Introduction to Strongly Correlated Electron Materials and to Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials.

Introduction to Strongly Correlated Electron Materials and to Dynamical Mean Field

Theory (DMFT).

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

Workshop on Quantum Materials Heron Island Resort New Queensland Australia1-4 June 2005

Outline• Introduction to strongly correlated electrons.• Introduction to Dynamical Mean Field Theory

(DMFT)• First Application. The Mott transition problem.

Theory and experiments.• More realistic calculations. LDA +DMFT. Pu

Am and the Mott transition across the actinide series. Pu and Am

• Cluster Extensions. Application to Cuprate Superconductors.

• Conclusions. Current developments and future directions.

Band Theory: electrons as waves.

Landau Fermi Liquid Theory.

Electrons in a Solid:the Standard Model

•Quantitative Tools. Density Functional Theory+Perturbation

Theory. 2 / 2 ( )[ ] KS kj kj kjV r r y e y- Ñ + =

Rigid bands , optical transitions , thermodynamics, transport………

Success story : Density Functional Linear Success story : Density Functional Linear ResponseResponse

Tremendous progress in ab initio modelling of lattice dynamics& electron-phonon interactions has been achieved(Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001)

C. Urano et. al. PRL 85, 1052 (2000)

The success of the standard model does NOT extend to strongly correlated systems . Anomalies cannot be understood within a RIGID BAND PICTURE,e.g. very resistive metals

Strong Correlation Anomalies : temperature dependence of the integrated optical weight up to high frequency. Violations of low energy optical sum rule. Breakdown of rigid band picture.

Breakdown of standard model

• Large metallic resistivities exceeding the Mott limit. Maximum metallic resistivity 200 ohm cm

• Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics.

• The quantitative tools of the standard model fail, e.g. alpha gamma transition in Cerium, Mott transition in oxides, actinides etc…

2 ( )F Fe k k l

h

Correlated Materials do big things

• Huge resistivity changes V2O3.

• Copper Oxides. (La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 .

• Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000

• (La1-xSrx)MnO3 Colossal Magneto-resistance.

Strongly Correlated Materials.

• Large thermoelectric response in NaCo2-xCuxO4

• Huge volume collapses, Ce, Pu……

• Large and ultrafast optical nonlinearities Sr2CuO3

• Large Coexistence of Ferroelectricity and Ferromagnetism (multiferroics) YMnO3.

Localization vs Delocalization Strong Correlation Problem

• Many interesting compounds do not fit within the “Standard Model”.

• Tend to have elements with partially filled d and f shells. Competition between kinetic and Coulomb interactions.

• Breakdown of the wave picture. Need to incorporate a real space perspective (Mott).

• Non perturbative problem.• Require a framework that combines both

atomic physics and band theory. DMFT.

Two paths for the calculation of electronic structure of materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

1( , ) Im[ ( , )] Im[ ]

( , )k

A k G kk

MODEL HAMILTONIAN AND OBSERVABLES

Limiting case itinerant electrons( ) ( )kk

A

( ) ( , )k

A A k

( ) ( ) ( )B AA

Limiting case localized electrons

Hubbard bands

Local Spectral Function

A BU

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

Parameters: U/t , T, carrier concentration, frustration :

Limit of large lattice coordination1

~ d ij nearest neighborsijtd

† 1~i jc c

d

†

,

1 1~ ~ (1)ij i j

j

t c c d Od d

~O(1)i i

Un n

Metzner Vollhardt, 89

1( , )

( )k

G k ii i

Muller-Hartmann 89

Mean-Field Classical vs Quantum

Classical case Quantum case

A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )

MFo n o n SG c i c is sw w D=- á ñ

1( )[ ]

1( )

( )[ ][ ]

nk

n kn

G ii t

G i

ww m

w

D =D - - +

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,


t c c c c U n n

10G-

Mean-Field Quantum Case

†0 0 0 0 0

H c c Un n

† †0 0 0 0

,

' ( )m l l ll

H t c A A c

Determine the parameters of the mediu t’ so as to get translation invariance on the average. A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)

† †

, ,


H t c c c c U n n

†, ' '

, ',

'm l l l ll l

H t A A

H=Ho +Hm +Hm0

DMFT as an approximation to the Baym Kadanoff functional

[ , , 0, 0, ]

[ ] [ ] [ ]

DMFT

atomij ij i ii ii i ii

Gii ii Gij ij i j

TrLn i t ii Tr G G

[ , ] [ ] [ ] [ ]ij ijG TrLn i t Tr G G

[ ] Sum 2PI graphs with G lines andU G vertices

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of atomic and band physics.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

1( , )

( )k

G k ii i

Pressure Driven Mott transition

How does the electron go from the localized to

the itinerant limit ?

R. Mckenzie, Science 278, 820-821 (1997).

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site

DMFT. High temperature universality

M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site

DMFT. High temperature universality

M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

V2O3:Anomalous transfer of spectral weight

Th. Pruschke and D. L. Cox and M. Jarrell, Europhysics Lett. , 21 (1993), 593

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

Transfer of optical spectral weight

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P

Metcalf Phys. Rev. Lett. 75, 105 (1995)

Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi

2000]

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator. M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

Insulatinganion layer

-(ET)2X are across Mott transition

ET =

X-1

[(ET)2]+1conducting ET layer

t’t

modeled to triangular lattice

t’t

modeled to triangular lattice

Single site DMFT and kappa organics Merino and McKenzie PRB 61, 7996

(2000)and 62 16442 (2000)

Ising critical endpoint! In V2O3

P. Limelette et.al. Science 302, 89 (2003)

. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et al., Phys. Rev.Lett. 90, 186403 (2003).

Conclusions.

• Three peak structure, quasiparticles and Hubbard bands.

• Non local transfer of spectral weight.• Large metallic resistivities.• The Mott transition is driven by transfer of

spectral weight from low to high energy as we approach the localized phase.

• Coherent and incoherence crossover. Real and momentum space.

• Theory and experiments begin to agree on a broad picture.

Collaborators References

• Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996).

• Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005).

• Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

Extensions of Single SiteDMFT and its applications to

correlated materials.

• More realistic calculations. LDA +DMFT. Pu Am and the Mott transition across the actinide series.



•Introduction to strongly correlated electrons.•Introduction to Dynamical Mean Field Theory (DMFT)•First Application. The Mott transition problem. Theory and experiments.

Two paths for calculation of electronic structure of

strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

Dynamical Mean Field Theory• Basic idea: reduce the quantum many body problem to

a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.[A. Georges and GK Phys. Rev. B 45, 6497, 1992].

• Merge atomic physics and band theory. Atom in a medium. Weiss field. = Quantum impurity model.

Solid in a frequency dependent potential. Incorporate band structure and orbital degeneracy to achive a realistic

description of materials. LDA +DMFT. Realistic combination with band theory: LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).

• .

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys.

Cond. Mat. 35, 7359 (1997).

• The light, sp (or spd) electrons are extended, well described by LDA .The heavy, d (or f) electrons are localized treat by DMFT. Use Khon Sham Hamiltonian after substracting the average energy already contained in LDA.

• Add to the substracted Kohn Sham Hamiltonian a frequency dependent self energy, treat with DMFT. In this method U is either a parameter or is estimated from constrained LDA

• • Describes the excitation spectra of many strongly correlated solids. .

Spectral Density Functional• Determine the self energy , the density and the

structure of the solid self consistently. By extremizing a functional of these quantities. (Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, PRB 2005). Coupling of electronic degrees of freedom to structural degrees of freedom. Full implementation for Pu. Savrasov and Kotliar Nature 2001.

• Under development. Functional of G and W, self consistent determination of the Coulomb interaction and the Greens fu

LDA+DMFT Self-Consistency loop

G0 G

Im p u rityS o lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

Edc

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

Pu in the periodic table

actinides

Mott Transition in the Actinide Series

Lashley et.al.

Pu phases: A. Lawson Los Alamos Science 26, (2000)

LDA underestimates the volume of fcc Pu by 30%.

Within LDA fcc Pu has a negative shear modulus.

LSDA predicts Pu to be magnetic with a 5 b moment. Experimentally it is not.

Treating f electrons as core overestimates the volume by 30 %

Pu is not MAGNETIC, alpha and delta have comparable

susceptibility and specifi heat.

Total Energy as a function of volume for Total Energy as a function of volume for Pu Pu W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

Double well structure and Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low

Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ]

DMFT and the Invar Model A. Lawson et. al. LA UR 04-6008 (LANL)

A. C. Lawson et. al. LA UR 04-6008

F(T,V)=Fphonons+F2levels

=125 K =.5 = 1400 KD

Invar model A. C. Lawson et. al. LA UR 04-6008

Phonon Spectra

• Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure.

• Phonon spectra reveals instablities, via soft modes.

• Phonon spectrum of Pu had not been measured.

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

Mott Transition in the Actinide Series

Lashley et.al.

Mott transition into an open (right) and closed (left) shell systems. AmAt room pressure a localised 5f6 system;j=5/2.

S = -L = 3: J = 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

Americium under pressureAmericium under pressure

Density functional based electronic structure calculations: Non magnetic LDA/GGA predicts volume 50% off. Magnetic GGA corrects most of error in volume but gives m~6B (Soderlind et.al., PRB 2000). Experimentally, Am has non magnetic f6 ground state with J=0 (7F0)

Experimental Equation of State (after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Mott transition in open (right) and closed (left) shell systems.

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

J=0

???

Tc

Am under pressure: J.C. GriveauJ. Rebizant G. Lander G. Kotliar PRL (2005)

LDA+DMFT spectra. Notice the rapid occupation of the f7/2 band,

(5f)7

Photoemission Spectrum from Photoemission Spectrum from 77FF00 Americium Americium

LDA+DMFT Density of States

Experimental Photoemission Spectrum(after J. Naegele et.al, PRL 1984)

S. Savrasov et. al. Multiplet Effects

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

J. C. Griveau et. al. (2004)

H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under pressure Griveau et. al.

Superconductivity due to valence fluctuations ?

Conclusions and Outlook

• Motivation: Mott transition in Americium and Plutonium. In both cases theory (DMFT) and experiment suggest a more gradual transformation than postulated in earlier theories.

• DMFT: Physical connection between spectra and structure. Studied the Mott transition from both ends, Studied open and closed shell cases. .

• DMFT: method under construction, but it already gives quantitative results and qualitative insights. It can be systematically improved in many directions. Interactions between theory and experiments.

• Pu: simple picture of alpha delta and epsilon. Interplay of lattice and electronic structure near the Mott transition.

• Am: Rich physics, mixed valence under pressure ? Superconductivity near the Mott transition.

•

Actinides and The Mott PhenomenaEvolution of the electronic structure between the atomic limit and

the band limit in an open shell situation.The “”in between regime” is ubiquitous central theme in

strongly correlated systems.Actinides allow us to probe this physics in ELEMENTS. Mott

transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] . Revisit the problem using a new insights and new techniques from the solution of the Mott transition problem within DMFT in a model Hamiltonian.

Use the ideas and concepts that resulted from this development to give physical qualitative insights into real materials.

Turn the technology developed to solve simple models into a practical quantitative electronic structure method .

More important, one would like to be able to evaluate from the theory itself when the approximation is reliable!! And captures new fascinating aspects of the

immediate vecinity of the Mott transition in two dimensional systems…..

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltonian

cluster exterior exteriorH H

Medium of free electrons :

impurity model.

Solve for the medium using

Self Consistency

G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

Site Cell. Cellular DMFT. C-DMFT. G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

tˆ(K) hopping expressed in the superlattice notations.

•Other cluster extensions (DCA Jarrell Krishnamurthy, M

Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes , causality

issues, O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)

http://www.physics.rutgers.edu/~kotliar/papers/PRB05108.pdf



0

1 2

( , ) ( )

( )(cos cos ) ( )(cos .cos ) .......latt k

kx ky kx ky

Cluster Extensions of Single Site DMFT

U/t=4.

Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev.

Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

Finite T Mott tranisiton in CDMFT Parcollet Biroli and GK PRL, 92, 226402. (2004))

Evolution of the spectral function at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Evolution of the k resolved Spectral Function at zero frequency. (QMC

study Parcollet Biroli and GK PRL, 92, 226402. (2004)) ) ( 0, )vs k A k

Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W

U/D=2 U/D=2.25

Momentum Space Differentiation the high

temperature story T/W=1/88

† †

, ,


t c c c c U n n

Cuprate superconductors and the Hubbard Model . PW Anderson 1987

.

• Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition.

• Earlier studies (Katsnelson and Lichtenstein, M. Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett

85, 1524 (2000) ) used QMC as an impurity solver and DCA as cluster scheme.

• We use exact diag ( Krauth Caffarel 1995 with effective temperature 32/t=124/D ) as a solver and Cellular DMFT as the mean field scheme.

CDMFT study of cuprates

Superconductivity in the Hubbard model role of the Mott transition and influence of the super-

exchange. (M. Capone V. Kancharla. CDMFT+ED, 4+ 8 sites t’=0) .

D wave Superconductivity and Antiferromagnetism t’=0 M. Capone V. Kancharla (see also VCPT Senechal and

Tremblay ). Antiferromagnetic (left) and d wave superconductor (right) Order Parameters

Follow the “normal state” with doping. Evolution of the spectral

function at low frequency.( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Hole doped case t’=-.3t, U=16 t n=.71 .93 .97

Color scale x= .37 .15 .13

K.M . Shen et. al. Science (2005).

For a review Damascelli et. al. RMP (2003)

Approaching the Mott transition: CDMFT Picture

• Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state!

• D wave gapping of the single particle spectra as the Mott transition is approached.

• Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, 226402. (2004) .

CDMFT one electron spectra n=.96 t’/t=.-.3 U=16 t

• i

Experiments. Armitage et. al. PRL (2001).Momentum dependence of the low-energy

Photoemission spectra of NCCO

Comparison with Experiments in Cuprates: Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k

hole doped electron doped

K.M. Shen et.al. 2004 P. Armitage et.al. 2001

2X2 CDMFT Civelli et.al. 2004

Conclusions


• General phenomena, but the location of the cold regions depends on parameters. Study the “normal state” of the Hubbard model is useful.

• On the hole doped normal and superconducting state can be connected to each other as in the RVB scenario. High Tc superconductivity may result follow from doping a Mott insulator phase but it is not necessarily follow from it. One may not be able to connect the Mott insulator to the superconductor if the nodes are in the “wrong place”.

To test if the formation of the hot and cold regions is the result of the

proximity to Antiferromagnetism, we studied various values of t’/t, U=16.

Introduce much larger frustration: t’=.9t U=16t

n=.69 .92 .96

Approaching the Mott transition:


• General phenomena, but the location of the cold regions depends on parameters.

• With the present resolution, t’ =.9 and .3 are similar. However it is perfectly possible that at lower energies further refinements and differentiation will result from the proximity to different ordered states.

Evolution of the real part of the self energies.

Fermi Surface Shape Renormalization ( teff)ij=tij+ Re(ij

Fermi Surface Shape Renormalization

• Photoemission measured the low energy renormalized Fermi surface.

• If the high energy (bare ) parameters are doping independent, then the low energy hopping parameters are doping dependent. Another failure of the rigid band picture.

• Electron doped case, the Fermi surface renormalizes TOWARDS nesting, the hole doped case the Fermi surface renormalizes AWAY from nesting. Enhanced magnetism in the electron doped side.

Understanding the location of the hot and cold regions.

o Qualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near (,/2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping.

o Electron doped case, has in the underlying normal state quasiparticles leave in the ( 0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (/2, /2).

How is the Mott insulatorapproached from the

superconducting state ?

Work in collaboration with M. Capone

Evolution of the low energy tunneling density of state with doping. Decrease of spectral weight

as the insulator is approached. Low energy

particle hole symmetry.

Alternative view

• DMFT is a useful mean field tool to study correlated electrons. Provide a zeroth order picture of a physical phenomena.

• Provide a link between a simple system (“mean field reference frame”) and the physical system of interest. [Sites, Links, and Plaquettes]

• Formulate the problem in terms of local quantities (which we can usually compute better).

• Allows to perform quantitative studies and predictions . Focus on the discrepancies between experiments and mean field predictions.

• Generate useful language and concepts. Follow mean field states as a function of parameters.

• Controlled approach!

Conclusions

Outline• Introduction to strongly correlated electrons.• Introduction to Dynamical Mean Field Theory

(DMFT)• First Application. The Mott transition problem.

Theory and experiments.• More realistic calculations. LDA +DMFT. Pu

Am and the Mott transition across the actinide series. Pu and Am



Collaborators References

• Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996).

• Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005).

• Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

Evidence for unconventional interaction underlying in

two-dimensional correlated electronsF. Kagawa,1 K. Miyagawa,1, 2 & K. Kanoda1, 2

Understanding the result in terms of cluster self energies (eigenvalues)

(0, )

~ ( , )

(0,0)

A

B

A

Systematic Evolution

Dynamical Mean-Field Theory


†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )


1( )[ ]

1( )

( )[ ][ ]

nk

n kn

G ii t

G i

ww m

w

D =D - - +

D

å

† †

, ,


t c c c c U n n


Classical case Quantum case


†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )


1( )[ ]

1( )

( )[ ][ ]

nk

n kn

G ii t

G i

ww m

w

D =D - - +

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,


t c c c c U n n

10G-

DMFT as an approximation to the Baym Kadanoff functional

[ , , 0, 0, ]

[ ] [ ] [ ]

DMFT

atomij ij i ii ii i ii

Gii ii Gij ij i j

TrLn i t ii Tr G G

[ , ] [ ] [ ] [ ]ij ijG TrLn i t Tr G G

[ ] Sum 2PI graphs with G lines andU G vertices

CDMFT vs single site DMFT and other cluster methods.

Cellular DMFT

43

21









Estimates of upper bound for Tc exact diag. M. Capone. U=16t, t’=0, (

t~.35 ev, Tc ~140 K~.005W)

RVB phase diagram of the Cuprate Superconductors

• P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987)

• Connection between the anomalous normal state of a doped Mott insulator and high Tc.

• Baskaran Zhou and Anderson Slave boson approach. <b> coherence order parameter. singlet formation order parameters.

RVB phase diagram of the Cuprate Superconductors. Superexchange.

• The approach to the Mott insulator renormalizes the kinetic energy Trvb increases.

• The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero.

• Superconducting dome. Pseudogap evolves continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

Problems with the approach.

• Numerous other competing states. Dimer phase, box phase , staggered flux phase . Different decouplings, different answers.

• Neel order• Stability of the pseudogap state at finite temperature.

[Ubbens and Lee] • Missing incoherent spectra . [ fluctuations of slave

bosons ]• Temperature dependence of the penetration depth [Wen

and Lee , Ioffe and Millis ] • Theory:[T]=x-Ta x2 , Exp: [T]= x-T a. • Mean field is too uniform on the Fermi surface, in

contradiction with ARPES.

DMFT : What is the dominant atomic configuration ,what is the fate of the atomic moment ?

• Snapshots of the f electron :Dominant configuration:(5f)5

• Naïve view Lz=-3,-2,-1,0,1, ML=-5 B, ,S=5/2 Ms=5 B . Mtot=0

• More realistic calculations, (GGA+U),itineracy, crystal fields ML=-3.9 Mtot=1.1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett., 84, 3670 (2000)

• This moment is quenched or screened by spd electrons, and other f electrons. (e.g. alpha Ce). Contrast Am:(5f)6

Anomalous Resistivity

PRL 91,061401 (2003)

The delta –epsilon transition

• The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase.

• What drives this phase transition?

• LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

Epsilon Plutonium.

Phonon entropy drives the epsilon delta phase transition

• Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

• At the phase transition the volume shrinks but the phonon entropy increases.

• Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Total Energy as a function of volume for Total Energy as a function of volume for Pu Pu W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

Expt. Wong et. al.

Alpha and delta Pu

. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

One Particle Local Spectral Function and

Angle Integrated Photoemission

• Probability of removing an electron and transfering energy =Ei-Ef,

f() A() M2

• Probability of absorbing an electron and transfering energy =Ei-Ef,

(1-f()) A() M2

• Theory. Compute one particle greens function and use spectral function.

e

e

QP in V2O3 was recently found Mo et.al

organics• ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene

(ET)2 X

Increasing pressure ----- increasing t’ ------------

X0 X1 X2 X3• (Cu)2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2• Spin liquid Mott transition

Vanadium Oxide Transport under pressure. Limelette etal

Failure of the StandardModel: Anomalous Spectral Weight Transfer

Optical Conductivity o of FeSi for T=20,40, 200 and 250 K from Schlesinger et.al (1993)

0( )d

Neff depends on T

RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction

1

10

1( ) ( )

V ( )n nk nk

D i ii

w ww

-

-é ùê ú= +Pê ú- Pê úë ûå

0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

†

0 0

( ) ( , ') ( ') ( , ') o o o oc Go c n n Ub b

s st t t t d t t ¯+òò

† †

, ,


t c c c c U n n

()

1 100 0 0( )[ ] ( ) [ ( ) ( ) ]n n n n Si G D i n i n iw w w w- -P = + á ñ

,ij i j

i j

V n n

0 0( , ')Do n nt t+










Quantum case


†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD†

( )( ) ( )MFo n o n SG c i c is sw w D=- á ñ

1( )[ ]

1( )

( )[ ][ ]

nk

n kn

G ii t

G i

ww

w

D =D - -

D

å

† †

, ,


t c c c c U n n

1( )] ( )

( )[ ]

1( )[ ]

( )]

[

[[ ]

n n nn

nk n n k

i i iG i

G ii i t

w m w ww

ww m w

+ - S =D -D

D =+ - S -å

Realistic DMFT loop

( )k LMTOt H k E® -LMTO

LL LH

HL HH

H HH

H H

é ùê ú=ê úë û

ki i Ow w®

10 niG i Ow e- = + - D

0 0

0 HH

é ùê úS =ê úSë û

0 0

0 HH

é ùê úD =ê úDë û

0

1 †0 0 ( )( )[ ] ( ) [ ( ) ( )HH n n n n S Gi G G i c i c ia bw w w w-S = + á ñ

110

1( ) ( )

( ) ( ) HH

LMTO HH

n nn k nk

G i ii O H k E i

w ww w

--é ùê ú= +Sê ú- - - Sê úë ûå

Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme. Causality issues O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)




Introduction to Strongly Correlated Electron Materials and to Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials.

Documents

band theory

transport slide

correlated materials

resistive metals slide

electron materials

x cu x o

x ba x cuo

mott transition problem