Introduction to Strongly Correlated Electron Materials, Dynamical Mean Field Theory (DMFT) and its extensions. Application to the Mott Transition.

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

•Physics of Condensed-Matter Systems. Princeton Center for Complex Materials Princeton. July 18-21 (2005).

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Summary

Strongly Correlated Electron Systems require a new starting point or (non-Gaussian) reference system for their description.

DMFT provides such a reference frame, mapping the full many body problem on the lattice to a much simpler system, a quantum impurity model in a self consistent medium. DMFT a first stab at a problem.

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Summary Application. Pressure and temperature driven Mott

transition. Universal aspects of the Mott transition in transition metal oxides.

Three peak structure in the one particle density of states. QP and Hubbard bands. Mott transition is driven by transfer of spectral weight [non rigid band picture ].

Low energy quasiparticle coherence scale.Coherence-incoherent crossover. Place where gap

closure occurs differs from the place where coherence disappears. Uc1 vs Uc2.

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Summary

Zeroth order picture to confront with experiments in a wide range of materials.

Realistic extensions. Interface with band theory. Illustrate with the physics of actinides.

Plaquette as a reference frame. Cluster DMFT. Superconductivity as a result

of proximity to a Mott insulator singlet state [Anderson’s RVB picture ]

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Some 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)

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Weakly correlated electrons:band theory.

Fermi Liquid Theory. Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors,….).

In a certain low energy regime, adiabatic Continuity to a Reference Systen of Free Fermions with renormalized parameters. Rigid bands , optical transitions , thermodynamics, transport………

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Standard Model of Solids Qualitative predictions: low temperature

dependence of thermodynamics and transport.

Optical response, transition between the bands.

Filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals.

Kinetic Boltzman equations for QP. scattering off phonons or disorder, ee. int etc.

( ) 1Fk l

~H constR~ const S T ~VC T

2 ( )F Fe k k l

h

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Quantitative Tools of Electronic Structure. Kohn Sham reference system

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

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

| ' | ( )

LDAxc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å

Static mean field theory. Derived from a functional which gives the total energy. Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.

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The electron in a solid: particle picture.Ba

Array of hydrogen atoms is insulating if a>>aB.

Mott: correlations localize the electron

e_ e_ e_ e_

Superexchange

Ba

Think in real space , solid collection of atoms

High T : local moments, Low T Anderson superexchange. spin-orbital order ,RVB.

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….)H H H+ H H H motion of H+ forms the lower Hubbard band

H H H H- H H motion of H_ forms the upper Hubbard band

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 in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem.•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA or LDA+U or Hartree Fock work well.•Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands.

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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.

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Model Hamiltonians: Hubbard model

† †

, ,

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

t c c c c U n n

U/t

Doping or chemical potential

Frustration (t’/t)

T temperature

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Strongly correlated systems are usually treated with model Hamiltonians

Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom.

In practice other methods (eg constrained LDA , GW, etc. are used)

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One Particle Spectral Function and Angle Integrated Photoemission

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

f() A() M2

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

(1-f()) A() M2

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

e

e

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

( , )k

A k G kk

Photoemission and the Theory of Electronic Structure

Limiting case itinerant electrons( ) ( )k

k

A

( ) ( , )k

A A k

( ) ( ) ( )B AA Limiting case localized electrons

Hubbard bands

Local Spectral Function

A BU

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Strong Correlation effects appear in 3d- 4f (and sometimes 5f) systems. Because their wave functions are more localized. Many compounds. Also p electron in organic materials with large volumes can be strongly correlated.

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

Breakdown of the Standard Model. Strong Correlation Anomalies cannot be understood within the Breakdown of standard model of solids. Metallic “resistivities beyond the Mott limit.

2 ( )F Fe k k l

h

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Failure of the StandardModel: Anomalous Spectral Weight Transfer as a function of T.

Optical Conductivity Schlesinger et.al (1993)

0( )d

Neff depends on T

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Correlated Materials do big things

Huge resistivity changes. Mott transition. 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.

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Strongly Correlated Materials.

Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Ando et.al.

NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999). Gigantic Volume Collapses. Lanthanide and

actinides. Large and ultrafast optical nonlinearities Sr2CuO3

(T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) )

……………….

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Strong correlation anomalies

Metals with resistivities which exceed the Mott Ioffe Reggel limit.

Transfer of spectral weight which is non local in frequency.

Dramatic failure of DFT based approximations (say DFT-GW) in predicting physical properties.

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Basic competition between kinetic energy and Coulomb interactions.

One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band and atomic limit.

The approach should allow to incorporate material specific information.

When the neither the band or the atomic description applies, a new reference point for thinking about correlated electrons is needed.

DMFT!

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DMFT

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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, 89

1( , )

( )k

G ki i

Neglect k dependence of self energy Muller-Hartmann 89

3 1~ [ ] ij ij ijG t

d

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DMFT mapping (Georges Kotliar 1992)

1( )[ ]

[ ]( )imp nn k imp nk

G ii t i

ww m w

é ùê úD = ê ú- + - S Dê úë ûå

Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from:

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Single site DMFT cavity construction: A. Georges, G. Kotliar, PRB, (1992)]

1 1( ) [ ( )]

( )n loc nloc n

i R G iG i

w ww

-D = + 1[ ]

[ ]k k

R zz e

=-å

†

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 Weiss field

2( ) ( )n loc ni t G iw wD =Semicircular density of states. Behte lattice.

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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 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

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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DMFT mapping (Georges Kotliar 1992)

1( )[ ]

[ ]( )imp nn k imp nk

G ii t i

ww m w

é ùê úD = ê ú- + - S Dê úë ûå

Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from:

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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,ED….)Analytical Methods•Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

G0 G

Im puritySo lver

S .C .C .

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Main Omission of this Course

Techniques for solving quantitatively the Anderson Impurity Model. G[G0]See Reviews.

Qualitative behavior of the solution of the Anderson Impurity Model. Kondo Physics.

Extension to describe ordered phases. Superconductivity. Antiferromagnetism. Etc…

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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)

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Extension to clusters. Cellular DMFT. C-DMFT. G. Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

tˆ(K) is the hopping expressed in the superlattice notations.

•Other cluster extensions (DCA, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK

cond-matt 0307587 (2003)

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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) ]

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Comments on DMFT.

Review of DMFT, technical tools for solving DMFT eqs. A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

CDMFT , instead of studying finite systems with open or periodic boundary conditions, study a system in a medium. Connection with DMRG, infer the density matrix by using a Gaussian anzats, and the periodicity of the system.

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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

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Strongly Correlated Electrons and DMFT. The challenge (besides learning to solve the

DMFT equations more accurately or more explicitly) is to identify which strong correlation phenomena can be capture from a local DMFT perspective using sites, linkes, plaquettes, etc as reference systems, and which aspects involve non local and non Gaussian fluctuations.

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V2O3 under pressure or

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NiSe2-xSx

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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)

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Pressure Driven Mott transition

How does the electron go from the localized to

the itinerant limit ?

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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)

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Insights from DMFT Low temperature Ordered phases . Stability depends on chemistry and crystal structureHigh temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration.

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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

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Qualitative single site DMFT predictions. Spectra of the strongly correlated metallic

regime contains both quasiparticle-like and Hubbard band-like features.

Mott transition is drive by transfer of spectral weight.

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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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

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Qualitative single site DMFT predictions: Optics Spectra of the strongly correlated metallic

regime contains both quasiparticle-like and Hubbard band-like features.

Mott transition is drive by transfer of spectral weight. Consequences for optics.

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Anomalous transfer of spectral weight in v2O3

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Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

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Anomalous Spectral Weight Transfer: Optics

0( ) ,eff effd P J

iV

Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996).

2

0( ) ,

ned P J

iV m

ApreciableT dependence found.

, ,H hamiltonian J electric current P polarization

, ,eff eff effH J PBelow energy

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Anomalous Resistivity and Mott transition Ni Se2-x Sx

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

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Single site DMFT and kappa organics

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Ising critical endpoint! In V2O3 P. Limelette et.al. Science 302, 89 (2003)

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Searching for a quasiparticle peak

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. ARPES measurements on NiS2-xSex

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

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Schematic DMFT phase Implications for transport.

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Anomalous Resistivity and Mott transition Ni Se2-x Sx

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

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Phase Diagram k Organics

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Transport in k organics: hysteresis. Limelette et. al.

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Ising endpoint finally found

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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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

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V2-xCrx O3

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Ising critical endpoint! In VCr2O3 Limelette et.al.

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Conclusion.

An electronic model accounts for all the qualitative features of the finite temperature of a frustrated system at integer occupancy.

The electronic degrees of freedom rather than the lattice drives the transition.

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Conclusion

Single site DMFT describes the main features of the experiments at high temperatures using a simple model.

Made non trivial predictions. Finite temperature conclusions are robust.

At low temperatures clusters will bring refinements of this picture.