Dynamical Mean Field Theory (DMFT) Approach to Strongly

Correlated Materials

G. KotliarPhysics Department and Center for

Materials TheoryRutgers

SCES04 Karlsruhe Germany

Outline

• Dynamical Mean Field ideas and techniques. Local Reference Systems [ sites, links and plaquettes]. Model Hamiltonians and First Principles Framework.

• Applications:temperature driven Mott transition, kappa organics, titanium sesquioxide, plutonium.

• Collaborators: (cluster extensions)G. Biroli M. Civelli,M Capone, V. Kancharla,O. Parcollet. (realistic studies) K. Haule, S. Savrasov, A. Lichtenstein, A. Poteryaev,N. Zein V. Udovenko.

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!

Weakly correlated electrons. FLT and DFT, and what goes wrong in

correlated materials. • Fermi Liquid . . Correspondence between a

system of non interacting particles and the full Hamiltonian.

• A band structure is generated (Kohn Sham system).and in many systems this is a good starting point for perturbative computations of the spectra (GW).

[ ( ) , ( ) ]r r

DMFT Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992). Figure from : G. Kotliar and D.

Vollhardt Physics Today 57,(2004)http://www.physics.rutgers.edu/~kotliar/RI_gen.html

The self consistent impurity model is a new reference system, to describe strongly

correlated materials.

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

band theory.

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

Local Self Energy

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).

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, Katsnelson and Lichtenstein periodized scheme, Nested

Cluster Schemes Schiller Ingersent ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

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.

How good is the local approximation ?

• It becomes exact as the coordination number increases or in the limit of infinite dimensions introduced by Metzner and Vollhardt. PRL 62,34, (1989).

• How good is it in low dimensions ? Promising recent developments from theory and experiments.

One dimensional Hubbard model .Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats,

[V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][ [M. CaponeM.Civelli V Kancharla C.Castellani and GK Phys. Rev. B 69, 195105

(2004) ]

U/t=4.

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.

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 Self-Consistency loop. S. Savrasov and G. Kotliar

(2001) and cond-matt 0308053

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

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

Next Step: GW+EDMFT S. Savrasov and GK.(2001). in New Theoretical Approaches to Strongly Correlated Systems, A.M. Tsvelik

Ed., Kluwer Academic Publishers 259-301, (2001))

.P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002) Phys. Rev. Lett. 91, 037209 (2003) Biermann et.al. PRL 90,086402 (2003)

W

W

Pressure Driven Mott transition

Schematic DMFT phase diagram and DOS of a partially frustrated integer filled Hubbard

model and pressure driven Mott transition.

Physics Today Vol 57, 53 (2004)

Schematic DMFT phase diagram of a partially frustrated integered filled Hubbard model. M. J. Rozenberg, G. Kotliar, H.

Kajueter, G. A. Thomas, D. H. Rapkine, J. M. Honig, and P. Metcalf, Phys. Rev. Lett. 75, 105, 1995

A different paradigm: the area of influence of a quantum critical point

Energy Landscape of a Correlated Material and a top to bottom

approach to correlated materials.

Energy

Configurational Coordinate in the space of Hamiltonians

T

Spectral Evolution at T=0 half filling full frustration figure from X.Zhang M. Rozenberg G.

Kotliar (PRL 70,16661993)

• Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features.

• Mott transition is driven by transfer of spectral weight.

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)

Consequences for the optical conductivity Evidence

for QP peak in V2O3 from optics.

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

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)

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

2000]

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).

Anomalous Resistivity and Mott transition Ni Se2-x Sx

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

Ising critical endpoint! In V2O3 P.

Limelette et.al. (2003)

Insulatinganion layer

-(ET)2X are across Mott transitionET =

X-1

[(ET)2]+1conducting ET layer

t’t

modeled to triangular lattice

X- Ground State

U/t t’/t

Cu2(CN)3Mott insulator

8.2 1.06

Cu[N(CN)2]Cl Mott insulator

7.5 0.75

Cu[N(CN)2]Br SC 7.2 0.68

Cu(NCS)2SC 6.8 0.84

Cu(CN)[N(CN)2]

SC 6.8 0.68

Ag(CN)2 H2O SC 6.6 0.60

I3SC 6.5 0.58

Prof. Kanoda U. Tokyo

Mott transition in layered organic conductors S Lefebvre et al.

cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)

• Theoretical issue: is there a Mott transition

in the integer filled Hubbard model, and is it

well described by the single site DMFT ?

Double Occupancy vs U

• CDMFT Parcollet, Biroli GK PRL (2004) Study frustrated t

t’ model t’/t=.9

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

Evolution of the k resolved Spectral Function at zero

frequency. (Parcollet Biroli and GK)

Uc=2.35+-.05, Tc/D=1/44

U/D=2 U/D=2.25

( 0, )vs k A k

Near the transition k dependence is strong.

• Qualitative effect, formation of hot regions!• D wave gapping of the single particle

spectra as the Mott transition is approached. New paradigm for thinking about the approach to the Mott insulator.

• Square symmetry is restored as we approched the insulator.

• Experimental predictions! Photoemission ?

Lattice and cluster self energies

Mechanism for hot spot formation: nn self energy ! General phenomena.

Conclusion.

• Mott transition survives in the cluster setting. Role of magnetic frustration.

• Surprising result: formation of hot and cold regions as a result of an approach to the Mott transition. General result ?

• Unexpected role of the next nearest neighbor self energy. CDMFT a new window to extend DMFT to lower temperatures.

Ti2O3 : Coulomb or Pauling

C.E.Rice et all, Acta Cryst B33, 1342 (1977) LTS 250 K, HTS 750 K.

Ti2O3.

• Isostructural to V2-xCrxO3. Al lot of the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Is this true in Ti2O3?

• Band Structure Calculations good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996) .Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, et. al. Phys. Rev. B. f55 , 16122 (1997).

U = 2, J = 0.5, W = 0.5β = 20 eV-1, LT structure

U = 2, J = 0.5, W = 0.5 β = 10 eV-1, HT

structure

2site-Cluster DMFT with intersite Coulomb2site-Cluster DMFT with intersite Coulomb

A. Poteryaev

Pauling and Coulomb Ti2O3[S. Poteryaev S. Lichtenstein and GK PRL (2004)

Dynamical Goodenough-Honig Pauling picture

Conclusion Ti2O3• 2 site cluster DMFT describes the MIT in Ti2O3.

Different from V2O3 [A. Poteryaev]• Coulomb interactions, frequency dependent

enhancement of the a1g-a1g hopping, [Haldane Ph.D thesis, Q Si and GK 1993 ].Dynamical Pauling-Goodenough mechanism triggers the MIT at low enough temperatures.

• Coulomb and Pauling synergistically cooperate. General paradigm for other materials which have dimers in the unit cell ? VO2 ?

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

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)

DMFT –Hubbard I. Phonons in fcc DMFT –Hubbard I. 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 mapping onto “self consistent impurity models” offer a new “reference frame”, to think about correlated materials and compute their physical properties.Formal parallel with DFT.

• .Plaquettes-Kappa organics-Hot and cold regions.

• Titanium sesquioxides. Dynamical Pauling Goodenough mechanism.

• Sites. Phonons in Plutonium. Mott transition across the actinide series.