Strongly Correlated Electron Systems a Dynamical Mean Field Perspective

Strongly Correlated Electron Systems a Dynamical Mean

Field Perspective

G. KotliarPhysics Department and Center for

Materials TheoryRutgers

5th International Conference on Inelastic X-Ray Scattering. Argonne National Labs Chicago September 20 2004

Outline

• Introduction to the concepts of the dynamical mean field method.

• Application: The temperature driven Mott transition. Theoretical predictions, and experiments. IXS ?

• Application: elemental Pu. DMFT predictions, and a key IXS experiment.

Electronic states in weakly and strongly correlated materials

• Simple metals, semiconductors. Fermi Liquid Description: Quasiparticles and quasiholes, (and their bound states ). Computational tool: Density functional theory + perturbation theory in W, GW method.

• Correlated electrons. Atomic states. Hubbard bands. Narrow bands. Many anomalies.

• Need tool that treats Hubbard bands, and quasiparticle bands, real and momentum space on the same footing. DMFT!

Strongly Correlated Electron Systems Display remarkable phenomena, that cannot be understood within the standard model of

solids. Resistivities that rise without sign of saturation beyond the Mott limit, (e.g. H. Takagi’s work on Vanadates), temperature

dependence of the integrated optical weight up to high frequency (e.g. Vandermarel’s work on Silicides).

Correlated electrons do “big things”, large volume collapses, colossal magnetoresitance, high temperature superconductivity . Properties are

very sensitive to structure chemistry and stoichiometry, and control parameters large non linear susceptibilites,etc……….

THE WHY

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

THE HOWTHE HOW

DYNAMICAL MEAN FIELD THEORY.

"Optimal Gaussian Medium " + " Local Quantum Degrees of Freedom " + "their interaction "

is a good reference frame for understanding, and predicting physical propertiesof correlated materials. Focus on local quantities, construct functionals of those quantities, similarities with

DFT.

How to think about their electronic states ?

How to compute their properties ?Mapping onto connecting their

properties, a simpler “reference system”. A self consistent impurity

modelliving on SITES, LINKS and

PLAQUETTES......

Need non perturbative tool.

Two paths for ab-initio 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.

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltonian

cluster exterior exteriorH H

Dynamical Mean Field Theory (DMFT) Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992). C.DMFT G. Kotliar et. al. Phys. Rev. Lett 87,186401 (2001).

One dimensional Hubbard model 2 site (LINK) CDMFT compare with Bethe Anzats, [V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.CaponeM.Civelli V Kancharla C.Castellani and GK P. R B 69,195105 (2004) ]

U/t=4.

A rapidly convergent algorithm ?

Functional formulation. Chitra and Kotliar (2001), Savrasov and Kotliarcond- matt0308053 (2003).

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( ')G R Ry r y r=- < > ( ') ( ) ( ') ( )R R R R Wf r f r f r f r< >- < >< >=

Ir>=|R, >

[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

1 1 1 10

1 1[ , ] [ ] [ ] [ , ]

2 2 C hartreeG W TrLnG Tr G G G TrLnW Tr V W W E G W

Double loop in Gloc and Wloc

Impurity model representability of spectral density functional.

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin

and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988).

• The light, SP (or SPD) electrons are extended, well described by LDA .The heavy, D (or F) electrons are localized treat by DMFT.

• LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term)

Kinetic energy is provided by the Kohn Sham Hamiltonian (sometimes after downfolding ). The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.

What did we learn ? Schematic DMFT phase diagram and DOS of a partially frustrated integer filled

Hubbard model and pressure driven Mott transition.

M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75,

105 (1995)

T

How do we know there is some truth in this

picture ? Qualitative Predictions Verified • Two different features in spectra. Quasiparticles

bands and Hubbard bands.• Transfer of spectral weight which is non local in

frequency. Optics and Photoemission.• Two crossovers, associated with gap closure

and loss of coherence. Transport.• Mott transition endpoint, is Ising like, couples to

all electronic properties. • Recently numerical approaches in two

dimensions found the first order line(M. Imada), C-DMFT 4 site studies (Parcollet et. al.).

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)

Anomalous transfer of optical spectral weight V2O3

:M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996).

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

Optical transfer of spectral weight , kappa organics. Eldridge, J., Kornelsen, K.,Wang, H.,Williams, J.,

Crouch, A., and Watkins, D., Sol. State. Comm., 79, 583 (1991).

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

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

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

2000]

Ising critical endpoint found! In V2O3 P.

Limelette et.al. (Science 2003)

Ising critical endpoint found! In V2O3 P.

Limelette et.al. (Science 2003)

Why does it work: Energy Landscape of a Correlated Material and a top to bottom

approach to correlated materials.

Energy

Configurational Coordinate in the space of Hamiltonians

T

Single site DMFT. High temperature universality vs low temperature

sensitivity to detail for materials near a temperature-pressure driven

Mott transition

What did we gain?

• Conceptual understanding of how the electronic structure evolves when the electron goes from localized to itinerant.

• Uc1 Uc2, transfer of spectral weight, ….• A general methodology which was extended to

clusters (non trivial!) and integrated into an electronic structure method, which allows us to incorporate structure and chemistry. Both are needed away from the high temperature universal region.

• Mott transition across the 5f’s, a very interesting playground for studying correlated electron phenomena.

• DMFT ideas have been extended into a framework capable of making first principles first principles studies of correlated materials. Pu Phonons. Combining theory and experiments to separate the contributions of different energy scales, and length scales to the bonding

• In single site DMFT , superconductivity is an unavoidable consequence when we try to go move from a metallic state to a Mott insulator where the atoms have a closed shell (no entropy). Realization in Am under pressure ?

DMFT Phonons in fcc DMFT Phonons in fcc -Pu: connect -Pu: connect bonding to energy and length scales.bonding to energy and length scales.

  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)

Where do we go now ?• One can study a large number of experimentally

relevant problems within the single site framework.

• Continue the methodological development, we need tools!

• Solve the CDMFT Mott transition problem on the plaquette problem, hard, but it is a significant improvement, the early mean field theories while keeping its physical appeal.

• Study material trends, make contact with phenomenological approaches, doped semiconductors (Bhatt and Sachdev), heavy fermions , 115’s(Nakatsuji, Pines and Fisk )……

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)

Pu in the periodic table

actinides

Mott transition in the actinide series (Smith Kmetko phase diagram)

Electronic Physics of Pu

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Elastic Deformations

In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c44/c’=1.2, in Pu C44/C’ ~ 6 largest shear anisotropy of any element.

Uniform compression:p=-B V/V Volume conserving deformations:

F/A=c44 x/L F/A=c’ x/L

Anomalous Resistivity

2 ( )F Fe k k l

h

Maximum metallic resistivity 2

Fe k

h

Specific heat and susceptibility.

Delta phase of Plutonium: Problems with LDA

o Many studies and implementations.(Freeman, Koelling 1972)APW methods, ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999).all give an equilibrium an equilibrium volume of the volume of the phasephaseIs 35% lower than Is 35% lower than experiment experiment this is the largest discrepancy ever known in DFT based calculations.

• LSDA predicts magnetic long range (Solovyev et.al.) Experimentally Pu is not magnetic.

• If one treats the f electrons as part of the core LDA overestimates the volume by 30%

DFT Studies of Pu • DFT in GGA predicts correctly the volume of

the phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that Pu is a weakly correlated system. Experimentally, there are clear signs of electron correlation in Pu .

.

Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001)

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)

Lda vs Exp Spectra

Alpha and delta Pu

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 until recently.

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

Expt. Wong et. al.

Expts’ Wong et. al.

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)

Conclusion• DMFT. Electronic Structure Method in

Development. a) quantitative results b) qualitative understanding by linking real materials to impurity models. Concepts to think about correlate materials.

• System specific. Many materials to be studied, realistic matrix elements for each spectroscopy. Optics. IXS.

• Interplay of theory and experiment. DMFT can enhance joint theoretical- experimental advances in the field of correlated electron materials.

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.

Transverse Phonon along (0,1,1) in epsilon Pu in self

consistent Born approximation.