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)

A. Georges and G. Kotliar PRB 45, 6479 (1992). G. Kotliar,S. Savrasov, G. Palsson and G. Biroli, PRL 87, (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, (2001) ) with two sites in the Hubbard model in one dimension. [V. Kancharla C. Bolech and GK PRB 67, (2003)][[M.CaponeM.Civelli V Kancharla C.Castellani and GK P. R B 69, (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, (2000), Sun and Kotliar (2003)(2004) Biermann et. al. (2003) Ex. Ir>=|R,  > Gloc=G(R , R  ’)  R,R’ ’ Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc. Sum of 2PI graphs 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 %

Total Energy as a function of volume for Pu (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein Savrasov and Kotliar (2005) W (ev) vs  (a.u ev)

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

DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory Experiment ( 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) volume sexp.LDALDA+U  28Å Å 3  34.4Å Å 3   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  smaller lattice constant  Tk around K

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 (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, (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 (2005). Calculated phonon spectra of plutonium X. Dai, S. Y. Savrasov, G.Kotliar, A. Migliori, H. Ledbetter and E. Abrahams,Science 300, pp (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 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, (2002) GW self energySelf energy correction beyond GW Coordination Sphere

Cellular DMFT. C-DMFT. Site  Cell. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (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 (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 = [ - ] -1 = G + [ - ]

Mott transition in layered organic conductors S Lefebvre et al.

Single site DMFT and kappa organics

. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) Mo et al., Phys. Rev.Lett. 90, (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 = E i - E f Q = k i - k f

oStart from functional of G and W (Chitra and Kotliar (2000), Ambladah et. al. oMake local or cluster approximation on  oFURTHER 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 V 0 (  ) 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)

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

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. A Lichtenstein and M. Katsnelson PRB 57, 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 Georges, A., 2004, Electronic Archive,.lanl.gov, condmat/

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  U/t  Doping  or chemical potential  Frustration (t’/t)  T temperature Mott 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.

oFURTHER 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. oReplace W(  ) or V 0 (  ) 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. A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988).

J. Tobin et. al. PHYSICAL REVIEW B 68, ,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!!!!