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Introduction to Strongly Correlated Electron Materials and to Dynamical Mean Field Theory (DMFT).

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### 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 Australia
- 1-4 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.

Electrons in a Solid:the Standard Model

Band Theory: electrons as waves.

Landau Fermi Liquid Theory.

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

- Quantitative Tools. Density Functional Theory+Perturbation Theory.

Success story : Density Functional Linear Response

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)

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

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

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 mohm 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…

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

Crystal structure +Atomic positions

Model Hamiltonian

Correlation Functions Total Energies etc.

MODEL HAMILTONIAN AND OBSERVABLES

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

Local Spectral Function

Limiting case itinerant electrons

Limiting case localized electrons

Hubbard bands

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)

Mean-Field Quantum Case

H=Ho

+Hm

+Hm0

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)

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)

Pressure Driven Mott transition

R. Mckenzie,

Science 278, 820-821 (1997).

How does the electron go from the localized to the itinerant limit ?

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)

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

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)

modeled to triangular lattice

t

t

t’

t’

k-(ET)2X are across Mott transitionET =

Insulating

anion layer

X-1

conducting

ET layer

[(ET)2]+1

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-xSexMatsuura 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.
- Cluster Extensions. Application to Cuprate Superconductors.
- Conclusions. Current developments and future directions.

- 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

Crystal structure +Atomic positions

Model Hamiltonian

Correlation Functions Total Energies etc.

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

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 d Pu to be magnetic with a 5 ub 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 Pu W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)

Non magnetic correlated state of fcc Pu.

Zein Savrasov and Kotliar (2004)

Double well structure and d 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 ]

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

C44 (GPa)

C12 (GPa)

C'(GPa)

Theory

34.56

33.03

26.81

3.88

Experiment

36.28

33.59

26.73

4.78

DMFT Phonons in fcc d-Pu

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

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

.g T

Log[2J+1]

???

Uc

S=0

U

U

g ~1/(Uc-U)

Experimental Equation of State

(after Heathman et.al, PRL 2000)

“Soft”

Mott Transition?

“Hard”

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~6mB

(Soderlind et.al., PRB 2000).

- Experimentally, Am hasnon magnetic f6ground state with J=0(7F0)

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

S

S

g T

Tc

Log[2J+1]

???

Uc

J=0

U

U

g ~1/(Uc-U)

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

Photoemission Spectrum from 7F0 Americium

LDA+DMFT Density of States

S. Savrasov et. al. Multiplet Effects

F(0)=4.5 eV

F(2)=8.0 eV

F(4)=5.4 eV

F(6)=4.0 eV

Experimental Photoemission Spectrum

(after J. Naegele et.al, PRL 1984)

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 Phenomena

Evolution 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…..

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)

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) ]

U/t=4.

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

Evolution of the spectral function at low frequency.

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.

Evolution of the k resolved Spectral Function at zero frequency. (QMC study Parcollet Biroli and GK PRL, 92, 226402. (2004)) )

U/D=2.25

U/D=2

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

.

- 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.

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.

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.

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) .

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

P. Armitage et.al. 2001

K.M. Shen et.al. 2004

2X2 CDMFT

Civelli et.al. 2004

Conclusions

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

Approaching the Mott transition:

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

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.

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 (p/2,p/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.

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

- 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!

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.

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 unconventionalinteraction underlying intwo-dimensional correlated electronsF. Kagawa,1 K. Miyagawa,1, 2 & K. Kanoda1, 2

Dynamical Mean-Field Theory

A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)

Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

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)

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)

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.
- k, D singlet formation order parameters.

RVB phase diagram of the Cuprate Superconductors. Superexchange.

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

- 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.

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:r[T]=x-Ta x2 , Exp: r[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 mB, ,S=5/2 Ms=5 mB . Mtot=0
- More realistic calculations, (GGA+U),itineracy, crystal fields G7 +G8, 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.

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 Pu W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)

Non magnetic correlated state of fcc Pu.

Zein Savrasov and Kotliar (2004)

ARPES measurements on NiS2-xSexMatsuura 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

e

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

f(w) A(w) M2

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

(1-f(w)) A(w) M2

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

n

n

e

k organics

- ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene
- K (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

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)

Neff depends on T

RESTRICTED SUM RULES

Below energy

ApreciableT dependence found.

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

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)

Mean-Field Classical vs Quantum

Quantum case

A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)

Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

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)

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