex β plane and regions of confidence described in the text. . 81
3.18 The residue distribution after subtracting the Gaussian part withβ = 2.18 on a 44 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.19 The residue distribution after subtracting the Gaussian with β =2.18 on a 64 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.20 The residue distribution after subtracting the Gaussian with β =2.348 on a 64 lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.21 The plot shows a typical pattern of the SU(2) zeros. The zerosof two different volumes, 44(filled squares) and 64 (filled circles),are plotted against their respective f ′′(x) = 0 zeros (empty squaresfor the 44 and empty circles for the 64). The density of states areapproximated using Chebyshev Polynomial of order 44. The fittingrange is over x ∈ [0, 2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.22 The plot is the zeros (empty cirdles) of two volumes: 44 and 64.Their f ′′(x) = 0 zeros (filled circles) are also plotted for comparison.Both of the actual zeros and the f ′′(x) = 0 zeros tend to approachto the same limit which is distant away from the real axis. . . . . . 86
3.23 The lowest zeros obtained using various orders of approximationat two different volumes: 44 and 64. . . . . . . . . . . . . . . . . . . 87
3.24 The locations of the zeros calculated using the entropy densityfunction f at volume 44 scale with Np = 6 × L4. The red pointcorresponds to the f ′′ = 0 zero. . . . . . . . . . . . . . . . . . . . . . 89
xiii
3.25 The zeros βL obtained using variousNp = 6 × L4 using the entropydensity function from two different volumes: 44,64. The plot showsthe distance from βL to the f ′′ = 0 zero β∞ versus L in the logarithmicscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.26 f (s) − βs for β = 1.00175, 100177 and 1.00179 on a 64 lattice. Thehorizontal lines is drawn to indicate the asymmetry of the heights.The error bars are provided with the same scale as f (s) − βs. . . . . 91
3.27 The double peak distribution of three volumes: 44, 64 and 84. . . . 92
3.28 Zeros of the real (point +) and imaginary (point x) part of Z forU(1) using the density of states for 44 and 64 lattices. . . . . . . . . 93
3.29 The shift in the imaginary part of the zeros due to the cuts of theintegration described in the text. . . . . . . . . . . . . . . . . . . . . 94
3.30 The lowest zeros from three volumes 44,64 and 84(from left to right).The error bars have taken account of both of the Monte Carlo sta-tistical error(seeds) and the Chebyshev interpolation error(orders).The three guidelines are the fits for the first, second and third low-est zeros, using only the zeros of 64 and 84. They intersect the realaxis approximately at the same point β = 1.01134. The diamondson the real axes(Imβ=0) are the double-peak β’s from Table.(3.6). . 96
3.31 The log-log plot of the imaginary part of the lowest zero vs thelattice size L = 4, 6, 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.33 The orders of power law from the fits are plotted versus the lowestsize L that is included in the fit. . . . . . . . . . . . . . . . . . . . . . 99
3.34 The first derivative of f (x) from different volumes. . . . . . . . . . 100
3.35 The locations of the two real roots of f ′′(x) = 0 are plotted versusthe lattice size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.1 The log-log plot of log(∆I/I)− 12A as a function of the number of grids
used in the Trapezoidal integration. We use A = 0.004069 and cutthe integral range beyond the precision goal at ym = 582.692 Thefit gives a linear relation of −4.248355148205 + 2.0000000000000x,which agrees with ln((2π2)/(Ay2
m)) = −4.2483551482054. . . . . . . 108
xiv
1
CHAPTER 1INTRODUCTION
1.1 Motivation and Overview
One very important property of the strong interactions is that the coupling
is running with the energy scale at which the interaction takes place [70, 35]. It
was discovered that for models like QCD (quantum chromodynamics) with the
SU(3) gauge group, the beta function, which describes how fast the coupling is
running with the scale of reference, is negative with 16 or less flavors of quarks
(asymptotically free). There has been a renewed interest [71, 36] in the scenario
of the beta function vanishing at a nontrivial infrared fixed point, in addition
to the ultraviolet fixed point at the zero coupling. This results in the loss of
confinement and the recovery of conformality. It was speculated [49, 6] that
the generalization of the fixed points to the complex coupling plane can help to
describe such a scenario. Along the same consideration, the generalization of
the renormalization group (RG) transformation flows to the complex plane was
proposed [24] and it was found out [61] that the flows, coming out of the weakly
coupled fixed point, are controlled by the ”gate” formed by the complex zeros of
the partition function (Fisher’s zeros). It was observed that if the zeros remain
at a finite distance from the real axis when the volume increases, the flows end
at the strongly coupled fixed point and it leads to a confinement theory for the
non-Abelian gauge fields on a lattice. Complex zeros that are approaching the
real axis will block the flow and therefore, a deconfining phase will appear. So to
understand the confining or deconfining nature of the gauge models, the analysis
of the Fisher’s zeros in lattice gauge theory plays an important role. As the two
simplest but contrastingly different examples, the SU(2) and U(1) lattice gauge
theories with a fundamental Wilson action are the ideal candidates for such a
2
purpose. The SU(2) model is believed [56, 16] to be free of a deconfining phase
transition in the continuum limit while the U(1) model was shown [47, 18] to
have a phase transition but the order of which remains controversial. A clear
picture on the structure of the Fisher’s zeros of the SU(2) pure gauge theory at
zero temperature will put us in a better position to understand the physics when
finite temperature or multiple favors of quarks are introduced.
The locations of the Fisher’s zeros will help the understanding of the RG
flows, while the distances of these zeros will help to understand the nature
of critical behaviors, for instance, the order or strength of phase transitions.
Although the U(1) gauge theory on a lattice has been investigated extensively in
the last decade and it is commonly accepted to possess a phase transition, there
are still arguments regarding the order of the transition [50, 47, 46, 18]. Fisher’s
zero provides an efficient tool for such investigations. It was shown [45, 43] that
the critical exponents which will characterize the order of the phase transition can
be computed through the finite size scaling of the Fisher’s zeros and the strength
of the transition can be revealed by the angle that the zeros pinches the real axis.
So it is very interesting to know, with precise calculations of the Fisher’s zeros for
the U(1) gauge model, whether we can end the controversy of the order of the
transition.
1.2 Introduction to Lattice Gauge Models
In this section, the basic formalism of Lattice Gauge Theory will be briefly
given for the sake of convenience in the following discussions. We will mainly
follow the conventional notations. More general and intensive discussions can
be found in [53, 52, 48, 76, 17, 69, 41, 42, 80].
In Lattice Gauge Theory, a lattice is usually referred to a discrete approxi-
mation of the space-time continuum. In particular, for the discussion that does
3
not involve finite temperature where the temporal direction is treated differently,
a lattice will typically mean a symmetric grid L×L×L×L in the four dimensional
hyper cubic lattice where L denotes the number of sites with the same equal
spacing a in each direction.
The Lattice is not a physical reality, but rather a treatment that builds the
theoretical models on a solid mathematical basis. It has many advantages. First
it has a built-in cutoff which removes the ultraviolet divergence in the quantum
field theory. Second, it makes the degrees of freedom countable and allows us
to use the methods of statistical mechanics, which provides a deep connection
between these two disciplines. Thirdly and more importantly, it naturally results
in a confining behavior in the strong coupling limit. However, the lattice spacing
a is artificial. Actual physics should be expected when the continuum limit a→ 0
and the infinite volume limit are taken.
1.2.1 Feynman Path Integral
In quantum mechanics, the transition amplitude of a particle from (t1, q1) to
(t1, q2) can be described by a statistical-like formula
⟨q2|e−iH(t2−t1)|q1⟩ =∫ q2
q1
D[q] eiS, (1.1)
where the action S(q) =∫Ldt is playing the role of Boltzmann factor. The inte-
gration∫D[q] on the right hand side of the equation means summing over all the
possible paths. However the definition is not complete unless an implementation
of ”summing all the possible paths” is given. Time and space extends can be split
into equal spacing grids so that the enumeration of paths is possible.
It can be generalized to the quantum fields where we can define the vacuum
4
to vacuum expectation to be
Z ≡ ⟨0|e−iHt|0⟩ =∫D[ϕ] eiS, (1.2)
where the integral now is summing over all the possible fields configurations
which is well defined if these fields are now confined on the sites of the space
time lattice. Therefore the action can be defined as S = a4 ∑xL. Z is called
partition function. The physical observables become the expectation which is
averaging over all possible field configurations, for example,
⟨0|T[ϕ(x1)ϕ(x2)...ϕ(xn)]|0⟩ = 1Z
∫D[ϕ]ϕ(x1)ϕ(x2)...ϕ(xn) exp(iS). (1.3)
To complete this analogy with statistical mechanics, we will need to appeal
to the Euclidean time, i.e., τ = −it which differs the physical time t by a Wick
rotation, and will replace iS by −S′.
1.2.2 U(1)
Inspired by Wegner’s Ising lattice gauge theories[74, 53], K. Wilson[76]
generalized it to the continuous gauge groups. Instead of a spin that admits only
value ”up” and ”down” in the Ising model, an internal planar angle parameter
θ(n) is attached to each site of the lattice, n. To compare this angle at different sites,
any two neighboring sites are then bridged by tagging the U(1) group elements
Uµ(n) = exp(iθµ(n)) to the link (n,n + µ) where µ is one of the four positive
directions departing from site n. In the negative direction on the same link, we
define θ−µ(n + µ) = −θµ(n) and it follows that U†µ(n) = U−µ(n + µ).
A plaquette is the smallest square consisting of four links in a plane spanned
by any two directions µ, ν. In the following, we will only consider on a finite,
4-dimensional symmetric lattice L4 with periodic boundary condition and the
total number of plaquettes isNp = 6L4.
5
With the difference operator
∆µθ(n) := θ(n + µ) − θ(n),
it can be easily shown that the following expression which is the sum of all four
links of the plaquette (n, µν),
θµν(n) := ∆µθν(n) − ∆νθµ(n) (1.4)
= θµ(n) + θν(n + µ) + θ−µ(n + µ + ν) + θ−ν(n + ν)
is gauge invariant under the local gauge transformation θ(n)→ θ(n)+χ(n) where
χ is an arbitrary function. To see that, with the transformation, the link variable
will transform like eiθµ(n) → e−iχ(n)eiθµ(n)eiχ(n+µ) or θµ(n) → θµ(n) + ∆µχ(n). This is
in analogy to electrodynamics: Aµ ∼ θµ and Fµν ∼ θµν.
Now we can define the action
S =∑n,µν
(1 − Re(UUUU)) (1.5)
=∑n,µν
(1 − Re exp iθµν(n)
)=
∑n,µν
(1 − cosθµν(n)).
With the continuum limit being taken∑
n,µν →∫
d4x/a4 and a weak coupling
approximation 1 − cosθµν ≈ θ2µν/2 1, the Euclidean action of electrodynamics is
1No Einstein automatic summation rule here
6
recovered1g2 S =
14
∫d4xFµνFµν, (1.6)
where we have defined θµ = agAµ and θµν = a2gFµν and g is the coupling strength.
Note the auto-summation in FµνFµν gives a factor 2 because (n, µν) and (n, νµ) are
actually the same plaquette.
The partition function for the U(1) lattice gauge fields can then written as
Z(β) =∏
l
( ∫ 2π
0
dθl
2π
)exp(−βS), (1.7)
where β = 1/g2 and the product of the integration covers all the links l on the
lattice.
1.2.3 SU(2)
The construction can be extended to the non-Abelian case, specifically, the
SU(2) gauge fields. Instead of having a planar angle parameter sitting at every
site as in the U(1) case, we assign a ”vector” θ(n) in the internal space spanned
by the three Pauli matrices
τ1
2=
0 1
1 0
, τ22 =
0 −i
i 0
, τ3
2=
1 0
0 −1
, (1.8)
say, θ(n) = 12agτiAi(n) where Ai are the three parameters that uniquely determine
the vector. τi/2 are the three elements of the su(2) Lie algebra and satisfy [τi, τ j] =
2iϵi jkτk.
In analogy to the case of U(1), to specify the relative orientation of these
vectors at two neighboring sites, we need to define a SU(2) group element
Uµ(n) = exp[12 iagτiAi
µ(n)] at the corresponding link in the positive direction,
while Uµ(n)† = U−µ(n + µ) for the negative direction.
Under the local rotation exp(−iϑ(n)) (where ϑ = τiχi/2) in the internal
space, the link variable undergoes Uµ(n)→ exp(−iϑ(n)Uµ(n) exp(iϑ(n+µ), which
implies that the trace of the plaquette element TrUp is invariant. So the action can
be defined as
S =∑
p
[1 − 12
Re TrUp], (1.10)
where the summation is over all plaquettes.
To compare with the physics in the continuum, we can expand Eq. (1.9)
with respect to a, taking only the non-vanishing lowest order of a,
Up = exp[ia2gFµν +O(a2)], (1.11)
where we have substituted
Fµν = ∂µAν − ∂νAµ + ig[Aµ,Aν] (1.12)
and Aµ ≡ 12τiAi
µ(n). We can modify Eq.(1.11) further and replace the summation
by an integral and Eq. (1.10) becomes
4g2 S =
∫d4x
12
FµνFµν, (1.13)
which is the standard Yang-Mills action in the Euclidean space.
So we can write down the partition function for the SU(2) lattice gauge fields
using the action in (1.7),
Z(β) =∏
l
( ∫dUl
)exp(−βS), (1.14)
where β = 4/g2 for SU(2). The integration is over all the link variables Ul and dUl
8
is the Haar measure which depends on the parametrization of the SU(2) group.
SU(2) With an Adjoint Term
There are various evidences showing that there should be no phase tran-
sition between the strong and weak coupling regions [56, 16]. Since the action
should be a real function of the plaquette variable, it can be expanded using the
character expansion with all the irreducible representations of SU(2). If we only
consider the fundamental representation ( j = 1/2) and the adjoint representation
( j = 1), the modified total action is [12]
βS + βASA ≡ β∑
p
(1 − 12
Re TrUp) + βA
∑p
(1 − 1
3Re TrU(A)
p
), (1.15)
where U(A) is the adjoint representation of the fundamental representation which
is satisfying U†τkU = [UA]klτl. For SU(2), it is a three dimensional representation.
It can be shown that the fundamental and the adjoint representation are related
by
TrU(A)p = |TrUp|2 − 1. (1.16)
The partition function is just [12, 34]
Z(β) =∏
l
( ∫dUl
)exp(−βS − βASA). (1.17)
1.3 Fisher’s Zeros and Critical Phenomena
In general, the physical phenomena that the statistical mechanics formalism
is dealing with can be categorized in two different kinds. In the first kind,
the equilibrium descriptions of the thermodynamic functions are smooth and
continuous in responding to the changes of parameters. In the second kind, it
is characterized by irregularities or discontinuities. The system often exhibits a
cooperative nature and may undergo drastic changes from one state to another of
totally different type, for instance, the phase transition from a liquid to a gas. One
9
would suspect that such phenomena can be naturally described by the formalism
( or its extension) of the first kind where the statistical functions are smooth and
free of singularities. For example, the grand partition function is a polynomial of
the fugacity and will not admit a real zero for a finite volume. Singularities only
happen in the V,N → ∞ limit. Lee and Yang showed remarkably [57, 79] that
it is possible to connect the phase transition of the system to the distribution of
zeros of the grand partition function in the complex fugacity plane. For the Ising
model with ferromagnetic interactions, the zeros lie on the unit circle |z| = 1 where
z = e−4βH which is in analogy to the fugacity ( H is the external field). Singular
behavior of the free energy density in the V,N→∞ limit might be interpreted by
the accumulation of complex zeros near the real axis.
As a natural generalization, it was proposed by Fisher [33] that similar
phenomenon may also be true with the partition function zeros in the complex
temperature plane. He showed it with the evidence from the zero-field Ising
model on the two dimensional lattice with a nearest-neighbor coupling K. The
partition function zeros are lying on double circles | tanh(K ± 1)| =√
2 when the
thermodynamic limit is taken. The logarithmic divergence of the specific heat
is connected to the linearly vanishing density of zeros near the real axis [33, 63],
which was confirmed in various other models [62, 72]. The partition function
zeros in the complex temperature ( or coupling) plane are usually called Fisher’s
zeros, to be distinguished from the Lee-Yang zeros.
The finite size scaling of both of the Lee-Yang and Fisher’s zero were in-
vestigated by Itzykson, Pearson and Zuber [40] through the scaling properties
of the distances among the zeros. They showed that the critical exponents were
governing the scaling law of the distances. Following this, Janke et al [43, 44]
10
analyzed the roles of the zero density function in the scaling of the singular phe-
nomena in various discrete models and proposed a way to determine the order
of the phase transitions as well as the transition strength through scaling of the
the zero density. With all these theoretical setups, it is interesting to know how
the Fisher’s zeros can help us to understand the critical behaviors in the discrete
models as well as the continuous field models.
The analysis using Monte Carlo technique on the U(1) lattice gauge field was
pioneered by Creutz [18]. He approached the continuous U(1) gauge field through
the ZN symmetric groups which take U(1) as the limit of N →∞. He found that a
phase transition of order higher than 1 persisted for N ≥ 6 and survived the U(1)
limit. However a hysteresis structure was observed in [47] on a L = 16 lattice
which suggested a first-order transition. It was revisited in [46] through the
simulations on the spherical lattice using finite size scaling of the Fisher’s zeros.
They found the critical exponent ν = 0.365(8) which is corresponding to a L−2.74
scaling of the imaginary part of the zero, apparently excluding the possibility of a
first-order transition. However a high-statistic analysis on the cumulants showed
[50] that the critical exponents ν is size-dependent and is actually ”rolling” toward
a first-order value (0.25). Therefore a reliable classification of such a transition
using improved methodology is yet to be seen to put the controversy at rest.
It has long been known that the pure SU(2) lattice gauge field with a Wilson
action in the fundamental representation is free from a deconfining phase transi-
tion, which suggests a possible theory of uniform confinement including both of
the weak and strong coupling regimes. In a series of papers [28, 27, 29], Falcioni
et al. studied the location of the complex zeros using single-point reweighting
and the high-order strong coupling expansion method. They spotted the zero
around β = 2.225 + i0.155 on a 44 lattice. What remains interesting is how the
11
zeros depend on the volume and what is the structure of the zero that is charac-
terizing such a system baring no transition in the continuum limit. It was shown
[24] with numerical evidences from the finite-volume large-N O(N) model and
the hierarchical Ising model that the Fisher’s zeros are located at the boundary
of the complex basin of attraction of the infra-red fixed points. With the general-
ization of the renormalization flow to the complex β-plane, it was shown that the
Fisher’s zeros serve as a ”gate” to control the flows which are starting from the
weak coupling fixed point. A confinement theory may be recovered if the zeros
are not blocking the flows from reaching the strong coupled fixed point.
In this thesis, we intend to address part of the questions raised above by
studying the Fisher’s zeros using the density of states method. The thesis is
organized as follows.
In Chapter 2, we give the definition of the density of states and analyze its
general properties. Based on the saddle point approximation, we reconstruct the
density of states from series expansion ( strong and weak coupling expansions).
We analyze the limitation of the perturbative approaches, with the presence
of complex singularities. We then appeal to the numerical calculations using
the Monte Carlo method to approximate the gauge integrations. We collect
sampling data indexed by β. For the SU(2) model, we reconstruct the numerical
representation of the density of states from these spectra of data through the
Ferrenberg-Swendsen’s reweighting method. For the U(1) gauge model, the
density of states are obtained using the Multi-canonical simulations (calculated
by Alexei Bazavov). We also study the volume dependence of these density of
states.
Chapter 3 is devoted to the Fisher’s zeros. We start with the single point
reweighting which is the conventional method to locate the zeros. We point out
12
its limitation for the SU(2) model by showing the fact the zeros are mostly lying
outside the so-called circle of confidence [2]. We examine this carefully with
quasi-Gaussian toy models. By the assumption of saddle point approximation
which is valid for the SU(2) model, we find that there exists a simple connection
between the Fisher’s zeros and the roots of the second order derivative of the
entropy density function. The latter corresponds to the zeros when an infinite
number of plaquettes is taken while using the finite-volume entropy density
function. We develop a series of tools such as the numerical evaluation of the
partition function and the Cauchy’s loop integration method to find Fisher’s zeros
where the entropy density function is approximated by analytic functions. We
discuss the stability of the polynomial approximation and present the results of
the zeros for the SU(2) model at two different volumes 44 and 64. We apply both
of the polynomial approximation method and the discrete reweighting method
to find the lowest three zeros of the U(1) gauge model at three different volumes
44, 64 and 84. We then discuss the scaling properties of these zeros.
The conclusions are provided in Chapter 4.
13
CHAPTER 2THE DENSITY OF STATES
2.1 The Density of States
In the partition functions of the U(1) and SU(2) model, the integration is
over all the link variables. We are mainly interested in the global properties of
the gauge fields on a lattice, in particular, the zeros of the partition function. It is
convenient to work with the density of states which is defined as the number of
configurations per unit interval in the total action S space. The density of states
wraps all the link variables into one, the total action S. The degrees of freedom
of the system are reduced from 4L4 to 1 which makes the problem into a one
dimensional problem. So now the partition function has a form [22]
Z(β) =∫ 2Np
0dS n(S) e−βS, (2.1)
where Np is the total number of plaquettes. We use the function of density of
states n(S) to abstract the complex integration over the links by using the delta
function,
n(S) =∏
l
∫dθl
2πδ(S −
∑p
(1 − cosθp))
(2.2)
and β = 1/g2 for the case of U(1);
n(S) =∏
l
∫dUl δ
(S −
∑p
(1 − 12
Re TrUp))
(2.3)
and β = 4/g2 for the case of SU(2).
The same concept has been used in the discussions for spin models[3] and
gauge models [2] where it is sometimes called the spectral density. It is often
more convenient to use the average action x ≡ S/Np instead of the total action S
as the variable. Correspondingly we will often use f (x), the logarithm of n(S),
n(S) = eNp f (x). (2.4)
14
In this paper, we will call f (x) the entropy density function. Note that in Eq. (2.1),
the integration bound is from 0 to 2Np. This is true for both U(1) and SU(2) lattice
with an even number of sites in each direction. The first bound is quite obvious
since it is the case when all links are taken to be the identity 1 of the group. The
second bound is not so trivial. Since both U(1) and SU(2) contain the Z2 group
as a subgroup and therefore have the element −1. When L is even, it is possible
to construct a path A (not necessarily connected) which touches each plaquette
once and only once [60]. Changing U→ −U on each link variable along this path
Awhile leaving rest of the links unchanged will result in TrUp → −TrUp for each
plaquette, which makes S to be 2Np. This property also implies a symmetry of
Z(β). For instance, in the partition function (1.14), by switching the sign of β, we
have
Z(−β) = e2Npβ∏
l
( ∫dUl
)exp
[− β
∑p
(1 + (1/n)Re TrUp)]. (2.5)
By changing variable Ul → Ul ∗ (−1) at each link along the path A, the integral
is not affected. But this will result in TrUp → −TrUp which makes the expression
after the product sign in Eq. (2.5) identical to Z(β), i.e.,
Z(−β) = e2Npβ Z(β). (2.6)
Similar discussion applies to the case of U(1) gauge fields and in principle any
gauge group which has the Z2 group as a subgroup. This symmetry has an
interesting relation with the Dyson instability [8] (see also [67]). As a consequence,
there is an obvious symmetry in n(S), say,
n(2Np − S) = n(S), (2.7)
or, by our definition of f (x),
f (x) = f (2 − x) . (2.8)
To analyze the property of f (x), we should take a look at the simplest example,
15
the SU(2) fields on a single plaquette.
SU(2) on One Plaquette
A simple but inspiring example is from the SU(2) gauge fields on a single
plaquette, the partition of which, by suitable choice of gauge, can be reduced to a
integration over just one link [60]. We can easily write out their density of states
by
n1pl.(S) =2π
√S(2 − S) . (2.9)
The behavior of the partition function at large β is related to the property of n(x)
near x = 0. In this one-plaquette example, the function n(x) ∝√
x where x is
small. The entropy density function f (x) has a logarithmic singularity at x = 0.
The same thing happens on the other end x = 2. By expanding√
2 − x in n(x) and
do the integration term by term, one can expand the partition function Z(β) in
terms of 1/β explicitly [58]. The large order behavior of the expansion is, of course,
determined by the large order behavior of the expansion in n(x) and is dictated by
the branch cut at x = 2. However, if the integration bounds are pushed to infinity
which is usually taken for mathematical simplicity, the expansion over 1/β will
suffer a zero radius of convergence [25, 60]. If we stick with the finite bounds
of integration [0, 2], we end up with a theory with finite radius of convergence
[58], but the trade-off is that the coefficients need to be expressed in terms of the
incomplete gamma functions. From the perturbative point of view, it is more
economical to expand n(x) instead of the underlying partition function.
The region in 1 < x < 2 is hardly explored in the physical world simulation
because any average plaquette ⟨x⟩ in this region corresponds to a negative β. The
simulations near β = 0 may generate certain number of configurations near x ≥ 1
region. But the bound x = 2 is beyond physical reach in reality. However, due
to the symmetry, this region can be investigated by taking β → −∞ [60], which
16
falls in the common agreement that the large order behavior of the weak coupling
series can be approached in terms of the behavior at small and negative coupling.
2.2 Saddle Point Approximation
With the notation of the average action x, we can rewrite the partition
function into
Z(β) =∫ 2
0dx exp
[Np( f (x) − βx)
]. (2.10)
The total number of plaquettes isNp = 6×L4, which is typically large and justifies
the saddle point approximation. By definition, the saddle point x0 is given by
f ′(x0) = β. (2.11)
x0 depends on the value of β and can be complex. If the saddle point is lying
on the real axis, then the approximation is straightforward. One can expand the
function f (x)−βx at x0 ( assuming there is only one saddle point in the integrating
path) by
f (x) − βx = f (x0) − βx0 + f ′′(x0)(x − x0)2/2 + ....
We ignore the higher orders and make it into a Gaussian distribution which works
well forNp →∞. We then have
Z(β) ≈ eNp[ f (x0)−βx0]
√2π
−Np f ′′(x0), (2.12)
where f ′′(x0) < 0. The same formula holds for the case when x0 is a complex
number if Re f ′′(x0) < 0. Let us discuss the case that only a saddle point and no
pole is presented in the region close to the real axis, to avoid complicating the
discussion. By Cauchy’s theorem, we can deform the integration path through
the saddle point along the so called path of steepest descent. It is a path along
which Im[ f ′′(x0)(x− x0)2/2] keeps to be a constant and along which Re ( f (x)− βx)
reaches maximum at x0. Let x − x0 = teiθ and f ′′(x0) = | f ′′(x0)|eiθ0 , then the path is
17
determined by
2θ + θ0 = nπ, n = 0, 1, 2, ... (2.13)
In practice, instead of finding the curve of the steepest descent, we take a linear
approximation, by integrating only along the straight line that is tangent to the
path of the steepest descent at x0. The real part is then approximated by a Gaussian
function, and then similarly we have
Z(β) ≈ eNp[ f (x0)−βx0]e−i(θ0−π)/2
√2π
−Np| f ′′(x0)| , (2.14)
but that is just Eq. (2.12).
In many occasions we want to work with the free energy density
F (β) ≡ − 1V
ln Z(β) (2.15)
where V is the total number of lattice sites V = L4. The average plaquette is
defined by
P(β) ≡ ⟨x⟩ = − 1Np
∂ ln Z(β)∂β
=∂(F /6)∂β
. (2.16)
With the saddle point approximation, the free energy density can be written
as
F /6 ≈ βx0 − f (x0) +1
2Npln
[− f ′′(x0)
]+ const.. (2.17)
Note that f ′′(x0) = ∂β/∂x0. In Eq(2.16), we replace the derivative over β
using the relation∂∂β=
1f ”(x0)
∂∂x0, (2.18)
then the average plaquette can be approximated by
⟨x⟩ = P(β) ≈ x0 +1
2Np
f ′′′(x0)f ”(x0)2 . (2.19)
The second central moment or fluctuation can also be calculated in a similar
18
manner,
⟨∆x2⟩ = − 1Np
∂2(F /6)∂β2 (2.20)
≈ − 1Np f ”(x0)
[1 − 1
2Np
f (4) f ” − 2 f ′′′2
f ”(x0)3
],
where ∆x = x − ⟨x⟩. In general, for n > 2
⟨∆xn⟩c ∝[
f ”(x0)2]3−2nN1−n
p , (2.21)
where ⟨∆xn⟩c are the central moments.
2.3 Series Expansions of n(S)
When it comes to a theory with interactions, the perturbative expansions
with respect to the interacting strength often play an important role. The most
significant discovery about the strong interaction is that quarks which are in-
teracting inside a hadron or meson are asymptotically free and the coupling is
running. The perturbative expansions in weak and strong coupling limits are
both of great interest[76, 77, 4].
In the pure SU(2) and SU(3) gauge models where no quarks are present,
there is no numerical evidence of a phase transition between the weak and strong
coupling regime, which makes us to think that there is a way to match the strong
and the weak coupling expansions. However there is a difficulty. Fig. (2.1)
[22] shows a plot of the average plaquette P as a function of β = 4/g2 which is
computed through three different methods: the weak and strong coupling series
expansions as well as the numerical simulation results. Neither the strong nor
the weak expansions works in the region around β ∼ 2, constrained by their finite
radius of convergence. We know that a finite region of convergence is defined by
a singularity on the boundary. So it should be the singularities in the complex
β-plane that prevent both of the expansions to go further and give a overlapping
19
coverage. [51]
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5 1 1.5 2 2.5 3 3.5
P
β
Plaquette
weak order 3weak order 4weak order 5
strong order 3strong order 5strong order 7
numerical
Figure 2.1: Weak and strong coupling expansions of the average plaquette P forSU(2) at various orders in the weak and strong coupling expansion compared tothe numerical values.
In this section, we will investigate these expansions in the context of the
density of states which can be compared with the simulation results closely.
Strong and weak expansions for both of the SU(2) and U(1) gauge fields are
discussed.
2.3.1 Strong Coupling Expansions
The strong coupling (small β) expansions of various lattice gauge theories
including U(1) and SU(2) have been intensively studied in [4] and a follow-up
correction [5]. They gave the results of the free energy density F up to the order
of β16. Note that by our convention, their βB = β/2 and their action is shifted by
20
a constant Np. As a result, we have an additional term β in the expression of the
free energy density, F /6. We shall write it out explicitly,
F /6 ≈ const. + β +∑m=1
a2mβ2m , (2.22)
where the coefficients a2m have been converted accordingly and summarized
in Table. (2.1). From the relation to the average plaquette P = ∂F /∂β, we
automatically have the expansion version of P
P(β) ≈ 1 +∑m=1
2ma2mβ2m−1 . (2.23)
When β is small, the most probable region is around x = 1. So strong
coupling approximation is equivalent to the expansion of f (x) about x = 1. Thus
it is convenient to use the variable y which is defined by y ≡ x − 1 and the
corresponding entropy density function g(y) ≡ f (1 + y). The symmetry in Eq.
(2.8) ensures that g(y) is an even function in −1 < y < 1 and therefore only even
orders of the expansions present, explicitly,
g(y) ≈∑m=0
g2my2m , (2.24)
We will base our discussion on the approximation for P(β) in Eq. (2.19)
which is justified in the large volume limit. We should start with the following
two relations
g′(y0) = β, (2.25)
P(β) ∼ x0 = 1 + y0(β), , (2.26)
where the first equation is the saddle point condition and the saddle point y0 is a
function of β implicitly. Let’s look at the lowest order. Eq. (2.24) and Eq. (2.25)
imply that 2g2y0 ≈ β. Then combined with Eq. (2.23) and relation Eq. (2.26), it
21
gives
y0 ≈ β/2g2 ≈ 2a2β , (2.27)
which means g2 = 1/(4a2). In general, Eq. (2.25) can be inversed perturbatively so
that the saddle point y0 can be taken as a series function of β, say y0(β). We then
can compare this expansion with the strong coupling expansion Eq. (2.23) order
by order through the relation Eq. (2.26). We match the coefficients of the two
sides and will end up with a set of relations between the expansion coefficients
g2m and a2m. We should write the leading several orders below,
g4 = −a4/(16a42),
g6 = (4a24 − a2a6)/(64a7
2),
g8 = (−24a34 + 12a2a4a6 − a2
2a8)/(256a102 ),
...
We can solve these equations starting from the lowest orders and hence will
be able to determine all the g2m by the values of a2m from the strong coupling
expansions which have been worked out in [5]. The results can be found in
Table.(2.1) [22] and Table.(2.2) for the SU(2) and U(1) model respectively.
Fig. (2.2) and Fig. (2.3) [22] show various orders of these strong coupling
expansions compared to the numerical simulation results which will be discussed
in Section. 2.4. Worse agreement between the expansion and numerical data in
the region away from x = 1 appears in the U(1) model, which implies a smaller
region of convergence. This is arising from the closer singularities to the real axis
in the U(1) model than the case of SU(2).
Similar to the one-plaquette model, in general, n(S) vanishes at S = 0 and
S = 2Np which correspond to the logarithmic divergence of f (x) at x = 0 and 2 (
or g(y) at y = ±1). A more general expression of f (x) which takes this logarithmic
divergence into account would be
h(y) ≡ g(y) − A(ln(1 − y2)) (2.28)
23
-2
-1.5
-1
-0.5
0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(x)
x
SU(2) strong coupling 64
order 2order 4order 6order 8order 10order 12order 14order 16
numerical
Figure 2.2: Numerical value of f (x) compared to the strong coupling expansionat successive orders for the SU(2) lattice gauge fields.
-1
-0.8
-0.6
-0.4
-0.2
0
0.4 0.6 0.8 1
f(x)
x
U(1) strong coupling 64
order 2order 4order 6order 8order 10order 12order 14order 16
numerical
Figure 2.3: Numerical value of f (x) compared to the strong coupling expansionat successive orders for the U(1) lattice gauge fields.
where A is associated with the coefficient of the logarithmic term in the weak
coupling expansion. We will derive the value of A in Section. 2.3.2 and just cite
24
the result here. In the V → ∞ limit, AU(1) = 1/4 for U(1) and ASU(2) = 3/4 for
SU(2). The expansion of h(y) is
h(y) ≃∑m=0
h2my2m . (2.29)
We can easily write down the coefficients h2m which are merely the g2m with
the expansion coefficients of the logarithm term subtracted. The values of h2m
are also listed in Table.(2.1) and Table.(2.2). In the case of SU(2), the coefficients
g2m and h2m are compared on a logarithmic scale in Fig. (2.4) [22]. The two sets
of coefficients merge rapidly to the same order of magnitude, indicating that the
singularities are hardly affected by these logarithmic poles.
-2
-1
0
1
2
3
4
5
6
7
0 2 4 6 8 10 12 14 16 18
Ln|coeffs.|
2m
strong coupling expansion
ln|g2m|: ln|h2m|:
Figure 2.4: The expansion coefficients h2m and g2m in the log scale are plottedversus the order 2m.
The goodness of the strong coupling expansion can be seen from the error
diagrams of different orders compared with the numerical results. First let us
25
look at the average plaquette P(β). The error diagrams of P for the SU(2) and
U(1) models are shown in Fig. (2.5) and Fig. (2.6) which display the logarithm of
the difference between perturbative P of successive orders and those calculated
directly from data for variousβ’s. The similar error diagrams can also be examined
from how the expansion of f (x) departs itself from the numerical data. The data
here means the numerical construction of the density function f (x) from Monte
Carlo simulations using either simple patching method or the multi-histogram
reweighting method ( which will be covered at length in Section 2.4.2). To remove
the dependence of the overall constant, we normalize the numerical f (x) at x = 1
by setting f (1) = 0. The successive orders of expansion of f (x) subtracting from
the numerical f (x) are plotted in the logarithmic scale in Fig. (2.7) and Fig. (2.8)
for SU(2) and U(1) respectively.
-15
-10
-5
0
5
10
15
0 0.5 1 1.5 2 2.5 3 3.5 4
Log|
∆P|
β
SU(2) 64 Strong Coupling
order 2order 4etc...
order 14order 16
num. error
Figure 2.5: Logarithm of the absolute difference between the numerical dataand the strong coupling expansion of P at successive orders for the SU(2) model(error diagram). The numerical data are taken from the volume 64. The error onnumerical P are obtained from 50 bootstraps.
The properties on the convergence of the expansion of P and f can be
26
-20
-15
-10
-5
0
5
10
0 0.5 1 1.5 2
Log|
∆P|
β
U(1) 44 Strong Coupling
order 2order 4etc...
order 14order 16
num. error
Figure 2.6: The error diagram of P at successive orders for the U(1) Model. Thenumerical data are from the volume 44. The error is estimated through 20 seedsof data.
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
0.4 0.6 0.8 1
Log|
∆f|
x
SU(2) 64 strong coupling
order 2order 4etc...
order 14order 16
num. error
Figure 2.7: The error diagram of f (x) at successive orders for the SU(2) Model.The numerical data are from the volume 64. The error on numerical f are obtainedfrom 50 bootstraps.
27
-20
-15
-10
-5
0
5
0.4 0.6 0.8 1
Log|
∆f|
x
U(1) 84 strong coupling
order 2order 4order 6order 8
order 10order 12
num. error
Figure 2.8: The error diagram of f (x) at successive orders for the U(1) Model. Thenumerical data are from the volume 84. The error is estimated through 20 seedsof data.
analyzed from these error diagrams. From Fig. (2.5) [22], we see that the larger
orders get worse error beyond β ∼ 2 and improve in the region β < 2.0. The
curves make a cross between β = 1.5 and 2. This is a sign of finite radius of
convergence [59]. Similarly, the larger order errors for f cross at some x between
0.4 and 0.6 which are close to the values of P at which the error curves cross. So
both of the graphs show the evidence of finite radius of convergence and suggest
there are singularities near the crossing region of the error diagrams. A similar
discussion applies to the U(1)’s error diagrams (Fig. (2.6) and Fig. (2.8) ) around
β ∼ 1 and P ∼ 0.5.
2.3.2 Weak Coupling Expansions
Now let’s continue the discussion to the weak coupling (large β) expansion
of f (x). For the sake of convenience, we switch the notation back to x, f (x). Again,
the discussion is based on the saddle point approximation which should be valid
28
in the infinite volume limit. We shall start with the expansion of the average
plaquette P using the form
P(β) ≃∑m=1
bmβ−m . (2.30)
As we have seen, f (x) has two logarithmic singularities at x = 0 and 2 where the
weak coupling is expanded about. Thus for f (x) around x = 0 we have the ansatz
f (x) ≃ A ln(x) +∑m=0
fmxm . (2.31)
Eq. (2.25) and Eq. (2.26) are still valid here in the infinite volume limit. For the
lowest order (zeroth order) where we only keep the logarithmic term, the saddle
point condition Eq. (2.26) implies that x0 = A/β. On the other hand, Eq. (2.25)
gives P(β) ≈ x0. As a direct consequence, we have A = b1. Disregarding the
volume correction, the perturbation can be performed order by order without
much difficulty. Thus all the coefficients fm can establish relations to the known
coefficients bm. The lowest several orders of equations are like
f1 = b2/b1, (2.32)
f2 = (b3b1 − b22)/(2b3
1),
f3 = (2b32 − 3b1b2b3 + b2
1b4)/(3b51).
In the infinite volume limit, these coefficients can be worked out easily if we have
the weak coupling expansion coefficients bm.
However we need to take into account that at finite volume, the density
function f (x) receives sizable corrections from the volume in the weak coupling
region (small x). Fig. (2.9) [22] shows such a fact. The saddle point approximation
of P should also be corrected with the 1/V effects. The relation Eq. (2.26) should
be replaced by the one with the first-order volume correction, explicitly
P ≈ x0 +1
12Vf ′′′(x0)f ′′(x0)2 , (2.33)
where V = Np/6. To compare the expansion results with the data which are
29
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.2 0.4 0.6 0.8 1
∆ln n(S)/Np
S/Np
SU(2) Diff 44 and 64
Figure 2.9: The difference of f (x) for two different volumes 44 and 64 is plottedas a function of x = S/Np. We see that in the weak coupling region (small x), thevolume effect is apparent and cannot be ignored in the expansion.
calculated at finite volumes, we have to take the volume correction into account.
Assuming the weak coupling expansion coefficients with volume correc-
tions have been worked out as
bm = b̄m + b′m/V + b′′m/V2 + ...
(2.34)
We should examine how this will make a difference in the coefficients A and
fm. At the lowest order, we have f (x) = A ln x and by Eq. (2.33), we have
P ≈ x0(1+ 1/(6VA)). Additionally, with Eq. (2.25), we have x0 = A/β. Combining
with the first order of the expansion P ≈ bm/β, we can have the expression for A
A = b̄1 + (b′1 −16
)/V. (2.35)
For higher orders, we can follow the same virtue to work out all the relations
between fm and bm order by order. The lowest several order will now be corrected
30
as
b1 = A +1
6V, (2.36)
b2 = (A +1
6V) f1 ,
b3 = (A +1
6V)( f 2
1 + 6A f2) − 4A2 f2 ,
...
U(1)
In [38], the average plaquette P(1/β) 1, was expanded up to the fourth order
of 1/β where the coefficients of the first and the second orders are exact due to
the simple loop diagrams. The third and fourth order involve two diagrams
for which they applied numerical calculations. The first order coefficient reads
b1 = (1/4) − 1/(4V). By Eq. (2.35), we can easily see that
A ≈ 1/4 − (5/12)(1/V) . (2.37)
The second order with correction is b2 = (1/32) − (1/16)(1/V), so with Eq. (2.37),
we have
f1 ≈ 1/8 − (1/8)(1/V) . (2.38)
For order three and four, numerical results exist for the volume 44 and 64. The
results are summarized in Table (2.3). The coefficients of the infinite volume limit
as well as that of the two lowest volumes 44 and 64 are given. The upper half is
the list of the expansion coefficients of the average plaquette P with respect to 1/β
[38]. The lower half is the corresponding expansion coefficients of f (x). Volume
effects from 44 and 64 are also given.
1They used a notation which differs from ours by a constant 1 due the to different conventionin the action.
31
V = ∞ V = 64 V = 44
b114
12955184
2551024
b2132
2171747375208971104256
650252097152
b3 0.01311 0.01309 0.01296
b4 0.00752 0.00749 0.00739
A 14
388315552
7633072
f118
129510368
2552048
f2 0.07363 0.07359 0.07314
f3 0.07638 0.07605 0.07515
Table 2.3: The weak coupling expansion for U(1) gauge fields.
To compare with the numerical results, there is a overall constant f0 in the
expansion which cannot be determined using the analysis discussed above, and
hence need to be fitted from the data. We take the numerical f (x) at some small x
to fix f0. The curves of successive orders are plotted against the numerical data in
Fig. (2.10). The error diagrams of f and P are given in Fig. (2.11) and Fig. (2.12).
SU(2)
An expression of the lowest order of the weak coupling expansion b1 on the
SU(2) lattice can be found in [37, 15]. For the case Nc = 2 and D = 4, we have
explicitly
b1 = (1/4)(1 − 1/(3V)) . (2.39)
Similarly, using Eq. (2.35) we have
A ≈ 3/4 − (5/12)(1/V) . (2.40)
32
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6
f(x)
x
U(1) weak coupling 84
order 0order 1order 2order 3
numerical
Figure 2.10: The successive orders of f (x) expansion from U(1) weak couplingexpansion are plotted in contrast with the numerical f (x) from multi-canonicalsimulations.
-14
-12
-10
-8
-6
-4
-2
0
0 0.2 0.4 0.6 0.8 1
ln|Error|
x
U(1) f(x): weak, 4x4x4x4
order 0order 1order 2order 3
num. error
Figure 2.11: The error diagram of f (x) at successive orders of weak couplingexpansion for the U(1) model.
The first order volume correction (−1/(3V)) in the expression of b1 comes from
the absence of zero mode (−1/V) in a sum given in [37] plus the contribution of
33
-14
-12
-10
-8
-6
-4
-2
0
0 1 2 3 4 5 6 7 8 9
ln|Error|
β
U(1) Plaquette 4x4x4x4
order 1order 2order 3order 4
num. error
Figure 2.12: The error diagram of P at successive orders of weak coupling expan-sion for the U(1) model.
the zero mode with periodic boundary conditions (+2/(3V)) calculated in [15].
No analytical expression is available for orders greater than 1. Numerical values
for the higher orders are available for some specific volume in the literature. An
estimate of b2 can be found in [37] while b3 was given in [1]. Unfortunately, in
these references, the volume 64 is not yet calculated. So rough extrapolations
from the existing data is needed. It shows that for V = 64 uncertainties can be
controlled to be as small as 0.0002 for b2 and 0.0008 for b3. For β ≥ 3, these effects
are close to the numerical errors of P for large β. In the following, we will use the
approximate values b2 = 0.1511 and b3 = 0.1427 for the particular volume V = 64.
We are not aware of any calculation of bm for m ≥ 4 for SU(2). The results are
summarized in Table (2.4). We did a numerical extrapolation for the high order
coefficients and compare the successive orders in Fig. (2.13).
34
m bm fm
1 0.7498 0.2015
2 0.1511 0.0999
3 0.1427 0.0796
4 0.1747 0.0791
Table 2.4: Weak coupling coefficients de-fined in Sec. 2.3.2 for the SU(2) model.The choice of b1 corresponds to V = 64 .
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
f(x)
x
SU(2) 64 weak coupling
order 1order 2
etc.order 15
numerical
Figure 2.13: Numerical value of f (x) compared to the weak coupling expansionat successive orders. The coefficients are calculated from the volume 64.
We can also compare the error diagrams for successive orders in the weak
coupling expansion of P with the numerical values in the case V = 64. The results
are shown in Fig. (2.15) [22]. In the region where the curves are smooth, the error
decreases with the order and appears to accumulate. This is very similar to the
35
-12
-10
-8
-6
-4
-2
0
0 0.2 0.4 0.6 0.8
ln|
∆f|
x
SU(2) 64 weak coupling
order 1order 2
etc.order 15
num. error
Figure 2.14: The error diagram of f at successive orders for the SU(2) model.
case of SU(3) [2]. However, the results show dependence on the estimates for
m ≥ 4 in the SU(2) model. We should note that for large β, the noise in the error
is similar to the numerical error on P. However, this would not be the case if we
had not included the contribution of the zero mode to b1 as shown in the second
part of Fig. 2.15 which is clearly indicating a persisting discrepancy.
We now look at the error diagram of the f (x) compared to the numerical
values at the volume V = 64. The results are shown in Fig. 2.14. In these graphs
we have taken f0 = −0.14663 which maximizes the length of the accumulation
line on the left of Fig. (2.14).
2.4 Numerical Calculation of n(S)
In this section, we will describe the numerical methods that we used to
construct the density of states, which is essential to our discussion of the Fisher’s
zeros.
We used Monte Carlo simulation to approximate the gauge integration
36
-14
-12
-10
-8
-6
-4
-2
0
2 3 4 5 6 7 8
ln|Error|
β
SU(2) Plaquette 6x6x6x6
order 1order 2
etc.order 15
num. error
-14
-12
-10
-8
-6
-4
-2
0
2 3 4 5 6 7 8
ln|
∆P|
β
SU(2) 64 no zero mode
order 1order 2
etc.order 15
num. error
Figure 2.15: The error diagram of P at successive orders for the SU(2)model(above). The error for the numerical P is calculated using 50 bootstraps ofthe data. The graph below is the case without the contribution of the zero modein b1.
37
over the links. We generated configurations at various values of β for multiple
volumes. The program for the SU(2) gauge fields was written by our collaborator
Alex Velytsky. The data were generated, with the collaborating effort by Alan
Denbleyker, for four different lattices: L = 4, 6, 8, 10 in four dimensions. We
then used the Ferrenberg-Swendsen’s multi-histogram reweighting method to
reconstruct the density of states through a spectrum of data. For the U(1) gauge
field, the numerical density of states were provided by our collaborator Alexei
Bazavov using multi-canonical method [11] in three different lattice sizes: L =
4, 6, 8. We used relatively small volumes because considering the precision we
desired, we need larger data samples which is demanding for our relatively small
cluster.
In this section, the numerical setup will be briefly discussed. It is followed
by a detailed description on the implementation of the Ferrenberg-Swendsen’s
method for numerical reconstruction of the density of states for the SU(2) gauge
field. The results are compared with the series expansion that we discussed in
the previous section. The volume dependence of f (x) for the U(1) model will also
be studied.
2.4.1 The Computational Setup
For the SU(2) gauge field, we used Monte Carlo simulation method to
generate the configurations which are the important samples of the full set of
gauge link configurations. We have generated the data for the volumes V =
44, 64, 84 and 104 with various seeds of the random numbers. The spectrum of
β range from 0.001 to 100 and we have used multiple spacings. Small steps
∆β = 0.0015, 0.02, 0.025 is taken for small β, while for large β such as β > 10, big
steps ∆β > 1 are taken to save the computation hours. Each data set consists of
100,000 configurations with the first 200 equilibrium steps removed. Table (2.5)
38
shows the numerical data lineup for the SU(2) gauge field.
L Num. of β’s Unit Configs seeds
4 284 100,000 20
6 447 100,000 15
8 937 100,000 2
10 980 100,000 1
Table 2.5: SU(2) data setup.
We should look at the autocorrelation which is important for the discussion
on the histogram reweighting method. The correlation is defined as
Γ(t) = ⟨(xi − ⟨x⟩)(xi+t − ⟨x⟩)⟩. (2.41)
Fig. (2.16) shows the typical correlation near the critical region of β. The
absolute value of the slope will give the autocorrelation length which is equal to
the integrated autocorrelation if Γ(t) satisfies perfectly an exponential law. The
integrated autocorrelation can be computed by
1 + 2τ =∞∑
t=−∞
Γ(t)Γ(0). (2.42)
Fig. (2.17) shows how the integrated autocorrelation is distributed with
respect to β for the four volumes V = 44, 64, 84, 104. We can see that the 1 + 2τ
reaches a peak at volume 64. As the volume increases, the distribution becomes
flat and hence there is no sign of divergent tendency of the autocorrelation length
which is an important signature for the critical behavior. We should expect no
phase transition for the SU(2) model which has been widely accepted. However
39
çççççççççççççççççççççççççççççççççççççç
ççççççççççççç
ççççççç
ç
çç
ç
çççç
ç
ççç
çç
ç
Slope = 0.091
0 10 20 30 40 50 60 70
-16
-14
-12
-10
-8
tlnHÈG
tÈL
Figure 2.16: The log-log plot of the autocorrelation vs t at β = 2.20 for a 44 SU(2)lattice.
it is still interesting to know how the distribution of the zeros ( as we have
known there must be a singularity near the transition region of the strong and
weak coupling domains ) will characterize such a system. From the graph, we
can see that the distribution peaks at around β = 2.20, 2.34, 2.40 for the volume
V = 44, 64, 84 (there is no obvious peak for the volume V = 104). This gives us a
hint on where the singularity might be located along the real axis.
For the U(1) gauge field, Alexei Bazavov has obtained the density of states
using Multi-Canonical method [11] for three different volumes: V = 44, 64, 84.
Each volume is simulated with 20 seeds of random numbers to provide a partial
set of statistical variety. Each seed will be weighted through the number of
tunnelings between the two attractors (the double-peak nature). The sampling
importance is evaluated by the tunneling numbers. The full set of tunneling
numbers as well as the fluctuations estimated from the seeds can be found in [9].
40
0
4
8
12
16
20
1.8 2 2.2 2.4 2.6 2.8
1+2
τ c
β
Correlation Time : SU(2)
44
64
84
104
Figure 2.17: The integrated autocorrelation time is plotted verse β for four dif-ferent volume V = 44, 64, 84, 104. The peak of the distribution is found at aroundβ = 2.20(for 44), 2.34 (for 64) and 2.40 (for 84). The peak is not obvious for thevolume 104.
V Num. of bins seeds
44 1000 20
64 1000 20
84 1000 20
Table 2.6: U(1) Density of States.
2.4.2 Histogram Reweighting
Monte Carlo data are usually collected by varying parameters like tempera-
ture or coupling strength in discrete steps. The Metropolis algorithm often results
in configurations distributed narrowly around the average value. Hence a single
41
run of such simulations will only cover a small piece of the configuration space
(or phase space) and gives zero statistics for the rest of the space. Since the parti-
tion function always plays a role as the normalization constant, the conventional
method is not capable to address the connection between the runs and therefore
is not able to measure statistical quantities such as free energy or entropy.
However, when these Monte Carlo runs have overlaps in the configuration
space, it is possible to assign weights to these runs according to their statistical
significances at each position in the configuration space and make connections
between the neighboring datasets [10, 32, 31, 55]. The approach is often called the
Ferrenberg-Swendsen’s method. There is also an alternative algorithm developed
by Wang and Landau [73]. However we will focus on Ferrenberg-Swendsen’s
method and provide a practical approach to obtain the initial values which are
important for the convergence of the iterations.
Figure 2.18: The overlapping of neighboring data sets.
Consider the simplest case where there are only two distributions involved.
Let S denote a point in the phase space, for example, the total action for a particular
configuration, and the partition function is Z(β) =∑
S exp[−βE(S)]. Generically we
can write the effective partition combining the two runs (distributions associated
42
with β1 and β2) into ∑S
W(S) exp(−β1E(S) − β2E(S)),
where W(S) is the weight factor at S, connecting both of the distributions.
If there is no overlapping be between the distributions, like the situation
shown in the left graph in Fig. (2.18), W(S) can be taken to be exp[β2E(S)] for all S
on the left to the middle point and exp[β1E(S)] on the right, then the description
remains consistent. However it is not so trivial when the two distributions have
an overlapping region. The weight W(S) has to satisfy certain form to make the
two distributions logically compatible to each other and not violating the fact that
they are both satisfying the Boltzmann assumption Z(β) =∑
S exp[−βE(S)].
The statistical quantity that is connecting the two distributions is the free
energy A ≡ − ln Z(β), and the difference of the free energy can be written as
∆A = A2 − A1 = lnZ(β1)Z(β2)
= lnZ(β1)Z(β2)
∑S We−β1E(S)−β2E(S)∑S We−β1E(S)−β2E(S)
,
= ln⟨We−β1E(S)⟩⟨We−β2E(S)⟩ .
Suppose that there are n1 (n2) configurations in the distribution associated with
β1 (β2). We should consider the continuous version by taking∑
S exp[−βE(S)] →∫dqn(q) exp[−U(q)] where U(q) = βq and we absorb n(q) in the weight W(q).
Requiring the fluctuation of this free energy difference to be stationary with the
weight function, i.e., δ⟨δ2∆A⟩/δW = 0, will result in the fact that W(q) has to
satisfy [10] ( see also Appendix)
W(q) =const
Z0n0
e−U0 + Z1n1
e−U1. (2.43)
Notice that to compute this weight function, one will need the implicit
information about the partition function Z(β) itself. So the equation is rather a
consistent constraint than a calculable implementation.
43
This formula can be generalized to the case of a number of distributions
[31, 55]. Let’s brief the setup of our actual calculations. We can carry out a series
of Monte Carlo simulations at β = β1, β2, ...βR. There are Nα configurations in
the αth set, with an integrated autocorrelation τα. Their configurations are then
binned into multiple histograms, Hβα(S) ≡ number of configurations in bin S. In
each histogram, if we divide it into a large number of sub bins, then each bin will
either contain zero or one configuration while the total expectation is a constant.
Thus the number of configurations in each histogram should fall in a Poisson
distribution and the errors can be estimated by,
⟨δ2Hα(S)⟩ = gα⟨Hα(S)⟩, (2.44)
where the gα = 1 + 2τα.
The partition function, simplified as the integration over S, is
Z(β) =∫
dS n(S)e−βS, (2.45)
where the density of states can be written as
n(S) = e f (S) (2.46)
Writing it into the form with probability density, one will have
1 =∫
P(S)dS =∫
n(S)e−βα,S
Z(βα)dS (2.47)
So the probability in a particular bin can be estimated by the histogram of a
dataset βα at S,
P(S)∆S =⟨Hα(S)⟩
Nα, (2.48)
where ∆S is the bin width of bin S. Comparing the above two equations, we
easily have
n(S) =⟨Hα(S)⟩Nα∆S
eβαS−Fα , (2.49)
where we have defined the average free energy Fα = − ln Z(β) .
44
The right hand side of last equation should be independent of choice of any
particular dataset βα. However this property is only valid in principle. For S far
away from the average value ⟨S⟩α of a particular dataset, the exponential term is
almost infinitely large. To keep n(S) finite, we need an infinitely large number
Nα of configurations. With one dataset β, it is impossible to have histograms that
cover all the S due to the limited data size. Actually one simulation at βwill give
empty histograms for most of the S, leaving only those near ⟨S⟩β non-vanishing.
To find the density of state near S we need the Monte Carlo run at a β such
that ⟨S⟩β is close to S. Multiple Monte Carlo runs at nearby β’s will give us piece-
wise information about n(S). As the case of two datasets, in the region where the
datasets overlap, we can combine the contributions from each of these datasets
by assigning a proper weight (function of S) in the following manner,
n(S) =R∑α=1
Wα(S) nα(S),
where nα(S) is calculated in Eq. (2.49) using dataset βα and by normalization,∑Wα(S) = 1.
By minimizing the error in n(S), it can be found that the weight should
and we can write out the density of state explicitly (we assume each dataset
consists of the same number of configurations)
n(S) =∑α(Hα(S)/gα)∑
α exp(Fα − βαS). (2.51)
Notice that it depends on the input of the free energy Fα which is not explicitly
available, but it can be calculated by the relation
exp(−Fα) =∑
S
n(S) ∆S e−βαS. (2.52)
45
So the determination of n(S) (or Fα) has to go through iterations. It is clear from
these two equations that the fluctuations in either of Fα or n(S) will increase after
it goes through the iteration cycle. So the initial values of the iterations need to
be chosen carefully otherwise it can easily result in divergence. Because of the
positive feedback feature, if the iteration converges, it converges slowly.
It is very important to start with a precise set of initial values. We found
that it is effective to use the average plaquettes to calculate the initial values of
the free energy Fα. Let’s fix the overall constant at β0 by Z(β0) = 1, or F0 = 0, then
we have the relation
F(β) =∫ β
β0
dF(β) =∫ β
β0
(−Z′(β)Z(β)
)dβ =∫ β
β0
⟨S⟩βdβ . (2.53)
where the average plaquettes P = ⟨S⟩β can be calculated from the data. By
integrating through the datasets from the lowest β, we have a spectrum of initial
values for the free energy. We found that in most occasions, this approach results
in precise values and leads to convergence. We shall discuss this method in detail
below.
Detailed Implementation For the SU(2) Model
First, each dataset is binned into certain number of histograms. It was
argued that there is an optimized choice on the number of histograms ( or bin
width), however we didn’t find it is really decisive. Typically, the width of the
distribution varies significantly with β. We make sure that the bin width is not
too large such that distributions of large β’s are not reduced to only several bins.
Second, we need to prepare the initial values for the iterations. The differ-
ence between successive β’s are usually optimized to maximize the overlapping
between the neighboring datasets, and hence may not be selected to be equally
46
spaced. To get the initial free energy density F (βα) through numerical integra-
tion, an algorithm of integration with non-equal steps should be used to improve
over the simple trapezoidal integration. we will use the generalized Simpson’s
Rule [66]. We set F (β0) = 0, then the higher F (βα) can the obtained by∫ βα
β0
Pdβ ≈ 12
[ f1 + fα +α−2∑i=0
f (βi, βi+1, βi+2)], (2.54)
where
f1 =12
(P1 + P0)(β1 − β0),
fα =12
(Pα + Pα−1)(βα − βα−1),
f (βi, βi+1, βi+2) = (βi+2 − βi)[Pi +
βi+2 − βi
βi+1 − βi
Pi+1 − Pi
2
]+
16
(2β2i+2 − βiβi+2 − β2
i + 3βiβi+1 − 3βi+1βi+2)[Pi+2 − Pi+1
βi+2 − βi+1− Pi+1 − Pi
βi+1 − βi
].
(2.55)
We found that the integration is effective and the iteration converges much faster.
Thirdly, to do the iteration, using the entropy density function f (x) =
(1/Np) ln n(x) is convenient. To prevent numerical overflow, the summation in
both of the equations involved is performed using the logarithmic formalism,
i.e., factoring out the major term and treating the rest as small contributors to the
log function. In order to determine the convergence of the iteration, we monitor
the average Chi-Square of the average plaquettes with 90% of the datasets,
χ2(i) =1
ncut
ncut∑α=0
(⟨x⟩βα(i) − ⟨x⟩βα)2
σ2α
, (2.56)
where ⟨x⟩βα and σα are the average plaquettes and its fluctuations from the data,
while ⟨x⟩βα(i) is calculated using the density of states at iteration i. We choose a
ncut to ignore the largest 10% of β’s because typically we use large β steps there
and the overlapping of the datasets is low, which makes the f (x) fluctuating but
causing no notable effect to the small-β region.
47
1e-007
1e-006
1e-005
0.0001
0.001
0.01
0.1
100 1000 10000
∆χ2
Iterations
Obtaining Initials: Itegration Methods Compare
TrapezoidalGen. Simpson
Figure 2.19: A comparison between the trapezoidal and the generalized Simp-son’s rules that are used to obtained the initial values of F (βα). The data is fromthe 44 lattice and skipping every 5 datasets. ∆χ2 is the χ-square difference be-tween successive iterations ( defined in the text ), the slope of which indicatesthe convergent rate of the iteration. It is obvious that when the datasets aresparse, the generalized Simpson’s rule gives better initial values and improvesthe convergence significantly.
Finally and most importantly, the iteration should be determined to be con-
vergent. The density of states over the histograms forms a non-linear dynamical
system of huge number of degrees of freedom. The convergence is hard to de-
termine and the criterion for the convergence is not yet available. However what
can be seen is that this system is not uniformly stable, i.e., the convergence is
conditional, depending on the histogram bin width, initial values, β steps, and
the overlapping between the datasets. The convergence, if it exists, is typically
reached slowly (Fig. (2.22)), which is a characteristic of such a system. Although
not proved, the convergence can be judged empirically by the asymptotic behav-
ior of the χ2 ∼iterations curve. In practice, we monitor the difference between
the successive χ2’s and look for a plateau over the iterations. In general for a
48
large enough number of iterations, if the absolute difference of the successive χ2
is decreasing over the iterations, i.e., the convergent rate is decreasing, it usually
results in a convergence.
Comments:
• If convergent, the iterations usually do not depend on the histogram bin
width ( or bin number of x ). However the bin width should be monitored
and adjusted so that it optimizes the smoothness/noise of the histograms
while it is reasonable for the computational convenience.
-4e-005
-2e-005
0
2e-005
4e-005
6e-005
8e-005
0.0001
102 103 104
∆ χ2
Iteration Number
Successive Average χ2 Difference: SU(2) 44
2005001000
Figure 2.20: This plot is a comparison of reweighting with various x bin num-ber: 200(circles), 500(squares) and 1000(diamonds). They all result in the sameconvergence down to the noise level of the data.
• Although the initial values may affect the convergence, if two sets of initial
values for such a system both result in convergence respectively, then they
will approach to the same convergent values if they do not differ too much
Figure 2.21: Two slightly different sets of initials are fed to the reweighting of aSU(2) data on a 44 lattice and reach the same convergence. The plot is ∆ f (x) atiteration 10000 (middle two curves) and 200(outer two curves). Here ∆ f meansthe difference of f (x) to the stabilized value ( at iteration 15000).
• The most decisive factor that will affect the reweighting is the overlapping
between the datasets.
First, even a convergence is reached, the insufficient overlapping between
different β’s may result in incorrect values, since the reweighting is to reach
the compatibility between overlapping data sets based on their statistical
levels, a sparse data lineup doesn’t have enough constraints to decide the
density to the desired precision. See Fig. (2.23) for an example.
Second, insufficient overlapping often leads to uncontrollable divergences.
This usually becomes worse as the volume increases because the distribu-
tions of the data become more and more narrow and therefore result in
less overlapping. A detail example is shown in Fig. (2.24) and Fig. (2.25)
where fewer datasets make the iteration diverge, and the iteration is made
50
-22
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
0.4 0.6
ln |f(x)-f(x)con|
x
Iterations with diff. initials
5200...2400
optim 20optim 200
Figure 2.22: An example of the iteration process. All iterations are comparedwith the convergent values f (x)con. Two different initials are compared here. Forthe first one we started with initial values f (x) = 0, for all x. The convergence isslow (upper five curves). For the second we used the integration method to getthe initials. The method is efficient for the iterations (lower two curves).
convergent with more datasets.
To increase the overlapping, we can either add intermediate datasets or
increase the statistics of each dataset. However, either way will be expensive
and hence an optimized selection of the values of β’s is often designed.
51
-0.0003
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.2 0.4 0.6 0.8 1
∆f(x)
x
Reweight with Different Numbers of β: 44
n=142n=57n=29
Figure 2.23: Convergent iterations may not always lead to satisfactory results.Reweightings with different numbers of β’s are compared. The resulting f (x)’sare then subtracted from the result with 284 β’s (normalized at x = 1). The solid,dashed, double-dashed curves are using 142,57 and 29 β’s respectively. Althoughall lead to convergence, the reweighting with fewer datasets show much biggerundesired fluctuations.
52
0.6
0.65
0.7
0.75
0.8
101 102 103 104 105
Average
χ2
Iteration Number
Convergent Overlapping : SU(2) 64
246 βs449 βs
Figure 2.24: The convergence is sensitive to the number of datasets. The reweight-ing using the same data set while including different numbers of datasets atvolume 64. The plot is showing the chi2 varies with iterations. The upper onewhich is using 246 β’s shows obvious divergence at iteration around 6000, whilethe lower one with 449 datasets indicates an strong sign of convergence. Similardifficulty appears with higher volumes such as 84 and 104.
Figure 2.25: The overlapping plots of the previous example. The boxes in eachgraph show two-σ spreading (centered at the average) of each data point. Thevertical increments have no meaning but to illustrate the neighboring overlappingin a clearer manner. The left graph corresponds to the diverging reweighting andshowing in sufficient overlapping among the data.
53
The error analysis of the Ferrenberg-Swendsen’s method was studied by
the same authors in [30]. However in this paper, we will use a simple method
to estimate the uncertainty due to the reweight procedure. For the two small
volumes 44 and 64, we generate 20 seeds of data which are independent replica of
the same observables. Then we use the reweighting method to find the density
of states for each of these replica, and estimate the uncertainty (Fig. (2.26)).
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0 0.2 0.4 0.6 0.8 1
∆f(x)
x
Error Due to Different seeds: 44, 19 seeds
Figure 2.26: The errors of the density of states due to the different seeds.
54
2.4.3 Volume Dependence of the density states
With the density of states precisely determined at various volumes, we can
investigate its volume dependence. We discussed the weak coupling expansion
in Section 2.3.2, however, here we are only interested in the volume effect in those
expansion coefficients. We should follow the same notation there, by expanding
f (x) in the following way
fV(x) = A ln(x) + ( f0 +f0′
V) + ( f1 +
f1′
V)x + ( f2 +
f2′
V)x2, (2.57)
where we already know A = 3/4−(5/12)(1/V) for SU(2) and A = 1/4−(5/12)(1/V)
for U(1). If we have f (x) calculated at two different volumes, say V1 and V2, then
by taking their difference, we can get rid of the volume-independent part of the
expansion coefficients and be able to to extract first order volume effect by
∆ f∆(1/V)
=fV1(x) − fV2(x)1/V1 − 1/V2
(2.58)
= − 512
ln(x) + f0′ + f1′ x + f2′ x2 +O(max(1/V21, 1/V
22)).
We make use of three different volumes: VL = L4, where L = 4, 6, 8 for
the U(1) model. We calculated the following two differences, ∆ fV4,V8 , ∆ fV6,V8
and do the data collapse using Eq. (2.58). The curves of ∆ fV4,V8 and ∆ fV6,V8 ,
with the small overall constant removed, overlap nicely (Fig. (2.27)). Fitting
the curves gives close results: f ′0 = 0.555, f ′1 = −0.105, f ′2 = 1.122 for ∆ fV4,V8 and
f ′0 = 0.520, f ′1 = −0.096, f ′2 = 1.137 for ∆ fV6,V8 . The difference between the f ′0 ’s is
about 0.035 which corresponds to the height of the flat part in Fig. (2.28).
55
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.1 0.2 0.3
∆f/
∆(1/V)
x
Volume dependence of f(x): U(1)
L=4,8L=6,8
Figure 2.27: The collapse of the difference of f (x) of three different volumes,∆ fV4,V8 and∆ fV6,V8 with the overall constant subtracted (described in the text). Thedifferences are then divided by their corresponding inverse volume difference.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.1 0.2 0.3 0.4 0.5
x
The difference of ∆fV4,V8 and ∆fV6,V8
Figure 2.28: The difference of the differences ∆ fV4,V8 and ∆ fV6,V8 . The flat partcorresponds to the overall constant 0.035.
56
CHAPTER 3FISHER’S ZEROS
3.1 Single Point Reweighting Method
A Monte Carlo simulation at a fixed value of the parameter β gives a partial
description of the density of state ( or partition function) in the vicinity of β.
This ”vicinity” includes the real neighborhood in the parameter space as well
as the analytic continuation into the complex plane . This idea leads to local
approximations of Z(β). Each dataset gives a normalized statistical description.
Therefore, it is possible to analyze the Fisher’s Zeros if they are within the reach
of such approximations [2, 29]. Suppose that Monte Carlo data are collected at a
value β0 and for a close-by value β = β0 +∆β the partition function can be written
as [2, 23, 19]
Z(β0 + ∆β) = Z(β0)⟨exp(−∆βS)⟩β0 , (3.1)
where S is the total action. We should be aware that when admitting a complex
β, Z(β0 +∆β) is an oscillating function with its frequency scaled by the number of
plaquettes. So it is convenient to use the reduced form which is defined as [19]
z(∆β) =Z(β0 + ∆β)
Z(β0)exp(∆β⟨S⟩β0) = ⟨exp(−∆β(S − ⟨S⟩β0)]⟩β0 (3.2)
z(β) should share the same zeros as Z(β0 + ∆β).
We write explicitly ∆β = βR + iβI, then the exponential reduces to the real
part ZR = ⟨exp(−βR∆S) cos(βI∆S)⟩ ( here ∆S = S − ⟨S⟩β0) and the imaginary part
ZI = ⟨− exp(−βR∆S) sin(βI∆S)⟩ which are both oscillating functions. Note that
all these expectations are taken with the data collected at β0. We then scan a
region (for example, a square) near β0 in the β-plane vertically and monitor the
successive values of the real and imaginary functions ZR, ZI. If the value of the
function, say the real part, switches sign then it means it reaches the value of 0
57
in between. By following this methodology, we can produce the zero contours of
both ZR and ZI. The intersections of these are the zeros of the total expectation,
and hence are the complex zeros of the partition function Z(β). Fig. (3.1) shows
an example of these contours.
0.04
0.06
0.08
0.1
0.12
2.25 2.3 2.35 2.4
Im
β
Re β
Finding Zeros: Method of Contour Plot
ReZ(β)=0 ImZ(β)=0
Figure 3.1: The plot shows the contours of both real and imaginary part of thepartition function. The intersections of these curves are the zeros of the partitionfunction.
This method can be generalized to multiple-point reweighting by assigning
weights to the neighboring datasets and combining them together [2]. This
method is proved to be effective when the complex zeros are close to the real axis,
especially for the statistical models that have a phase transition in the infinite
volume limit. Later we will demonstrate that this method works well in the U(1)
case.
However, the method is limited if the zeros have large imaginary compo-
nents, to be specific, if the zeros lie outside the circle of confidence [2]. This circle
of confidence is determined by the statistical level of the data. For a model with
58
a Gaussian-like distribution, the radius of the circle can be estimated by using
the Gaussian approximation. For example, in the SU(2) model, the probability
density of a particular data is narrowly distributed along S and resembles almost
the shape of a Gaussian distribution. For a model with an exact Gaussian dis-
tribution, the fluctuation on the reduced partition function z(β) = Z(β0 + β)/Z(β0)
can be calculated by [2]
σ2(|z(β)|) = exp(β2
R
A) − |z(β)|2, (3.3)
where A corresponds to the quadratic coefficient in the Gaussian distribution
exp(−AS2). A model with an exact Gaussian distribution is free of zero. So we
should only trust the region such that the Gaussian z(β) is at least one sigma
distinctive away from zero, which means that we have 84% confidence to exclude
a fake zero due to the fluctuation. Specifically, σ(|z̄(β)|) = σ(z(β))/√
(N) ≤ |z(β)|
where N is the data size. This will define a circle [2], β2R + β
2I ≤ A ln(N + 1).
Generally for a non-Gaussian model, this circle of confidence is replaced by a
level of confidence which can be calculated by the criteria [19, 20]
σz(β)/|z′(β)| < d, (3.4)
where d is a fraction of the typical distance between the zero contours of the
real and imaginary parts. One should be cautious with the zeros obtained with
the contour plot method. Due to the fluctuation of the data, it will often produce
intersections of the real and imaginary curves, but only those well within the
confidence region are candidates of the actual zeros. So it is always be good to
verify them with other independent methods.
3.2 Quasi-Gaussian Distribution: A Toy Model
We know that for the SU(2) model its probability distributions of the action
are very close to Gaussian. For a partition function with its probability density
59
resembling exactly a Gaussian, there should be no zero. So the zeros must arise
from the deviation from the Gaussian distribution. We should look at a simple
toy model for which we can calculate the result precisely.
We use the Metropolis algorithm to generate random numbers obeying a
normal distribution as follows [20]
P(S) = (2π)−1/2 exp(−S2/2) . (3.5)
Figure 3.2: The probability distribution of total action in the Gaussian toy modeldescribed in the text.
We generated 1,600,000 values of S. The average of this set gives 0.00087.
The values were then binned into 100 histograms as is displayed in Fig. (3.2)
which shows a very small deviation from the Gaussian distribution. The data
is highly correlated. To remove the autocorrelation, we simply skip a couple of
times of the correlation length which is around 8 from the inverse of the slope of
Fig. (3.3).
With the skipped data which consists of 40, 000 independent configurations,
we can now explore its zero using the afore-mentioned single-point reweighting
method. The zero-level contours of the real and imaginary parts of< exp(−β∆S) >
60
Figure 3.3: Logarithm of the autocorrelation versus the time series distance forthe original set of 1,600,000 values.
are shown in Fig. (3.4) [20]. The circle of convergence (Reβ)2+(Imβ)2 = ln(40000) ≈
3.262 is also plotted. The zero closest to the real axis is around β ≈ 1.5 + i3.0,
but it is outside the radius of confidence defined above, which fits well in our
expectation that the Gaussian Model should be free of zeros. The solid lines
in Fig. (3.4) correspond to the hyperbolas: βRβI = 2nπA (imaginary part) and
βRβI = 2(n+ 1)πA (real part). The hyperbolas will never intersect with each other
and hence a partition function due to a Gaussian distribution should not possess
a zero in the complex plane. We can see that these contours coincide well with
the hyperbolas.
We now look at the quasi-Gaussian distribution by adding small perturba-
tions with a cubic and a quartic term,
P′(S′) ∝ exp(−(1/2)S′2 − λ′3S′3 − λ′4S′4). (3.6)
Due to the perturbations, the variance of the distribution may differ from unity
61
Figure 3.4: Zeros of the real (circles) and imaginary (crosses) part for 40,000Gaussian configurations. The solid lines are the circle of confidence and thehyperbolas of the normal distribution.
as designed. So we will perform the following transformation to normalize the
distribution,
S = (S′ − ⟨S′⟩)/σS′ . (3.7)
Using the new variables, the probability density can be re-expressed as
P(S) ∝ exp(−λ1S − λ2S2 − λ3S3 − λ4S4) (3.8)
where the λ’s are related to the old λ′ by plugging Eq. (3.7) into Eq. (3.6).
We could then study how the zeros structure responds to the changes of the
controlling parameters λ′3, λ′4.
3.2.1 Example 1. λ′3 = 0.1, λ′4 = 0.01
We first consider the case λ′3 = 0.1, λ′4 = 0.01. It has been chosen in such a
way that we have both zeros inside and outside the Gaussian region of confidence.
With all these setup, the locations of all the imaginary and real zeros can be rather
precisely determined using numerical integration, which we will call the ”actual”
zeros. The actual zero level curves are compared with the MC ones in Fig. (3.5).
62
Figure 3.5: Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the first example. The small dots are the accuratevalues for the real (gray) and imaginary (black) parts. The exclusion regionboundary for d = 0.12 is represented by boxes (red). The solid line is the circle ofconfidence of the Gaussian approximation.
We found zeros at β = 1.3735 + i1.7104 and β = 1.3735 + i2.9478. The variance
of the original variable S′ is 1.0988, so the zeros in the original frame need to be
rescaled with this value. For the lowest zero, the MC zero is very close to the
actual zeros which is located near 1.3+i1.6 and the second lowest zero which is
at 0.9+i2.8, it shows a significant difference to the MC result, and is nevertheless
excluded by the circle of confidence and level of confidence in Fig. (3.5).
3.2.2 Example 2: λ′3 = 0.01, λ′4 = 0.002
In the second example we use λ′3 = 0.01, λ′4 = 0.002, which are much smaller
perturbations and the lowest actual zero should appear at β = 1.207 + i5.241, far
outside the Gaussian circle of confidence(Fig. (3.5)) [20]. But the Monte Carlo
method produces fake zeros with much smaller imaginary parts. However they
are all excluded by the circle of confidence. The result is further confirmed by the
level of confidence.
63
Figure 3.6: Zeros of the real (circles) and imaginary (crosses) part for 40,000configurations corresponding to the second example. The small dots are theaccurate values for the real (gray) and imaginary (black) parts. The exclusionregion boundary for d = 0.15 is represented by boxes. The solid line is the circleof confidence of the Gaussian approximation.
3.3 Approximation with Analytic Functions
Theoretically, once the density of states of a system is known, we should be
able to find almost all the statistical quantities including the zeros of the partition
function of the underlying system. That is the exact reason that we spent the
whole Chapter 2 in the discussion of the density of states. However more often
than usual, we can only get an approximate form of the density of states which
is subject to errors of different kinds. In this case, using an analytical function
to approximate the numerical density of states is necessary. These functions can
provide a starting point to explore the complex region based on the information
collected on the real axis.
64
3.3.1 LargeNp limit with a fixed f (x)
It has been speculated [21] that the condition Re f ′′(x) = 0 served as the
boundary to separate the actual zeros from the region where the Gaussian dis-
tribution dominates and is free of any zeros. In this section, we should further
examine the relation between the roots of f ′′(x) = 0 and the actual zeros. By
”actual zeros”, we mean the actual roots of the partition function at that partic-
ular volume. We will show that with some proper conditions being satisfied,
there exists a simple scaling relation between the actual zeros and the roots of the
f ′′(x) = 0.
We know the entropy density function f (x) has a volume dependence in the
weak coupling regime which has been discussed in Section 2.3.2. However it is
still interesting to know that if we scale f (x) − βx by an arbitrary Np where f (x)
is obtained at a finite volume, how will the Fisher zeros respond to such change.
We will show in the following that in the limit Np → ∞, the produced zeros will
approach to the points that correspond to the roots of f ′′(x) = 0 mapped to the
complex-βplane. The importance of this property is two-folded. First, it guides us
where to search for the actual zeros because empirically these points have almost
identical real parts as the actual zeros; second, at very high volumes, these points
are very close to the actual zeros and therefore can be good approximations.
We will only demonstrate it in a simplified case with several assumptions.
Let Z(β) =∫
dx exp[Np( f (x) − βx)]. Suppose that f (x) is analytic and free of
singularities in the complex x-plane except the bounds of the integration. Then
Z(β) should also be an analytic function. Assume that both Z(β) and f ′′(x) have
only zeros of order no more than one and in all cases the integration path can be
replaced by a simple path through a single saddle point, i.e., Eq. (2.12) is satisfied.
Then in the limit ofNp →∞, β∗ ≡ f ′(x∗) are the roots of Z(β)/(Z′(β)/Np), where x∗
65
are the roots of f ′′(x), i.e., f ′′(x∗) = 0.
In general, Z(β) scales with Np. To make the discussion unambiguous, we
should consider the scale insensitive version Z(β)/(Z′(β)/Np) which should have
the same zeros as Z(β) by assumption.
To prove that this is true, we will start with the saddle point approximation
which we assume valid all the time. We have
Z(β) = eNp( f (x0)−βx0)
√2π
−Np f ′′(x0), (3.9)
it is easy to see that
Z(β)Z′(β)/Np
= − 1
x0 + (1/2) f ′′′(x0)Np f ′′(x0)2
,
= − f ′′(x0)2
x0 f ′′(x0)2 + 12Np
f ′′′(x0), (3.10)
where we have used the saddle point condition f ′(x0) = β and the fact that
dx0/dβ = 1/ f ′′(x0).
We now consider the roots of f ′′(x) = 1/Nαp where α < 1 to ensure the
validity of the saddle point approximation. We will label them as xNp . For large
Np, we can show that for a x∗ which satisfies f ′′(x∗) = 0, there is a xNp close to
it. To see that, since f ′′′(x∗) , 0 by assumption, the inverses mapping of f ′′(x),
which is between the domain of f ′′ and the sheet of the Riemann surfaces that x∗
is located, is well defined. The inverse mapping ( f ′′)−1x∗ exists, then ( f ′′)−1
x∗ (1/Nαp )
gives one of the xNp and
xNp − x∗ ∝ ([( f ′′)x∗]−1)′(0)(1/Nαp − 0)
∝ 1f ′′′(x∗)
N−αp . (3.11)
We now let α > 1/2, so that the expression |NpxNp f ′′(xNp)2| ≪ | f ′′′(xNp)|
holds always true for large enough Np. Then we map them to the β-plane by
66
βNp = f ′(xNp). Using Eq. (3.10), we will have
Z(βNp)
Z′(βNp)/Np∼ −
f ′′(x′Np)2
f ′′′(x′Np)∝ N−2α
p . (3.12)
Since f ′ is a continuous mapping, both (βNp − β∗) and Z(βNp)/[Z′(βNp)/Np] are
arbitrarily small for largeNp, which means β∗ is the root of Z(βNp)/[Z′(βNp)/Np] in
the largeNp limit.
Note:
• At x∗, the mapping f ′(x) is singular and the inverse is ill-defined. But the
conclusion will not be affected.
• The saddle point approximation 3.10 cannot be applied to a integration path
with multiple saddle points, taking the U(1) gauge model as an example.
• The points from f ′′(x) = 0 are not corresponding to the zeros in infinite
volume limit due to the volume dependence of f (x) in practice. The zeros
of both the actual zeros and these roots move with volumes. However there
exists a clear scaling of the distance of the zeros calculated with aNp to the
f ′′(x) = 0 zeros. Since the volume effect decreases with the volumes, it is
interesting to know whether such a scaling approximate the actual scaling
of the system.
Let’s examine this phenomenon through the following toy model. We as-
sume there is such a system whose entropy density function can be described in
the following simple form
f (x) = 1 + 2(x − 511
) − 12
(x − 511
)2 − (x − 511
)4. (3.13)
f ′′(x) has two roots: x = 0.454545±0.288675i which can be mapped to the β-plane.
We know that for such a system, Z(β) as well as Z′(β) should have no singularities
at any volume. So these correspond to the zeros of Z(β) asNp →∞, and they are
67
at β = 2. ± 0.19245i.
Now we can plug f (x) back to the expression of the partition function and
useNp as the scale factor, then we can actually search for the zeros of the partition
function using the method we will endeavor to describe in the next subsection.
To better see the scaling of the zeros with the ”volume”, we add some unrealistic
”volumes” which admit half of the lattice spacing. In the following, we will call
these zeros obtained from the variation ofNp theNp-scaled zeros. The results are
summarized in Table. (3.1) and Fig. (3.7).
”L” ”Volume” Reβ Imβ
2.5 235 2.000000000 0.283832564
3 486 2.000000000 0.248845504
3.5 900 2.000000000 0.229913015
4 1536 2.000000000 0.218722786
4.5 2460 2.000000000 0.211668011
5 3750 2.001 0.2071
∞ ∞ 2 0.19245
Table 3.1: The results of the zeros with the increasing ”volumes”.
3.3.2 Chebyshev Approximation
An analytic form of f (x) will enable us to understand the properties that
are associated with the correspondence between the derivatives of the partition
function and the density of states. The other benefit of having an analytic form
68
0.192
0.2
0.22
0.24
0.26
0.28
0.3
2 3 4 5 6 8 ∞
Im
β
L
Imaginary Part of Lowest Zero vs Size
ç
ç
ç
ç
ç
ç
Slope = -2.6457
5.02.0 3.00.010
0.100
0.050
0.020
0.030
0.015
0.070
L
ÈImHΒ
L-Β¥LÈ
Figure 3.7: The left are the imaginary parts of the zeros in the toy model describedin the text in different ”volumes” plotted against the f ′′(x) = 0 zeros whichcorresponds to the zeros in infinite ”volume” limit. The right is the log-log plotof the difference of the zeros to the f ′′ = 0 zero vs the volume. We found thatIm(βL − β∞) ∝ L−2.6457.
is on the evaluation of Z(β). The original grids of the density of states sometimes
may not be sufficient to do precise numerical integrations (which is how we define
the partition function). It is especially true when the imaginary component of β
is large and, as a consequence, the partition function oscillates more frequently
than what the original bin width can resolve.
We will use the Chebyshev Interpolation for such purposes [14, 75, 65] . It
has been shown that, for the Chebyshev interpolation of numerical data, the Least
Square Fit method is more efficient and robust than the discrete and integration
methods [26, 7, 13]. In this paper, we will primarily follow this approach.
Given a range of interest [a, b], it can be mapped to [−1, 1] in which we can
expand the target function by
f (y) =Nc∑
n=0
cnTn(y) (3.14)
where Tn(y) = cos[n arccos(y)] are the Chebyshev polynomials of the first kind.
We then minimize the distance of the function to a data set or multiple data sets,
69
which will uniquely determine the coefficients cn by a set of linear equations.
3.3.3 The Ellipse of Convergence
When doing a complex continuation of a series function, we should keep in
mind that, like other polynomial approximations, Chebyshev interpolation with
a finite order and a mapped range may introduce artifacts such as fake zeros,
especially when we are taking the derivatives of the approximating function. We
should be careful that we stay in the valid region of the approximation which can
be quantified by the ellipse of convergence [65].
The Chebyshev approximation is really a relabeled Fourier series expansion
[13], differed by a further mapping arccos(x). To work in the complex plane,
the following relation is helpful: Tn(z) = (ωn + ω−n)/2, where z = (ω + ω−1)/2.
The connection is straightforward with the aid of intermediate parameter y,
cosh ny = Tn(z) and cosh y = z.
The convergence of a Chebyshev series can be analyzed through the variable
ω. We turn Eq. (3.14) intoNc∑
n=0
cn (ωn + ω−n) =Nc∑
n=0
cnωn +
Nc∑n=0
cnω−n. (3.15)
A sufficient but not necessary condition for the expression to converge is
that both of the series on the r.h.s of Eq. (3.15) are convergent. If there is a
singularity z0 which is the closest to the expansion point (the origin) in the z-
plane, then it corresponds to two in the ω-plane: ω0(±) = z0 ±√
z20 − 1 which
are conjugate about the unit circle. Both of the Lauren expansions in the r.h.s
of Eq. (3.15) will converge uniformly in the area defined by the two concentric
circles which are determined by ω0(±). Interestingly, these two concentric circles
correspond to the same ellipse in the z-plane, with the semi-major and semi-minor
to be |ω0| ± 1/|ω0|[65, 68]. The region inside the ellipse corresponds to the region
70
enclosed by the two concentric circles in the ω-plane.
The determination of the ellipse of convergence of a Chebyshev expansion
is not always straightforward. Fig. (3.8) shows the logarithm of the coefficients
in the Chebyshev series approximating the numerical f (x) on a 44 lattice using a
maximum order 40 and the range (0.3, 0.6). There is an obvious cut at order 15.
For order n > 15, the small slope indicates a very small radius of convergence. The
tail of the orders up to 15 can be conservatively fitted. A fit for order 4 ∼ 15 gives
a slope of −0.609, which corresponds to a radius of ωR = exp(−0.609) = 0.5435.
So the ellipse in the z-plane has a semi-major 1.1917 and focus length 1. Mapped
back to the fitting range [0.3, 0.6], the ellipse’s semi-major and semi-minor are
0.1788 and 0.0973 and it is centered at 0.45. The artifact of the approximation can
also be indicated by the roots of f ′(x) = β. The outer ring of roots of f ′(x) = β
appear to be independent of the values of β. Fig. (3.9) shows the roots of f ′(x) = β
where β = 2.2 + 0.02i, 2.2 + 0.04i, 2.2 + 0.06i and 2.2 + 0.08i. We can see that the
outer ring of roots of f ′(x) = β seem to have no dependence on β’s and they are
coincident with the ellipse of convergence defined above. However there are
three non-trivial solutions inside the ellipse. Two of them tend to merge as the β
approach the value f ′(x∗) where x∗ is the root of f ′′(x) = 0 (the cross in the graph).
As we can see from Fig. (3.10), the roots of f ′′(x) = 0 are well inside the ellipse
and the two non-trivial saddle points collapse right at x∗.
71
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á á
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á á
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á
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á
á
á á
á
á
á á á á áá
á
á
0 5 10 15 20 25 30
-20
-15
-10
-5
0
ordern
lnÈc
nÈ
Figure 3.8: The logarithm of the absolute coefficients of the Chebyshev approxi-mation discussed in the text versus the order number n.
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à
à
à
à
à
0.3 0.4 0.5 0.6-0.10
-0.05
0.00
0.05
0.10
Re x
Imx
Figure 3.9: The ellipse of convergence of a Chebyshev series ( described in thetext) and the roots of f ′(x) = β with the values given in the text. The two saddlepoints which are obviously inside the ellipse merge to the root of f ′′(x) = 0 (redcross) as the value of β approaches to the value corresponding to the root off ′′(x) = 0.
72
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áç
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ç
ç
ç
+
0.3 0.4 0.5 0.6-0.10
-0.05
0.00
0.05
0.10
Re x
Imx
Figure 3.10: The ellipse of convergence of a Chebyshev series and the roots off ′′(x) = 0(empty circles). The empty squares are the roots of f ′(x) = β where β isgiven by f ′(rootsof f ′′ = 0) ( the cross ).
3.3.4 The Moment Tests
To estimate the goodness of the approximation, we can look at the moments
from both the data and the approximations. We can define the following scale-
invariant moments,
M1 = ⟨x⟩,
M2 = Np⟨(x − ⟨x⟩)2⟩,
M3 = N2p ⟨(x − ⟨x⟩)3⟩,
M4 = N3p (⟨(x − ⟨x⟩)4⟩. − 3M2
2) (3.16)
Due to the fact that these variables are not all independent, to estimate the
errors on these moments, we can use either Jackknife or bootstrap method [64] to
address the correlations between them. We found out that the results from these
two method are similar. There is also a method introduced by Wolff [78] which is
73
to estimate the error bars more precisely. The results are shown in Fig. (3.11) and
Fig. (3.12), which indicate that the Chebyshev approximation is consistent with
the numerical data.
0.24
0.28
0.32
0.36
0.4
2 2.1 2.2 2.3 2.4
2nd Moment
β
4x4x4x4
MC3CH40CH34CH30
Figure 3.11: The second moment of the SU(2) model at volume 44. The numericalresult is plotted against various orders of Chebyshev approximation.
-0.5
0
0.5
1
1.5
2 2.1 2.2 2.3 2.4
3rd Moment
β
4x4x4x4
MCCH40CH34CH30
Figure 3.12: The third moment. Similar to Fig. (3.11).
74
3.3.5 Evaluation of the Partition Function with Nu-merical Integration
We now get to the detail on how to evaluate the partition function once we
have the approximate form of the density of states. The partition function Z(β)
is often used as a normalization and its value bares no direct physical meaning.
What we are evaluating here is actually Z′(β)/Z(β).
From the definition, the difficulty of the numerical calculation of Z(β) is that
when the imaginary part of β is introduced, it will scale withNp which makes the
integrand highly oscillating and it is typically hard to integrate numerically. In
the following, we will compare two different methods to resolve this problem.
3.3.5.1 Path of the Steepest Descent
The first method is to make use of the saddle points and the path of steepest
descent. Constructing the path of steepest descent can be numerically consuming
and we found that approximating the path by a line which is tangent to the
steepest descent path at the saddle point is very reliable. Let x0 be a saddle point,
f ′(x0) = β. Let f ′′(x0) = | f ′′(x0)|eiθ0 . Then we can deform the integration to the
path
x
Figure 3.13: Path of Steepest Descent.
75
x = x0 + t exp(i2
(θ0 − π)), (3.17)
where t is the new integration variable and is real. This method works very
effectively and accurately. However there is a limitation when applying to the
actual problem. It arises from the locations of the saddle points. We can see
from Fig. (3.9) that as the imaginary part of β increases, the two non-trivial
saddle points tend to approach to each other and at a certain value of β which
is close enough to the f ′′(x) = 0 zero (the cross), the saddle point approximation
Eq. (3.10) will break down because the integrating path now has to include two
saddle points.
3.3.5.2 The Improved Trapezoidal Integration
Ironically, simple integration algorithms like the Trapezoidal rule works
also very well in the special case of SU(2). The fundamental reason is that in
general, the probability density associated with the SU(2) at a real β is very close
to a Gaussian distribution. For an exact Gaussian distribution, let’s consider
I =∫ a
−adxe−
12 Ax2
cos(x). (3.18)
We can use the simple Trapezoidal integration with 2N points,
IN =
N∑n=−N
exp(−12
An2h2) cos(nh), (3.19)
where h = a/N.
By using the Jacobi Imaginary Transformation ( See Appendix for a proof),
we can show that the error of the integration goes like
∆II≡ I − IN
I∝ exp(− π
2
2Aa2 N2). (3.20)
It shows that the integration improved with the order of exponential of negative
N2 which makes the convergence of the integration very fast.
76
For a quasi-Gaussian distribution with small perturbations, empirical re-
sults show that it also exhibits similar rate of convergence.
3.3.6 Localization of Zeros Using Cauchy’s Theorem
There is a general algorithm to find the zeros of an analytic function by using
the Cauchy’s Integral Theorem [54, 39]. For simplicity, we will only consider the
special case when all the zeros are of order 1 which applies to our problem.
Suppose that an analytic function Z(β) has K simple zeros enclosed by a closed
contour C, then there are the following relations,
12πi
∮c(ln Z)′ βn dβ =
K∑i=1
(βi)n, n = 0, 1, 2, ... (3.21)
where βi are all the zeros in contour C. When n = 0, the summation on the right
hand side is just the number of zeros.
The partition function we are considering is an analytic function of β, which
is obtained through either naive summation or extended trapezoidal rule. In
principle, we could construct a large contour and perform the integral with n = 0
to determine the number of zeros enclosed. If there are K zeros, then we do
K integrations with n = 1, ...,K and solve the resulting K-order equations for
the zeros. However it turns out that higher order nonlinear equations are hard
to solve and are extremely sensitive to the error on the coefficients. The slight
changes in the coefficients can result in drastic shifts of the roots or even cause
a totally different root structure. So practically, we will only work with contours
that include at most two zeros. We scan the complex plane with rectangular
contours which enclose zeros of two or less. We found the method quite robust
and reliable.
77
Noticing that P(β) = ⟨x⟩β = (1/Np)(ln Z)′, we can rewrite the equation into
Np
2πi
∮cP(β) βn dβ =
K∑i=1
(βi)n, n = 0, 1, 2, ... (3.22)
What we need for the loop integrals is P(β), but we have to evaluate P(β)
through ratio of Z′(β) and Z(β) which we will obtain using the numerical integra-
tion technique developed in the last subsection. There is a common constant in
both of Z(β) and Z′(β) which will be canceled in the ratio. This constant needs
to be known in advance to prevent overflow or underflow in the evaluation of Z
and Z′. Unfortunately, this constant varies with β. As a result, since the difference
will be scaled by Np in the exponential, different points in the loop may require
canceling constants that are very different. However, it is remarkable that the
constants have a simple connection to P(β) itself. We know for large enough
volume, approximately, P(β) ≈ x0 at the saddle point x0 and the dominant term
in the exponential is close to the function evaluated at the saddle point x0, i.e.,
f (x0) − βx0. So the constant needed to calculate P(β) can be well approximated
by f (P(β)) − βP(β). Considering that we are calculating P(β) in a loop, if the step
is taken small enough, the canceling constant of the preceding point can be good
enough for the succeeding one.
There is a further reason to look carefully at the relation between P(β) and
β. If the integration over β is along a simple loop, what is the resulting loop of
P(β) in the complex x-plane? In general it follows the inverse mapping of the
relation f ′(x0) = β. If f ′(x0) is a simple polynomial, the inverses of the mapping
may possess several branch cuts.
When the loop is too close to a zero, a small step in β-plane may result in a
different branch in x-plane, which will cause a drastic change in the value of P(β).
The integration will fail, for example, the loop integral with n = 0 will deviate
obviously from an integer. So a new loop is needed to insure the reliability of the
78
A B
CD
2.10 2.15 2.200.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
ReΒ
ImΒ
AB
CD
0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50-0.10
-0.05
0.00
0.05
0.10
ReHxL
ImHxL
Figure 3.14: The plot is showing the correspondence of a loop in the β-plane tothe loop in the x-plane.
integration. We often apply a method called rerouting to go around the zero. An
illustrating graph is shown in Fig. (3.15). The basic idea is that when a path is too
close to a zero, the P(β) will appear to diverge. So we can set a reasonable zone (
for the case involving the Chebyshev Polynomials, this is set to be the ellipse of
convergence). Whenever the value P exceeds this zone, it will trigger a reroute
which is typically to construct a rectangular path to go around the region that is
too close to the zero.
3.4 SU(2) Zeros
We have now gathered enough tools to analyze the partition function zeros
of the SU(2) model. We know there is no phase transition for the SU(2) model in
the infinite volume limit. But it is interesting to know how its zero structure is
like. As we have mentioned in the introduction, the non-vanishing height of the
lowest zeros of the partition function may admit a confinement theory extending
to the weak coupling domain.
79
zero 1
zero 2
Figure 3.15: An actual loop in which two zeros are presented. The path is detouredto get a better integration.
3.4.1 Zeros From Single-Point Reweighting
We can first look at the result of using single-point reweighting method. Fig.
(3.16) [23] shows a contour plot at β = 2.18 with 1, 000, 000 configurations. Two
different methods are compared here. The single-point reweighting gives zeros
with imaginary part higher than 0.18. However these zeros are all outside the
circle of confidence. We also include here the quasi-Gaussian fitting result. We
fit the perturbation parameters which are discussed in Section 3.2 and it spotted
two zeros at β = 2.18 + i0.13 and β = 2.22 + i0.12 which are right on the border of
the circle of confidence. However they are excluded by the level of confidence.
So we can conclude that near Reβ ∼ 2.18, there is no zero below Imβ = 0.13.
We did a thorough test using the existing data, in order to estimate the
statistical fluctuations in the location of these complex zeros. We generated
200 bootstraps (each contains 40,000 configurations) at β = 2.225. For each of
the 200 sets, we calculated the zero contours of the real part of the partition
function. Using this procedure, 383895 zeros of the real part were located. The
distribution of these zeros are plotted using a 200 by 200 grid in the complex
80
Figure 3.16: The contour plot shows the zeros of a SU(2) gauge model on a 44
lattice. Blue and green lines are the zero curves of the imaginary and real part ofthe partition function from the Monte Carlo simulation at β = 2.18. The crossesand circles are the counterparts from a quasi-Gaussian approximation whichoverlaps the data in the bottom part. Both show isolated zeros of the partitionfunction. However they are all lying outside the radius of confidence, or morestrictly, they are all above the level of confidence ( the squares). Therefore thesezeros are artificial. The Monte Carlo data of a neighboring β doesn’t indicate theconsistent locations of these zeros.
β-plane. The results are shown in Fig. (3.17) [21]. The zeros of the real part
accumulate in several bands. Each band contains zeros of number from 60 to
more than 100. The circle of confidence in the Gaussian approximation for 40,000
independent configurations as well as level of confidence are also shown in this
graph for references. It is clear that as we get closer to the boundary of the region
of confidence, the fluctuation becomes larger. When it is beyond the circle of
confidence, it becomes merely random noise.
3.4.2 Histograms Analysis
We have seen that the Fisher zeros arise from the deviation of its distribution
from the Gaussian distribution. To determine the region where the zeros might
81
Figure 3.17: Distribution of eros of the real part of the partition function in thecomplex β plane and regions of confidence described in the text.
appear, we can examine its signal of such deviation. By looking at the coherent
pattern of the distribution with the major Gaussian part subtracted, we can tell
whether a zero is close-by. We use the histogram residue method. We first sort
the data into n number of bins. Define the residues
ri = (Ni −NPi)/√
(NPi) , (3.23)
where Ni is the number of configurations in i-th bin and Pi is the probability in the
i-th bin (here the probability is not the probability of the original distribution but
that of the Gaussian distribution determined by the average and variance of the
original distribution). N is the total number of configurations. Fig. (3.18,3.19,3.20)
[23] shows such a pattern. For the volume 44 at β = 2.18, we can see the strong
pattern departing from the pure Gaussian distribution which implies zeros pos-
sibly exist in the vicinity. In contrast, for the volume 64, the difference from the
Gaussian looks purely random which suggests that the Gaussian distribution
dominates here. However the coherent pattern appears for the volume 64 at
β = 2.34.
82
Figure 3.18: The residue distribution after subtracting the Gaussian part withβ = 2.18 on a 44 lattice.
Figure 3.19: The residue distribution after subtracting the Gaussian with β = 2.18on a 64 lattice.
3.4.3 Zeros using Analytic Approximation
We now use the analytic approximation of the entropy density function f (x)
and the series of tools we have developed to find the Fisher’s zeros. We have
83
Figure 3.20: The residue distribution after subtracting the Gaussian with β = 2.348on a 64 lattice.
discussed the structure and property of the roots of f ′′(x) = 0, which correspond
to the zeros in Large-Np limit. Now we can look at the actual location of the zeros
at these finite volumes. We use the numerical integration to evaluate ⟨x⟩β (which
is −Z′(β)/NpZ(β)), and then do loop integrations in the β-plane. We typically
break the region into a collection of boxes and search for zeros in each one. We
use Eq. (3.22) with n = 0 to determine the number of zeros in each box and decide
whether to do a refined integration. If the box contains no zeros, we skip; if the
box encloses more than two zeros, then we break it into two smaller boxes. If the
box contains only one zero, then the integration in Eq. (3.22) with n = 1 gives
directly the location of the zero; if the box contains two zeros, the we have the
following two equations
β1 + β2 = I1,
β21 + β
22 = I2,
where I1,2 are the loop integration described in Eq. (3.22) with n = 1, 2. The
solutions of these equations will give the locations of the two zeros. Fig. (3.21)
84
0
0.1
0.2
0.3
0.4
0.5
0.6
1.5 2 2.5 3
Im(
β)
Re(β)
SU(2) zeros
L=4 f’’=0 L=4 res.L=6 f’’=0 L=6 res.
Figure 3.21: The plot shows a typical pattern of the SU(2) zeros. The zeros oftwo different volumes, 44(filled squares) and 64 (filled circles), are plotted againsttheir respective f ′′(x) = 0 zeros (empty squares for the 44 and empty circles forthe 64). The density of states are approximated using Chebyshev Polynomial oforder 44. The fitting range is over x ∈ [0, 2].
shows a plot of all the zeros in the box of (1.5,3) (0,0.55) at the two different
volumes 44 and 64. The roots of f ′′(x) = 0 zeros (empty squares and circles) are
right beneath the strands of the actual zeros.
We are interested in both the actual zeros and the roots of f ′′(x) = 0. How-
ever the latter needs to be examined carefully with the ellipse of convergence
which is important for the analytic continuation to the complex plane. We have
shown that the effective orders of approximation can be determined through the
plot of the logarithm of the absolute value of the Chebyshev coefficients versus
the order. The ellipse determined through the slope of the plot should well en-
close the roots of f ′′(x) = 0 to make them reliable, as has been illustrated in Section
3.3.3.
To ensure the zeros we got are not artifacts from the approximation, we
85
should make sure of two aspects: first, the zeros should not vary with the ranges
of the approximation; second, the zeros should be stable with the variation of the
orders of approximation.
Table (3.2) shows the Fisher zeros obtained at the volume 44 with three
different ranges: [0.1, 0.8], [0.2, 0.7] and [0.3, 0.6]. The order is determined through
the ln |cn| ∼ n plots. The points corresponding to the roots of f ′′(x) are inside the
ellipse of convergence. The actual zeros are obtained through the Cauchy loop
integration method. All the error bars are estimated by the replica of numerical
f (x) computed from 20 seeds of Monte Carlo data (Table (3.2)). The similar results
for the volume 64 are also given in Table (3.3). We can see that the approximation
is rather stable with the change of fitting ranges (see Fig. (3.22)).
Range order Ell’s Major f ′′ - Reβ f ′′- Imβ Actual Reβ Actual Imβ
Table 3.2: Both of the actual zeros and the f ′′(x) zeros are shown with differentranges of fitting for the volume 44. The semi-major of the corresponding ellipsesare also given ( the focus length is half the range).
The approximation should also be stationary with respect to the variation
of the orders. Table (3.4) shows the Fisher zeros obtained using various orders of
approximation for the two volumes 44 and 64. We see that variation is within the
statistical fluctuation (less than one sigma), which is shown in Fig. (3.23).
86
Range order Ell’s Major f ′′ - Reβ f ′′- Imβ Actual Reβ Actual Imβ
Table 3.3: Both of the actual zeros and the f ′′(x) zeros are shown with differentranges of fitting for the volume 64.
0.04
0.06
0.08
0.1
0.12
0.14
0.16
2.16 2.2 2.24 2.28 2.32
Im
β
Re β
Zeros with diff. ranges
Actual Zerosf’’(x)=0 Zeros
Figure 3.22: The plot is the zeros (empty cirdles) of two volumes: 44 and 64. Theirf ′′(x) = 0 zeros (filled circles) are also plotted for comparison. Both of the actualzeros and the f ′′(x) = 0 zeros tend to approach to the same limit which is distantaway from the real axis.
We can notice that the imaginary part of the actual zeros decreases slightly
with volume. However, the imaginary part of the f ′′(x) = 0 zeros increases
with volume which shows the actual zeros will stay away from the real axis.
This corresponds to a crossover rather than a real phase transition in the infinite
volume limit [56, 16]. This fits in what is widely accepted on the critical behavior
about the SU(2) pure gauge model at the zero temperature.
87
Volume order Actual Reβ Actual Imβ
4 14 2.1970(25) 0.1516(35)
15 2.1974(35) 0.1516(39)
16 2.1976(37) 0.1502(51)
6 24 2.3135(17) 0.1224(37)
25 2.3159(28) 0.1203(41)
26 2.3161(46) 0.1194(32)
Table 3.4: The actual zeros of various orders of approximationfor the two volumes 44 and 64 discussed in the text.
0.08
0.1
0.12
0.14
0.16
2.16 2.2 2.24 2.28 2.32
Im
β
Re β
Zeros with diff. orders
44 order 1444 order 1544 order 1664 order 2464 order 2564 order 26
Figure 3.23: The lowest zeros obtained using various orders of approximation attwo different volumes: 44 and 64.
Due to the rapidly increasingNp and the noise level of the data, the searching
for the actual zeros of the SU(2) model at the volume 84 and higher becomes
difficult. However as the toy model in Section 3.3.1 hints, we might be able to
reveal the scaling property by scaling a finite-volume entropy density function
88
with an arbitraryNp. We can look at the distances of theseNp-scaled zeros to the
roots of f ′′(x) = 0 on the complex β-plane.
We use f (x) obtained from the volume 44. We change the value of Np and
use the Cauchy loop integral method to calculate the corresponding zeros. The
results are given in Table.(3.5). The difference between the Np = 6 × 44 zero and
theNp →∞ zero is about 0.076. The difference between theNp = 6× 64 zero and
theNp →∞ zero is about 0.027 which is close to what is actually calculated using
the f (x) at volume 64. The series of zeros are plotted against the f ′′(x) = 0 zero in
Fig. (3.24). Fig. (3.25) shows the log-log plot of the distances of the zeros to the
f ′′(x) = 0 zero using f (x) obtained from two different volumes: 44, 64. The 44 case
shows a power law of order −2.6207 while the 64 shows a power law of order
−2.62175. The two scalings are remarkably close which means that the volume
Table 3.5: The zeros with the increasing ”volumes” using the density of states f4
and f6 of the 44 and 64 lattices.
89
L = 4
5
6
78
9 10¥
2.197 2.198 2.199 2.200 2.201 2.202 2.203
0.06
0.08
0.10
0.12
0.14
Re Β
ImΒ
Figure 3.24: The locations of the zeros calculated using the entropy density func-tion f at volume 44 scale withNp = 6×L4. The red point corresponds to the f ′′ = 0zero.
0.006
0.012
0.024
0.048
0.096
4 5 6 7 8 9 10
|β L -
β ∞|
L
Scale of the zero distances: 44,64
Using 44 fUsing 64 f
Figure 3.25: The zeros βL obtained using various Np = 6 × L4 using the entropydensity function from two different volumes: 44,64. The plot shows the distancefrom βL to the f ′′ = 0 zero β∞ versus L in the logarithmic scale.
90
3.5 U(1) Zeros
We now move on to the discussion of the Fisher’s zeros of the U(1) model.
Unlike the case of the SU(2), the discrete reweighting method works very reliably
here in finding the zeros, especially for higher volumes, due to the fact that the
zeros tend to approach the real axis with a power law of the volume. We applied
both of the discrete reweighting and the analytic approximation methods. We
found that in general they agree with a very high accuracy. The results from the
U(1) model provide a contrasting example for case of SU(2) because these two
systems belong to different universalities.
3.5.1 Volume Dependence of the Double Peak
The plaquette distribution of the U(1) model appears to have a double peak
structure near the βcrit which is around 1. If the double peak persists at large
volume, we should expect a first-order transition. So we should first look at the
change of the double peak over the volumes.
To determine the separation of the double peaks, we can visually check the
distribution f (s) − βs (here s ≡ x which is the average action) for the value of
β such that the two peaks have equal heights. In Fig. (3.26) [9], we show that
f (s)− βs is slightly tilted to the left for β = 1.00175 and to the right for β = 1.00179.
So the two peaks make a tie around βS = 1.00177(2). With the same graphs, we
can also tell approximate values s1 and s2 which are the two maximum values of
s. The numerical results for the three volume are summarized in Table 3.6.
A plot of the double peak distribution is displayed in Fig. (3.27). Fig. (3.27)
makes clear that the dip between the peaks deepens and the peak separation
slightly decreases as the volume increases. The peak separation s2 − s1 scales
91
-0.7474
-0.74739
-0.74738
-0.74737
0.34 0.36 0.38 0.4 0.42
f(s)-
βs
s
U(1) 64
β=1.00175β=1.00177β=1.00179
Figure 3.26: f (s) − βs for β = 1.00175, 100177 and 1.00179 on a 64 lattice. Thehorizontal lines is drawn to indicate the asymmetry of the heights. The error barsare provided with the same scale as f (s) − βs.
L βS s1 s2
4 0.9793(1) 0.370(5) 0.445(5)
6 1.00177(2) 0.353(2) 0.411(2)
8 1.00734(1) 0.349(1) 0.395(1)
Table 3.6: βS, s1 and s2 defined in the textfor L = 4, 6 and 8.
92
0
5
10
15
20
0.3 0.35 0.4 0.45 0.5
P(s)
s
U(1) plaquette distr. at βS
44
64
84
Figure 3.27: The double peak distribution of three volumes: 44, 64 and 84.
approximately like L−0.7. Given that we have only a small set of volumes, this
statement should be used to estimate the range of β that is necessary to calculate
the density of states at larger volumes. The general scaling may receive con-
siderable correction from higher volumes, which will be revisited later in this
chapter.
3.5.2 Zeros using the Discrete Reweighting Method
With the numerical expression of the density of states, we can do a simple
summation to evaluate the partition function without doing an analytic approxi-
mation,
Z(β) ≃ ∆s∑
s
eNp( f (s)−βs) . (3.24)
Then we can find the zero contours of the real part and the imaginary part of the
partition function. These plots are shown, for an example, in Fig. (3.28). The
lowest two zeros can be resolved by this method rather precisely. The results are
given in Table.(3.7) and Table.(3.8). Out of density of states from 20 seeds of data,
the zeros can be located as precisely as down to 10−5.
93
0
0.02
0.04
0.06
0.08
0.1
0.97 0.975 0.98 0.985 0.99
Im
β
Reβ
U(1) 44
ReZ=0ImZ=0
0
0.01
0.02
0.03
0.04
0.05
0.99 0.995 1 1.005 1.01
Im
β
Reβ
U(1) 64 ReZ=0ImZ=0
Figure 3.28: Zeros of the real (point +) and imaginary (point x) part of Z for U(1)using the density of states for 44 and 64 lattices.
L first second third
4 0.9791(1) 0.9780(4) -
6 1.00180(5) 1.0007(1) 0.9993(5)
8 1.00744(2) 1.0068(2) 1.0061(4)
Table 3.7: Real part of the first three zerosfor L = 4, 6 and 8.
We now look at how the cuts of ranges that we use to evaluate the partition
function will affect the locations of the zeros. We first fix the right integration
bound to be at s = 0.95 which is close to the edge of the domain, and we change
the left bound incrementally from a low value. We monitor the change of the
location of the first zero ( typically the imaginary part of the zero because it is
more sensitive to the change). We can do the same to fix the left bound at s = 0.05.
Fig. (3.29) shows an example of how the zeros respond to the variation of the cuts
in the volume of 64. Typically, we will expect a drastic change of the locations of
94
L first second third
4 0.0259(1) 0.057(1) -
6 0.00758(2) 0.018(1) 0.027(2)
8 0.00306(2) 0.008(1) 0.012(1)
Table 3.8: Imaginary part of the firstthree zeros for L = 4, 6 and 8.
the zeros on the cuts, which indicates that the distribution decays rapidly with
the cuts and beyond certain values, the bounds of the integration have no effect
to the locations of the zeros.
ç ç ç ç ç ç ç ççç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ççç
0.27 0.28 0.29 0.30 0.31 0.32
- 5.5
- 5.0
- 4.5
- 4.0
- 3.5
s
logHÈImΒ-
ImΒ0ÈL
ççççççççççççççç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
0.41 0.42 0.43 0.44 0.45 0.46
- 5.5
- 5.0
- 4.5
- 4.0
- 3.5
s
logHÈImΒ-
ImΒ0ÈL
Figure 3.29: The shift in the imaginary part of the zeros due to the cuts of theintegration described in the text.
95
L Reβ σs σc Imβ σs σc
0.9791235 3.6e-5 5.3e-8 0.0260065 3.7e-5 3.9e-9
4 0.9777314 3.5e-4 7.1e-6 0.0572764 1.4e-4 3.3e-6
0.9752954 1.1e-3 2.9e-4 0.0831705 1.3e-3 3.2e-4
1.0017969 1.7e-5 1.7e-6 0.0075821 8.7e-6 1.4e-6
6 1.0007433 6.0e-5 2.3e-5 0.0182044 2.8e-5 4.0e-6
0.9988964 1.4e-4 2.7e-4 0.0271866 4.5e-4 1.5e-4
1.0074380 1.1e-5 7.7e-8 0.0030653 3.6e-6 6.8e-8
8 1.0068296 2.3e-5 2.1e-6 0.0077673 2.4e-5 3.3e-7
1.0060410 1.1e-4 1.2e-5 0.0115079 1.0e-4 8.5e-5
Table 3.9: The lowest three zeros in the three volumes 44,64 and 84. Col-umn 1-4 are, the real parts of the zeros, the estimate error σs from differentseeds of Monte Carlo runs and the error σc due to the orders of Cheby-shev interpolation( we used three different orders 40,44 and 50 for allthree volumes). Same for the imaginary part.
3.5.3 Zeros from analytic approximation
The other method we used is to approximate f (x) using analytic functions
and use Cauchy’s loop integral method as we did for the SU(2) model. We found
that the zeros obtained in the way are almost identical to those obtained using
discrete reweighting. To estimate the error due to the polynomial approximation,
we calculate them using three different orders of Chebyshev approximation.
The error received from the order of the approximation is very small compared
with the statistical error (typically about one magnitude off). The results are
summarized in Table.(3.9).
96
0
0.02
0.04
0.06
0.08
0.1
0.96 0.98 1 1.02
Im(
β)
Re(β)
U(1) zeros
Figure 3.30: The lowest zeros from three volumes 44,64 and 84(from left to right).The error bars have taken account of both of the Monte Carlo statistical er-ror(seeds) and the Chebyshev interpolation error(orders). The three guidelinesare the fits for the first, second and third lowest zeros, using only the zeros of 64
and 84. They intersect the real axis approximately at the same point β = 1.01134.The diamonds on the real axes(Imβ=0) are the double-peak β’s from Table.(3.6).
Fig. (3.30) shows how these zeros are distributed on the complex plane. The
precision of the zeros increase as the imaginary part of the zeros decreases. We
can see that the first, second and the third zeros of the three volumes 44, 64 and 84
crosses roughly at the same point βc ∼ 1.01134 on the real axis, which corresponds
to the transition point in the infinite volume limit.
97
0.01
4 8
Im
β 1
L
The imaginary part vs lattice size L: log-log
fitdata
Figure 3.31: The log-log plot of the imaginary part of the lowest zero vs the latticesize L = 4, 6, 8.
3.5.4 The Scaling of Zeros
It was suggested in [40] that the jth Fisher zero has an dependence on the
volume which can be expressed using the critical exponent ν by
β j(L) ∼(
jLd
)1/νd
, (3.25)
where d is the dimension. For the lowest zero, the scaling is Imβ1 ∼ L1/ν. Fig. (3.31)
shows a log-log plot of the imaginary part of the lowest zeros vs the volumes.
It gives a scaling of L−3.07 which corresponds to ν = 0.325. The result is in good
agreement with the result in [50] which was obtained using high-statistics finite
size scaling of cumulants for their smaller volumes.
Now let’s look at theNp-scaled zeros similar to those discussed in the SU(2)
98
case. It was suggested from the SU(2)Np-scaled zeros that this scaling has small
volume dependence. We use the entropy density function from the 84 lattice. We
change the value of Np = 6 × L4 with L varying from 8 up to 21 and search for
the zeros. The lowest zeros are plotted in Fig. (3.32). Unlike the case of SU(2),
the roots of f ′′ = 0 are not corresponding to theNp →∞ limit because the saddle
point approximation Eq. (2.12) breaks down due to the presence of three saddle
points on the real axis for β near the value of βS given in Table (3.6). There are
two real solutions of f ′′ = 0, but neither is the saddle point for the βS . Since the
zeros are close enough to the real axis, so we can just analyze the scaling of the
imaginary part of the zeros. We fit ln(Im(βL)) ∼ ln L with a linear function for the
scaling power. We analyze how the power changes with datasets included in the
fit by throwing out the smallest lattice size L one at a time until the last two. The
fit result of the power scaling versus the lowest L is plotted in Fig. (3.33). Clearly
the scaling is approaching to L−4 which corresponds to ν = 0.25 and indicates a
strong sign of a first order phase transition in the infinite volume limit.
3.5.5 The Derivatives of f (x)
We can look at the problem from a different angle. We have shown the
double peaks of the probability distribution of the U(1) model. If the width of the
double peaks is not vanishing in the V → ∞ limit, then we should expect a first-
order transition. However as the volume increases, it becomes harder to precisely
find the peaks and there is ambiguity on the definition of double peaks (equal
area or equal height). However we found that the real roots of f ′′(x) = 0 can be
good alternatives to the double peaks. These roots correspond to the locations
99
0
0.0006
0.0012
0.0018
0.0024
0.003
1.00736 1.0074 1.00744
Im
β
Re β
Np scaled zeros using 84 f(x): U(1)
Np scaled zerosIntersect 1.007354
Figure 3.32: TheNp scaled zeros discussed in the text.
-4.12
-4.1
-4.08
-4.06
-4.04
-4.02
-4
8 10 12 14 16 18 20
fit of -1/
ν
L of fit lower bound
Np scaled zeros using 84 f(x): U(1)
Figure 3.33: The orders of power law from the fits are plotted versus the lowestsize L that is included in the fit.
where the distribution function exp[Np( f (x)− βx)] changes convexity. Obviously
the distance between the roots is always smaller than the width of the double
peaks. If the roots’ separation survives the V →∞ limit, so does the width of the
100
double peaks.
We have the preliminary data for larger lattice sizes: L = 10, 12, 14, 20. Fig.
(3.34) shows the numerical derivative of f (x) with respective to x. The depth
between the extrema is flattening as volume increases. However the width of the
extrema will survive the V →∞ limit which is shown in Fig. (3.35). The two real
roots of f ′′(x) (the extrema of f ′(x)) are plotted versus L. The preliminary data
from higher volumes show a persisting width of about 0.014, which indicates a
scenario of first-order transition.
0.96
0.98
1
1.02
1.04
1.06
0.3 0.35 0.4 0.45
f’(x)
x
Numerical Derivatives of f(x)
L=4L=6L=8L=10L=12L=14L=20
Figure 3.34: The first derivative of f (x) from different volumes.
101
0.34
0.36
0.38
0.4
0.42
4 8 12 16 20
x
L
Real roots of f’’(x)=0
left rootright root
Figure 3.35: The locations of the two real roots of f ′′(x) = 0 are plotted versus thelattice size.
102
CHAPTER 4CONCLUSION
In this work, we studied the Fisher’s zero in the complex coupling plane
which is important to identify the type of phase transition in lattice gauge theory.
We used an general approach to find the zeros by analyzing the density of states
of the models. We used both perturbative and non-perturbative methods to con-
struct the density of states. With the numerical calculation of the entropy density
function, we were able to locate the zeros by using discrete reweighting method
and the Cauchy’s loop integral method with the Chebyshev approximation of
the entropy density function. We applied these methods on the SU(2) and the
U(1) lattice gauge theory with a fundamental Wilson action on the lattices. The
locations of the zeros were presented and the scaling of the zeros was discussed.
In Chapter 2, we calculated the density of states perturbatively. By using
the saddle point approximation, we constructed series expansions of the density
of states in both of the strong and weak coupling regimes by using the strong
and weak coupling expansions of the average plaquettes. We calculated the
series of the SU(2) and U(1) lattice gauge models and found that they exhibited
finite radii of convergence and suggested the complex singularities. We also
constructed the density of states numerically using Monte Carlo simulations
and the Ferrenberg-Swendsen’s histogram reweighting method. We analyzed
the volume dependence of the density of states and gave the low order volume
corrections for the U(1) lattice gauge model.
In Chapter 3, we calculated the Fisher’s zeros using both of the discrete
reweighting and analytic approximation method. We carefully examined our
numerical calculations by analyzing the artifacts which may arise from either the
statistical fluctuations or the approximation method. With the quasi-Gaussian
103
models, we showed that the artificial zeros can be excluded effectively by circle
of confidence [2] and concluded that for the SU(2) model at the volume 44, the
zeros should be located above Imβ = 0.13. By approximating the entropy density
function using Chebyshev polynomials, we calculated the locations of the lowest
zeros: β = 2.197(3) + i0.150(5) (for 44) and β = 2.319(5) + i0.119(6) (for 64). The
imaginary part of the zeros which correspond to the root of f ′′ = 0 tend to
increases with volumes, which shows that the zeros of the SU(2) will stabilize at a
distance from the real axis. TheNp scaled zeros showed that these zeros approach
to the f ′′ = 0 roots with a scaling of L−2.62 with respective to the lattice size L.
For the U(1) lattice gauge theory, we used both of the method to calculated the
locations of th lowest three zeros and found that they were in precise agreement.
By doing an extrapolation to the real axis, we found that the transition point in the
infinite limit is located at βc = 1.01134 and the imaginary parts of the zeros show
a scaling of L−3.07 for the three volumes 44, 64 and 84, which is consistent with the
result obtained using the finite size scaling of the cumulants [50]. Preliminary
results at larger volumes show a first order transition in the infinite volume limit,
which is also indicated by the scaling of theNp-scaled zeros.
104
APPENDIX ATWO-DATA-POINT REWEIGHTING
We will use an over line to denote sample average and a pair of brackets to
denote large data limit ( statistical average). Let
δ∆A = ∆A − ∆A, (A.1)
where ∆A = A2 − A1 and we define ∆A ≡ f (Z1,Z2). It is obvious that ⟨δ∆A⟩ = 0,
but the variance is not
(δ∆A)2 =
(∂ f∂Z0
)2
(δZ0)2 +
(∂ f∂Z1
)2
(δZ1)2 +
(∂ f∂Z0
) (∂ f∂Z1
)(δZ0)(δZ1). (A.2)
The third term is zero in statistical limit, assuming that the two Monte Carlo runs
APPENDIX BFERRENBERG-SWENDSEN’S FORMULA FOR THE WEIGHT
We assume the errors are mainly from the histograms, which approach to
the large statistical limit like
⟨δH(x))2⟩ = (1 + 2τc)⟨H(x)⟩ ≡ g⟨H(x)⟩. (B.1)
The error on the density of states is then
⟨(δn)2⟩ =⟨ R∑α=1
∂n(x)∂H̄α(x)
δHα(x)
2⟩
=
R∑α,γ
∂n(x)∂H̄α(x)
∂n(x)∂H̄γ(x)
⟨δHαδHγ⟩.
Since individual Monte Carlo runs are independent to each other, we can assume
the correlation to be zero for distinct data points,
⟨δHαδHγ⟩ = δαγ⟨(δHα)2⟩ = δαγg⟨Hα⟩. (B.2)
Notice that∂n(x)∂H̄α(x)
=Wα
(eβαx−Fα
Nα∆x
)(B.3)
and the fact that in the limit of infinite data
n(x) =eβαx−Fα
Nα∆x⟨H(x)⟩. (B.4)
So we have
⟨(δn)2⟩ =∑α
W2α
eβαx−Fα
Nα∆xgαn(x) (B.5)
Then we can use Lagrange’s Multiplier to minimize this error with respect
to Wα(x) using the constraint∑αWα(x) = 1. And we can get readily
Wα(x) =Nα/gα eFα−βαx∑′αNα′/gα′ eFα′−βα′x
. (B.6)
106
APPENDIX CTRAPEZOIDAL INTEGRATION FOR QUASI-GAUSSIAN FUNCTIONS
The problem is associated with the numerical integration of
I(A) =∫ ∞
−∞e−
12 Ax2
cos xdx (C.1)
using the simple trapezoidal rules which can be described by∫ b
af (x)dx ≈
N−1∑n=1
f (xi)h +f (x0) + f (xN)
2h . (C.2)
It has an error of order∼ 1/N2. However in the following we should prove that for
the integration in Eq.(C.1), the error of the trapezoidal approximation is of order
exp(−kN2) where k is a constant. As a result, trapezoidal rule works exceptionally
well in this type of integral.
To be precise, let us work with the integral with a proper cut
I =∫ ym
−ym
e−12 Ax2
cos xdx. (C.3)
Using 2N trapezes, we can write
IN = hN∑
n=−N
e−12 An2h2
cos(nh), (C.4)
where h = ym/N. It is actually a truncated form of the elliptic theta function
θ3(ν, τ) =∞∑
n=−∞qn2
cos(2nπν), (C.5)
where q = eiπτ. Use the Jacobi Imaginary transformation,
θ3(ν, τ) =√
(i/τ)e−πiν2/τθ3(ντ,−1τ
). (C.6)
107
Let us take τ = −Ah2/(2iπ), ν = h/(2π), we then have
IN→∞ ≈ h
√iτ
e−iπν2/τ∞∑
m=−∞
(e−iπ/τ
)m2
cos(2πmν)
=
√2πA
e−1
2A
(1 + 2e−
2π2
Ah2 cosh(2πAh
) + ...)
=
√2πA
e−1
2A
(1 + 2e−
2π2
Ah2(e
2πAh + e−
2πAh
)+ ...
)=
√2πA
e−1
2A
(1 + e−
2π2
Ah2 +2πAh + ....
). (C.7)
The exact result of the integration is I =√
2πA e−
12A , so now we have
∆I/I = exp(−η(h)2A
), (C.8)
where η(h) = π2
h2 − πh . Since h = ym/N, so ∆I/I ∝ exp(−kN2) where k = π2/(2y2mA).
In the following, we will do a numerical verification. We will use the fact
log(∆I/I) = − 2π2
Ay2m
(N − ym
2π)2 +
12A
(C.9)
and log-log plot of log(∆I/I) − 12A vs (N − ym
2π ) should follow a power law of two.
C.0.5.1 Generalization
1. Shifting
∫ ∞
−∞e−ax2/2 cos(x + δ)dx = cos(δ)
∫ ∞
−∞e−ax2/2 cos(x)dx. (C.10)
2. Sandwiched Polynomials.
In =
∫ ∞
−∞e−ax2/2xn cos(x)dx (C.11)
It is obvious that I2k−1 = 0 where k is an integer. For even orders,
I2n =
(−2∂∂A
)n
I0. (C.12)
108
ç
ç
ç
ç
ç
ççççççççççççççç
3.0 3.5 4.0 4.5 5.0 5.52
3
4
5
6
7
LogHN - ym�2ΠL
LogHLogHDI�IL-1�2AL
Figure C.1: The log-log plot of log(∆I/I) − 12A as a function of the number of
grids used in the Trapezoidal integration. We use A = 0.004069 and cut theintegral range beyond the precision goal at ym = 582.692 The fit gives a linear re-lation of −4.248355148205+ 2.0000000000000x, which agrees with ln((2π2)/(Ay2
m))= −4.2483551482054.
Let ξ(A) = 1/A2 − 1/A, the lowest several orders read
I2 = −ξI0 ,
I4 = −ξI2 + 2ξ′I0 ,
I6 = −ξI4 + 4ξ′I2 − 4ξ′′I0 ,
I8 = −ξI6 + 6ξ′I4 − 12ξ′′I2 + 8ξ′′′I0 .
For A << 1, the errors can be written as
∆I2 ∼ I2η(h)
1 − Ae−
12Aη(h) ,
∆I4 ∼(I4η(h) + I2
η2(h)A2
)e−
12Aη(h).
109
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