Strongly Correlated Electron Systems a Dynamical Mean

Field Perspective

G. KotliarPhysics Department and Center for

Materials TheoryRutgers

ICAM meeting: Frontiers in Correlated Matter Snowmass September 2004

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

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.

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

Hubbard model and pressure driven Mott transition.

Pressure driven Mott transition.

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. • An “exact numerical approach PRG “ recently

found the first order line(M. Imada), C-DMFT offers a consistency check.

Ising critical endpoint found! In V2O3 P.

Limelette et.al. (Science 2003)

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

2000]

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)

Big question: will we be nearly as successful in our attemps to understand and predict (some ) physical properties of correlated

materials, with DMFT, as we have been for weakly correlated materials using

( approximate DFT and perturbation theory in screened Coulomb interactions eg.GW )?

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 ?

Links, Ti2O3 : Coulomb and Pauling

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

Evolution of the k resolved Spectral Function at zero frequency. (Parcollet Biroli and

GK PRL, 92, 226402. (2004)) ) ( 0, )vs k A k

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

U/D=2 U/D=2.25

U/t=8, t’= -0.3Density= 0.88, 0.89, 0.9, 0.91, 0.922,

0.96, 0.986, 0.988, 0.989, 0.991, 0.993

U/t=16,t’= +0.9

Underlying normal state of the Hubbard model

near the Mott transition, (force the Weiss field to its paramagnetic value),

T=0 ED solution of the C-DMFT equations. M. Civelli, M. Capone, O.

Parcollet and GK

Approaching the Mott transition: plaquette Cdmft.

• 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.

• Study the “normal state” of the Hubbard model. General phenomena, but the location of the cold regions depends on parameters. [Civelli Capone Parcollet and Kotliar ]

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 )……

Mott transition into an open (right) and closed (left) shell systems. In single site DMFT, superconductivity must intervene before reaching the Mott insulating state.[Capone et. al. ] AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

Americium under pressure [J.C. Griveaux J. Rebizant G. Lander]

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)

Answer: cautiously optimistic yes, but it needs a lot of work.

• Focus on short distance intermediate energy scale properties. [Method is designed for that]

• Need analytic +numerical work. Connection with other approaches/DMRG

• Need adaptive k space.• One can already do a lot with single site

DMFT in many many many materials.• Plaquette equations are one order of

magnitude harder to solve.

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 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)

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.

Mott transition into an open (right) and closed (left) shell systems. In single site DMFT, superconductivity must intervene before reaching the Mott insulating state.[Capone et. al. ] AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

S=0

???

Americium under pressure [J.C. Griveaux J. Rebizant G. Lander]

Overview of rho (p, T) of Am

• Note strongly increasing resistivity as f(p) at all T. Shows that more electrons are entering the conduction band

• Superconducting at all pressure

• IVariation of rho vs. T for increasing p.

DMFT study in the fcc structure. S. Murthy and G. Kotliar

fcc

LDA+DMFT spectra. Notice the rapid occupation of the f7/2 band.

One electron spectra. Experiments (Negele) and LDA+DFT theory (S. Murthy and GK )

Conclusion Am

• Crude LDA+DMFT calculations describe the crude energetics of the material, eq. volume, even p vs V .

• Superconductivity near the Mott transition.Tc increases first and the decreases as we approach the

Mott boundary. Dramatic effect in the f bulk module. What is going on at the Am I- Am II boundary ???

Subtle effect (bulk moduli do not change much ), but crucial modifications at low energy.

Mott transition of the f7/2 band ? Quantum critical point ?:

H.Q. Yuan et. al. CeCu2(Si2-x Gex). Am under pressure Griveau et. al.

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.

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.

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.

RVB phase diagram of the Cuprate Superconductors

• P.W. Anderson. Baskaran Zou and Anderson. Connection between high Tc and Mott physics.

• <b> coherence order parameter.

• K, D singlet formation order paramters.

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

• High temperature superconductivity is an unavoidable consequence of the need to connect with Mott insulator that does not break any symmetries to a metallic state.

• Tc decreases as the quasiparticle residue goes to zero at half filling and as the Fermi liquid theory is approached.

• Early on, accounted for the most salient features of the phase diagram. [d-wave superconductivity, anomalous metallic state, pseudo-gap state ]

Problems with the approach.

• Numerous other competing states. Dimer phase, box phase , staggered flux phase , Neel order,

• Stability of the pseudogap state at finite temperature.

• Missing finite temperature . [ fluctuations of slave bosons , ]

• Temperature dependence of the penetration depth [Wen and Lee , Ioffe and Millis ] Theory:

[T]=x-Ta x2 , Exp: [T]= x-T a. • Theory has uniform Z on the Fermi surface, in

contradiction with ARPES.

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.

Study a model of kappa organics. Frustration.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Keeps all the goodies of the slave boson mean field and make many of the results more solid

but also removes the main difficulties. • Can treat coherent and incoherent spectra.• Not only superconductivity, but also the

phenomena of momentum space differentiation (formation of hot and cold regions on the Fermi surface) are unavoidable consequence of the approach to the Mott insulator.

• Can treat dynamical fluctuations between different singlet order parameters.

• Surprising role of the off diagonal self energy which renormalizes 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)

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.

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 ?

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

Approaching the Mott transition: plaquette Cdmft.

• 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..

• Square symmetry is restored as we approched the insulator

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.

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.

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

Dynamical Goodenough-Honig Pauling picture

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

U/t=8, t’= -0.3Density= 0.88, 0.89, 0.9, 0.91, 0.922,

0.96, 0.986, 0.988, 0.989, 0.991, 0.993

U/t=16,t’= +0.9

Underlying normal state of the Hubbard model

near the Mott transition, (force the Weiss field to its paramagnetic value),

T=0 ED solution of the C-DMFT equations. M. Civelli, M. Capone, O.

Parcollet and GK

U/t=16 t’=-.3 n=.95 and t’=.9 n=.95

Insights into the differences between electron and hole doped cuprates ?

• t’ <0 has an underlying normal state with QP around (pi/2, pi/2). This is a state which can naturally evolve into the d-wave superconductor.

• t’>o has the quasiparticles around (pi,0), does not connect smoothly with the SC.

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

Hubbard model and pressure driven Mott transition.