Dynamical Mean Field Theory of the Mott Transition

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

Jerusalem Winter School

January 2002

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OUTLINE OF THE COURSE

Motivation . Electronic structure of correlated materials, limiting cases and open problems. The standard model of solids and its failures.

Introduction to the Dynamical Mean Field Theory (DMFT). Cavity construction. Statistical Mechanical Analogies. Lattice Models and Quantum Impurity models. Functional derivation.

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Outline

The limit of large lattice coordination. Ordered phases. Correlation functions.

Techniques for solving the Dynamical Mean Field Equations. [ Trieste School June 17-22 2002]

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Outline The Mott transition. Early ideas. Brinkman Rice.

Hubbard. Slater. Analysis of the DMFT equations: existence of a Mott

transition.

The Mott transition within DMFT. Overview of some important results of DMFT studies of the Hubbard Model. Electronic Structure of Correlated Materials. Canonical Phase diagram of a fully frustrated Hubbard model. Universal and non universal aspects of the physics of strongly correlated materials.

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Outline

Analysis of the DMFT equations. Existence of a Mott transition. Analysis from large U and small U.

The destruction of the metallic phase. Landau analysis. Uc1 . Uc2.

The Mott transition endpoint. A new look at experiments.

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Outline

The electronic structure of real materials.

Examples of problems where DMFT gives new insights, and quantitative understanding: itinerant ferromagnetism, Fe, Ni. Volume collapse transitions, actinide physics. Doping driven Mott transition titanites.

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Outline

New directions, beyond single site DMFT.

Realistic Theories of Correlated Materials

ITP, Santa-Barbara workshop

July 29 – December 16 (2002)

O.K. Andesen, A. Georges,

G. Kotliar, and A. Lichtenstein

Contact: kotliar@physics.rutgers.edu

Conference: November 25-29, (2002)

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The promise of Strongly Correlated Materials

Copper Oxides. High Temperature Superconductivity.

Uranium and Cerium Based Compounds. Heavy Fermion Systems.

(LaSr)MnO3 Colossal Magnetoresistence.

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The Promise of Strongly Correlated Materials.

High Temperature Superconductivity in doped filled Bucky Balls (B. Battlog et.al Science)

Thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017).

Large Ultrafast Optical Nonlinearities Sr2CuO3 (T Ogasawara et.al cond-mat 000286)

Theory will play an important role in optimizing their physical properties.

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Drude 2ne

m

Sommerfeld

Bloch, Periodic potential

Bands, k in Brillouin zone

2 ( )F Fe k k l

h

How to think about the electron in a solid?

Maximum metallic resistivity 200 ohm cm

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Standard Model

High densities, electron as a wave, band theory, k-spaceLandau: Interactions Renormalize AwayOne particle excitations: quasi-particle bandsDensity Functional Theory in Kohn ShamFormulation, successful computational tool for total energy, and

starting pointFor perturbative calculation of spectra, Si Au, Li, Na

……………………

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Standard Model : Metals

2

~

H

V

const

const

R

T

S T

C T

Hall Coefficient

Resistivity

Thermopower

Specific Heat

Susceptibility

Predicts low temperature dependence of thermodynamics and transport

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Quantitative Tools:

Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, GW, transport.……………………

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Mott : correlations localize the electron

Ba

Array of hydrogen atoms is insulating if a>>aB

e_ e_ e_ e_

Superexchange

Ba

Think in real space , atoms

High T : local moments

Low T: spin orbital order

1

T

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Mott : Correlations localize the electron

Low densities, electron behaves as a particle,use atomic physics, real space

One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….)

Rich structure of Magnetic and Orbital Ordering at low T

Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.

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Localization vs Delocalization Strong Correlation Problem

•A large number of compounds with electrons which are not close to the well understood limits (localized or itinerant).•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA or LDA+U or Hartree Fock works well•Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands

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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455

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Failure of the Standard Model: NiSe2-xSx

Miyasaka and Takagi (2000)

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Failure of the standard model : Anomalous Resistivity:LiV2O4

Takagi et.al. PRL 2000

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Failure of the StandardModel: Anomalous Spectral Weight TransferOptical Conductivity of FeSi for T=,20,20,250 200 and 250 K from Schlesinger et.al (1993)

0( )d Neff

0( )d

Neff depends on T

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Strong Correlation Problem

Large number of compounds (d,f,p….). Departure from the standard model.

Hamiltonian is known. Identify the relevant degrees of freedom at a given scale.

Treat the itinerant and localized aspect of the electron

The Mott transition, head on confrontation with this issue

Dynamical Mean Field Theory simplest approach interpolating between that bands and atoms

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Hubbard model

† †

, ,

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

t c c c c U n n

U/t

Doping d or chemical potential

Frustration (t’/t)

T temperatureMott transition as a function of doping, pressure temperature etc.

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Limit of large lattice coordination

1~ d ij nearest neighborsijt

d

† 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, 891

( , )( )k

G k ii i

Muller-Hartmann 89

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Mean-Field : Classical

,ij i j i

i j i

J S S h S- -å åMF eff oH h S=-

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

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1( )

kk

D zz t

é ùê ú=ê ú-ê úë ûå

)

DMFT Impurity cavity construction: [A. Georges, G. Kotliar, PRB, (1992)]

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

s st t t t ­ ¯= +òò ò

† †

, ,

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

t c c c c U n n

0

†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ

1( ) ( ) [ ( )]n n ni G i R G iw w w-D =- +

Weiss field10 ( ) ( )n n nG i i iw w m w- = + - D

[R

[ ( )]R D z z=)

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Comments on DMFT

Exact in both atomic and band limits Weiss field is a function Multiple energy scales in a correlated

electron problem, non linear coupling between them.

Frezes spatial fluctuations but treats quantum fluctuations exactly, local view of the quantum many body problem.

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Example: semicircular DOS

2 1[ ( )]R D z t z

z= +

)

0

2 †( )( ) ( ) ( )n o n o n S Gi t c i c iw w wD = á ñ

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

s st t t t ­ ¯= +òò ò

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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: [A. Georges, G. Kotliar, PRB, (1992)]

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

s st t t t ­ ¯= +òò ò

† †

, ,

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

t c c c c U n n

0

†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ

10 ( ) ( )n n nG i i iw w m w- = + - D

0

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

Weiss field

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Solving the DMFT equations

G 0 G

I m p u r i t yS o l v e r

S . C .C .

•Wide variety of computational tools (QMC,

NRG,ED….)•Analytical Methods

G0 G

Im puritySo lver

S .C .C .

( )niwD

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Mean-Field : Classical vs Quantum

Classical case Quantum case

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

†

0 0 0

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

s st m t t tt ­ ¯

¶+ - D - +

¶òò ò

( )wD†

( )( ) ( )MFL o n o n HG c i c iw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=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

† †

, ,

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

t c c c c U n n

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Single site DMFT, functional formulation

Express in terms of Weiss field (semicircularDOS)

The Mott transition as bifurcation point in functionals oG or F[], (G. Kotliar EPJB 99)

[ , ] log[ ] ( ) ( ) [ ]ijn n nG Tr i t Tr i G i Gw w w-GS =- - S - S +F

† †,

2

2

[ , ] ( ) ( ) ( )†

( )[ ] [ ]

[ ]loc

imp

L f f f i i f i

imp

iF T F

t

F Log df dfe

[ ]DMFT atom ii

i

GF = Få

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DMFT for lattice hamiltonians

k independent k dependent G, Local Approximation Treglia et. al 1980

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How to compute

View locally the lattice problem as a (multiorbital) Anderson impurity model

The local site is now embedded in a medium characterized by

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How to determine the medium Use the impurity model to compute and

the impurity local Greens function. Require that impurity local Greens function equal to the lattice local Greens function.

Weiss field

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Response functions is q dependent but irreducible vertex is momentum independent

is q dependent but irreducible vertex is momentum independent

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Evaluation of the Free energy.

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Solving the DMFT equations

G 0 G

I m p u r i t yS o l v e r

S . C .C .

•Wide variety of computational tools (QMC,

NRG,ED….)•Analytical Methods

G0 G

Im puritySo lver

S .C .C .

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Review of DMFT, technical toolsfor solving DMFT eqs.., applications, references……

A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

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DMFT: Methods of Solution

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Mott transition: Early ideas. Half filling. Evolution of the one electron spectra [physical

quantity measured in photoemission and BIS] as a function of control parameters. ( U/t, pressure, temperature )

Hubbard, begin in paramagnetic insulator.

As U/t is reduced Hubbard bands merge.

Gap closure. Mathematical description, closure of equations of motion, starting from atoms (I.e. large U). Incoherent motion, no fermi surface.

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Mott transition: early ideas.

Brinkman and Rice. Gutzwiller.

Begin in paramagnetic metallic state, as U/t approaches a critical value the effective mass diverges. Luttinger fermi surface.

Mathematical description, variational wave function, slave bosons, quantum coherence and double occupancy.

*

1( )Z Uc U

m

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Slave bosons: mean field +fluctuations Fluctuations of the slave bosons around the

saddle point gives rise to Hubbard bands. Starting from the insulating side, in a

paramagnetic state, the gap closes at the same U, where Z vanishes.

No satisfactory treatement of finite temperature properties.

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Mott vs Slater

Mott: insulators in the absence of magnetic long range order.

e.g. Vanadium Oxide Nickel Oxide. Mott transition in the paramagnetic state .

• Slater: insulating behavior as a consequence of antiferromagnetic long range order. Double the unit cell to convert a Mott insulator into a band insulator.

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A time-honored example: Mott transition in V2O3 under pressure or chemical substitution on V-site

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Local view of the spectral function Partition function of the Anderson impurity

model : gas of kinks [Anderson and Yuval]

( )i i

Insulating state

Metallic state,

( ) si ( )i i gn Metallic state, proliferation of kinks.

Insulating state. Kinks are confined.

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Local view of the spectral function.

Consistent treatement of quasiparticles and collective modes.

Kinky paths, with may spin fluctuations: low energy resonance [Abrikosov Suhl Resonance]

Confined kinks, straight paths, Hubbard bands. [control the insulator partition function]

Strongly correlated metal has both.

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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Spectral Evolution at T=0 half filling full frustration

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Destruction of the metal2

*

1( )Z Uc U

m

The gap is well formed at Uc2, when the metal is destroyed.

Hubbard bands are well formed in the metal.

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Parallel development: Fujimori et.al

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Destruction of the insulator

Continue the insulating solution below Uc2. Coexistence of two solutions between Uc1

and Uc2 Mott Hubbard gap vanishes linearly at Uc1.

( 1)gap U Uc

( 1)gap U Uc

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Recent calculation of the phase diagram of the frustrated Half filled Hubbard model with semicircular DOS (QMC Joo and Udovenko PRB2001).

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Case study: IPT half filled Hubbard one band

(Uc1)exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar 1995) , (Uc1)IPT =2.4

(Uc2)exact =2.95 (Projective self consistent method, Moeller Si Rozenberg Kotliar PRL 1995 ) (Uc2)IPT =3.3

(TMIT ) exact =.026+_ .004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =.5

(UMIT )exact =2.38 +- .03 (QMC Rozenberg Chitra and Kotliar PRL 1991), (UMIT )IPT =2.5 For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude).

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Schematic DMFT phase diagram Hubbard model (partial frustration) Rozenberg et.al. PRL (1995)

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Kuwamoto Honig and AppellPRB (1980)

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Phase Diag: Ni Se2-x Sx

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Insights from DMFT Low temperatures several competing phases . Their relative stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT

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Insights from DMFTThe Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl parameters: doping, temperature,pressure…

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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg 2000)

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. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690

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Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )

Resistivity near the metal insulator endpoint ( Rozenberg et. Al 1995) exceeds the Mott limit

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Anomalous Resistivity and Mott transition Ni Se2-x Sx

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Anomalous resisitivity near Mott transition.

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Anomalous transfer of spectral weight in v2O3

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Anomalous transfer of spectral weight in v2O3

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Anomalous transfer of spectral weight in v2O3

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Anomalous transfer of spectral weight in heavy fermions [Rozenberg etal]

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Insights from DMFT Mott transition as a bifurcation of an

effective action

Important role of the incoherent part of the spectral function at finite temperature

Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. [Non local in frequency] Real and momentum space.

–

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Anomalous Resistivity:LiV2O4

Takagi et.al. PRL 2000

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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455

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Standard Model

Typical Mott values of the resistivity 200 Ohm-cm

Residual instabilites SDW, CDW, SC

Odd # electrons -> metal

Even # electrons -> insulator

Theoretical foundation: Sommerfeld, Bloch and Landau

Computational tools DFT in LDA

Transport Properties, Boltzman equation , low temperature dependence of transport coefficients2 ( )

MottF Fe k k l

h

Mott

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Failure of the “Standard Model”: Cuprates

Anomalous Resistivity

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DMFTFormulation as an electronic structure method (Chitra and Kotliar)Density vs Local Spectral FunctionExtensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)

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DMFT Spin Orbital Ordered StatesLonger range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar,)Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell et.al., C-DMFT Kotliar et. al ).

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Strongly Correlated Electrons

Competing Interaction

Low T, Several Phases Close in Energy

Complex Phase Diagrams

Extreme Sensitivity to Changes in External Parameters

Need for Quantitative Methods

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Landau Functional

† †,

2

2

[ , ] ( ) ( ) ( )†

† † † †

0

†

Mettalic Order Para

( )[ ] [ ]

mete

[ ]

[ , ] [ [ ] ]

( )( )

r: ( )

( ) 2 ( )[ ]( )

loc

LG imp

L f f f i i f i

imp

loc f

imp

iF T F

t

F Log df dfe

dL f f f e f Uf f f f d

d

F iT f i f i TG i

i

i

2

2

Spin Model An

[ ] [[ ]2 ]

alogy:

2LG

t

hF h Log ch h

J

G. Kotliar EPJB (1999)

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LDA functional

2log[ / 2 ] ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

n KS KS

LDAext xc

Tr i V V r r dr

r rV r r dr drdr E

r r

w r

r rr r

- +Ñ - -

+ +-

ò

ò ò

[ ( )]LDA r

[ ( ), ( )]LDA KSr V r

Conjugate field, VKS(r)

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Minimize LDA functional

[ ]( )( ) ( ) '

| ' | ( )

LDAxc

KS ext

ErV r V r dr

r r r

d rrdr

= + +-ò

0*2

( ) { )[ / 2 ]

( ) ( ) n

n

ikj kj kj

n KSkj

r f tri V

r r ew

w

r e yw

y +=

+Ñ -=å å

Kohn Sham eigenvalues, auxiliary quantities.

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Ising character of the transfer of spectral weight

Ising –like dependence of the photo-emission intensity and the optical spectral weight near the Mott transition endpoint

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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Spectral Evolution at T=0 half filling full frustration

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Parallel development: Fujimori et.al