Dynamical Mean Field Theory Approach to Strongly Correlated

Materials

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University• Sanibel Symposium St Simons Island Georgia. March 2005

• Introduction to Dynamical Mean Field Theory (DMFT) ideas.

• Applications to total energies. A case

Study. Metallic Plutonium.

• Applications to spectroscopies. Optical Conductivity in Cerium. Mott transition or volume collapse ?

• Conclusions. Future Directions.

Mean-FieldClassical vs Quantum

Classical case Quantum case

Review: Kotliar and D. Vollhardt Physics Today 57,(2004)

†

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 t

G i

ww m

w

D =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

10G-

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltonian

cluster exterior exteriorH H

2

4 3

1

A. Georges and G. Kotliar PRB 45, 6479 (1992). G. Kotliar,S. Savrasov, G. Palsson and G. Biroli, PRL 87, 186401 (2001) .

Auxiliary Quantum Impurity Model. Finite number of quantum degrees of freedom “cluster”, in a medium.

DMFT Self Consistency Condition

Lattice Hamiltonian

Reduce the system to a finite number of interacting degrees of freedom in a

free effective medium.

Solve the quantum impurity model to obtain “local “ auxilliary

quantities, e.g. Gloc, c

From the “local quantities” infer the lattice quantities G(k, )

k,

DMFT

Use the “local quantities” and periodicity to determine the

effective medium

Lattice quantities

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.CaponeM.Civelli V Kancharla C.Castellani and GK P. R B 69,195105 (2004)

U/t=4.]

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.

Functional approach Ambladah et. al. (1999) Chitra and Kotliar Phys. Rev. B

62, 12715 (2000), Sun and Kotliar (2003)(2004) Biermann et. al. (2003)  

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( )G x xy y=- < > ( ') ( ) ( ') ( )x x x x Wff ff< >- < >< >=

Ex. Ir>=|R, > Gloc=G(R, R ’) R,R’’

1 10

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

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Sum of 2PI graphs[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc

Practical Implementation, approximations

• 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) .

• Replace the dynamical interaction by a static Hubbard interactions. • Basis sets. LMTO’s full potential-ASA.• Use of approximate solvers. Interpolative solvers. Hubbard I.

Quantum Montecarlo. Exact Diagonalization…• Relativistic corrections. Spin orbit interaction. • Self consistent vs one shot determination of charge density. • LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin

and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).• Review G. Kotliar S. Savrasov K. Haule V. Oudovenko O.

Parcollet and C. Marianetti. (2005).

Pu in the periodic table

actinides

Pu phases: A. Lawson Los Alamos Science 26, (2000)

LDA underestimates the volume of fcc Pu by 30%

Predicts magnetism in Pu .

Gives negative shear constant.

Core-like f electrons overestimates the volume by 30 %

Total­Energy­as­a­Total­Energy­as­a­function­of­volume­for­function­of­volume­for­PuPu

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2005)

W (ev) vs (a.u. 27.2 ev)

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-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

• Pu strongly correlated element, at the brink of a Mott instability.

• Realistic implementations of DMFT : total energy, photoemission spectra and phonon dispersions of delta Pu.

• Double well in E vs V, clue to understanding Pu elastic anomalies.

• S. Savrasov, G. Kotliar, and E. Abrahams, Nature 410, 793 (2001).

Conclusion

• Spectral Density Functional. Connection between spectra and bonding. Microscopic theory of Pu, connecting its anomalies to the vicinity of a Mott point.

• Combining theory and experiment we can more than the sum of the parts. Next step in Pu, much better defined problem, discrepancy in (111 ) zone boundary, may be due to either the contribution of QP resonance, or the inclusion of nearest neighbor correlations. Both can be individually studied.

Case Study: elemental cerium

Various phases :

isostructural phase transition (T=298K, P=0.7GPa)

(fcc) phase

[ magnetic moment

(Curie-Wiess law) ]

(fcc) phase

[ loss of magnetic

moment (Pauli-para) ]

with large

volume collapse

v/v 15

( -phase a 5.16 Å

-phase a 4.8 Å)

volumes exp. LDA LDA+U 28Å3 24.7Å3

34.4Å3 35.2Å3

-phase (localized): High T phaseCurie-Weiss law (localized magnetic moment),Large lattice constantTk around 60-80K

-phase (localized): High T phaseCurie-Weiss law (localized magnetic moment),Large lattice constantTk around 60-80K

-phase (delocalized:Kondo-

physics): Low T phaseLoss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic momentsmaller lattice constantTk around 1000-2000K

-phase (delocalized:Kondo-

physics): Low T phaseLoss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic momentsmaller lattice constantTk around 1000-2000K

Qualitative Ideas.

• Johanssen, Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators, f-f hybridization matters.

• Allen and Martin. Kondo volume collapse picture. The dominant effect are changes in the spd-f hybridization.

Photoemission&experiment

•A. Mc Mahan K Held and R. Scalettar (2002)

•K. Haule V. Udovenko S. savrasov and GK. (2003) But both Mott scenario and Kondo collapse predict

similar features………

Resolution: Turn to Optics!

• Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior.

• See if they are simple spectators (Mott transition picture ) or whether a Kondo binding unbinding takes pace (Kondo collapse picture).

Optical Conductivity Temperature dependence.

Origin of the features.

Conclusion• The anomalous temperature dependence and the formation of a pseudogap, suggests that the

Kondo collapse picture is closer to the truth for Cerium.

• K. Haule V. S. Oudovenko S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett. 94 036401-036405 (2005).

• Possible experimental verification in Ce(ThLa) alloys.• Qualitative agreement with experiments. ( J.Y. Rhee,

X. Wang, B.N. Harmon, and D.W. Lynch, Phys. Rev. B 51, 17390 (1995), VanderEbb et. al. PRL 2u (2001) ) .

Goal of a good mean field theory• Provide a zeroth order picture of a physical phenomena.• Provide a link between a simple system (“mean field

reference frame”) and the physical system of interest.• Formulate the problem in terms of local quantities (which

we can compute better ).• Allows to perform quantitative studies, and predictions .

Focus on the discrepancies between experiments and mean field predictions.

• Generate useful language and concepts. Follow mean field states as a function of parameters.

• Exact in some limit [i.e. infinite coordination]• Can be made system specific, useful tool for material

exploration and for interacting with experiment.

Conclusions

• While DMFT is still a method under construction, it has already reached a stage where it has predictive power and can interact meaningfully with experiments.

• Applications to d electrons V2O3, Ti2O3, Fe, Ni, VO2, La1-xSrxTiO3, CrO2, SrRuO4, high temperature superconductors ………………..

• Future directions: more complex materials, applications to non periodic situations: surfaces, heterostructures……..

• Locality of the self energy ?• Simplifying the equations and going to longer scales

downfolding and renormalization groups.

Collaborators 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. (2005).

• Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

Collaborators References

• The alpha to gamma transition in Ce: K. Haule V. S. Oudovenko S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett. 94 036401-036405 (2005).

• Calculated phonon spectra of plutonium X. Dai, S. Y. Savrasov, G.Kotliar, A. Migliori, H. Ledbetter and E. Abrahams,Science 300, pp. 953-955 (2003);

• Electronic correlations in metallic Plutonium. S. Savrasov, G. Kotliar, and E. Abrahams, Nature 410, 793 (2001).

Locality of the self energy :Convergence in R space of first self energy correction for Si in a.u. (1

a.u.= 27.2 eV) N. Zein

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2

Sigma s GW

Sigma p GW

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 1 2

D Sigma s

D Sigma p

Lowest order graph in the screened coulomb interaction (GW approximation) treated self consistently reproduces the gap of silicon. [Exp : 1.17 ev, GW 1.24 ] W. Ku, A. Eguiluz, PRL 89,126401 (2002)

GW self energy Self energy correction beyond GW

Coordination Sphere Coordination Sphere

Cellular DMFT. C-DMFT. Site Cell. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87,

186401 (2001)

tˆ(K) hopping expressed in the superlattice notations.

•Other choices of medium “G0” , connection with other methods, causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003). Other correlation functions,

energies etc..

Ex. Single site DMFT Local Self Energy

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

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

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

= W

= [ - ]-11CV

= G

+ [ - ]KSV10KSG 1G

Mott transition in layered organic conductors S Lefebvre et al.

Single site DMFT and kappa organics

. ARPES measurements on NiS2-xSex

Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et al., Phys. Rev.Lett. 90, 186403 (2003).

DMFT:Realistic Implementations

• Focus on the “local “ spectral function A() (and of the local screened Coulomb interaction W() ) of the solid.

• Write a functional of the local spectral function such that its stationary point, give the energy of the solid.

• No explicit expression for the exact functional exists, but good approximations are available. LDA+DMFT.

• The spectral function is computed by solving a local impurity model in a medium .Which is a new reference system to think about correlated electrons.

Specific heat and susceptibility.

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf

o Start from functional of G and W (Chitra and Kotliar (2000), Ambladah et. al.

o Make local or cluster approximation on

o FURTHER APPROXIMATIONS: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) .

o Truncate the W operator act on the H sector only. i.e.

• Replace W() or V0() by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation with light electrons only. e.g.

M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998) T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein cond-matt (2004)

( , ', ) ( ') ( ) ( )( ( ) ) ( ')dcxc R H R Rr r r r V r r E rabe a ab bw d f w fS = - - S S -

( , ', ) ( ) ( ) ( ) ( ') ( ')R H R R R RW r r r r W r rabgde a b abgd g dw ff wff=S

1 10

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

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr PW E G W

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

LDA+DMFT Formalism : V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004). V. Udovenko S. Savrasov K. Haule and G. Kotliar

Cond-mat 0209336

or the U matrix can be adjusted empirically.• At this point, the approximation can be derived from

a functional (Savrasov and Kotliar 2001)

• FURTHER APPROXIMATIONS, ignore charge self consistency, namely set

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). See also . A­Lichtenstein­and­M.­Katsnelson­PRB­57,­6884­(1988).

Reviews:Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Blumer, A. K. �McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, 2003, Psi-k Newsletter #56, 65.

• Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron Correlations and Materials Properties 2, edited by A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New York), p. 428.

• Georges, A., 2004, Electronic Archive, .lanl.gov, condmat/ 0403123 .

loc[ ]G

[ ] [ ]LDAVxc Vxc

1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,intM PLocal Impurity Model

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

,loc locG W 1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,i n tM PImpurity Solver

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

Qualitative Ideas

• “screened moment alpha phase” Kondo effect between spd and f takes place. “unscreend moment gamma phase” no Kondo effect (low Kondo temperature).

• Mathematical implementation, Anderson impurity model in the Kondo limit suplemented with elastic terms. (precursor of DMFT ideas, but without self consistency condition).

Real Space Formulation of the DCA of Jarrell and collaborators.

Strongly correlated systems are usually treated with model

Hamiltonians

• In practice other methods (eg constrained LDA are used)

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

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in

single site DMFT.

Connection of bonding and spectra. M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)

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.

Small amounts of impurities stabilize the phase (A.

Lawson LANL)

Magnetic Susceptibilty of Pu vs Temperature. Smith and Boring LANL Science.

o FURTHER APPROXIMATIONS: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) .

o Truncate the W operator act on the heavy sector only. i.e.

o Replace W() or V0() by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation. FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein prb (2004) or the U matrix can be adjusted empirically.

• At this point, the approximation can be derived from a functional (Savrasov and Kotliar (2001), (2004)).

• FURTHER APPROXIMATIONS, ignore charge self consistency, namely set

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). See also . A­Lichtenstein­and­M.­Katsnelson­PRB­57,­6884­(1988).

( , ', ) ( ') ( ) ( )( ( ) ) ( ')dcxc R H R Rr r r r V r r E rabe a ab bw d f w fS = - - S S -

( , ', ) ( ) ( ) ( ) ( ') ( ')R H R R R RW r r r r W r rabgde a b abgd g dw ff wff=S

[ ] [ ]LDAVxc Vxc

loc[ ]G

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

Alpha and delta Pu Photoemission Spectra DMFT(Savrasov et.al.) EXP (Arko Joyce Morales Wills Jashley PRB 62,

1773 (2000))

X.Zhang M. Rozenberg G. Kotliar PRL 70,1666(1993). A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

Unfortunately photoemission cannot decide between the Kondo collapse picture and the

Mott transition picture.Evolution of the spectra as a function of U , half filling full frustration, Hubbard model!!!!