Electronic Structure of Strongly Correlated Materials a DMFT

Electronic Structure of Strongly Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Landelijk Seminarium voor Vaste Stof Fysica Nijmeigen Nov 16 2001

Outline n n n Introduction to the electronic structure of correlated electrons Dynamical Mean Field Theory Model Hamiltonian Studies. Universal aspects insights from DMFT System specific studies: LDA+DMFT Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The promise of Strongly Correlated Materials n n n Copper Oxides. High Temperature Superconductivity. Uranium and Cerium Based Compounds. Heavy Fermion Systems. (La. Sr)Mn. O 3 Colossal Magnetoresistence. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The Promise of Strongly Correlated Materials. n High Temperature Superconductivity in doped filled Bucky Balls (B. Battlog et. al Science) Thermoelectric response in Ce. Fe 4 P 12 (H. Sato et al. cond-mat 0010017). Large Ultrafast Optical Nonlinearities Sr 2 Cu. O 3 (T Ogasawara et. al cond-mat 000286) n Theory will play an important role in optimizing their physical properties. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

How to think about the electron in a solid? Drude Sommerfeld Bloch, Periodic potential Bands, k in Brillouin zone Maximum metallic resistivity 200 mohm cm THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Standard Model High densities, electron as a wave, band theory, kspace Landau: Interactions Renormalize Away One particle excitations: quasi-particle bands Density Functional Theory in Kohn Sham Formulation, successful computational tool for total energy, and starting point For perturbative calculation of spectra, Si Au, Li, Na ………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Standard Model : Metals Predicts low temperature dependence of thermodynamics and transport Hall Coefficient Resistivity Thermopower Specific Heat Susceptibility THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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. ………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott : correlations localize the electron n Array of hydrogen atoms is insulating if a>>a. B e_ n e_ e_ e_ Superexchange Think in real space , atoms High T : local moments Low T: spin orbital order RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

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. bands. Ni. O, Co. O Mn. O…. ) Rich structure of Magnetic and Orbital Ordering at low T Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre et

Failure of the Standard and Model: Ni. Se. Miyasaka S 2 -x x (2000) Takagi THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failure of the standard model : Anomalous Resistivity: Li. V 2 O 4 Takagi et. al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strong Correlation Problem n n n 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992) Mean-Field : Classical vs Quantum case Classical case THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Single site DMFT, functional formulation n n Local self energy (Muller Hartman 89) Express in terms of Weiss field (semicircular. DOS) The Mott transition as bifurcation point in functionals o. G[G] or F[D], (G. Kotliar EPJB 99) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

tools for solving DMFT eqs. . , applications, references…… n A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68, 13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights from DMFT q Low temperatures several competing phases. Their relative stability depends on chemistry and crystal structure q. High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration) Rozenberg et. al. PRL (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Kuwamoto Honig and Appell PRB (1980) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phase Diag: Ni Se 2 -x Sx THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights from DMFT q. The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase q. Control parameters: doping, temperature, pressure… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

. ARPES measurements on Ni. S 2 -x. Sex Matsuura et. Al Phys. Rev B 58 (1998) 3690 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivity and Mott transition Ni Se 2 -x Sx THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights from DMFT n n n 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. Real and momentum space. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n n Realistic Calculations of the Electronic Structure of Correlated materials Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials. Inspired by the LDA+U approach (Anisimov, Andersen and Zaanen) Anisimov Poteryaev Korotin Anhokin and Kotliar (1997). Lichtenstein and Katsenelson (1998) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Combining LDA and DMFT The light, SP (or SPD) electrons are extended, well described by LDA n The heavy, D (or F) electrons are localized, treat by DMFT. n LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles or viewed as parameters n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK). n n DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. GDFT[r(r)] Introduce local orbitals, ca. R(r-R)orbitals, and local GF G(R, R)(i w) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a Legendre transformation, G[r(r), G(R, R)(iw)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Density Functional n n n The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed expanding around the atomic limit. No explicit expression exists. DFT is useful because good approximations to the exact density functional GDFT[r(r)] exist, e. g. LDA, GGA A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA+DMFT functional F Sum of local 2 PI graphs with local U matrix and local G THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA+DMFT Self-Consistency loop E U DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comments on LDA+DMFT • • Static limit of the LDA+DMFT functional , with F= FHF reduces to LDA+U Removes inconsistencies and shortcomings of this approach. DMFT retain correlations effects in the absence of orbital ordering. • • Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Small amounts of Ga stabilize the d phase THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Problems with LDA o o o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. Many studies (Freeman, Koelling 1972)APW methods ASA and FP-LMTO Soderlind et. Al 1990, Kollar et. al 1997, Boettger et. al 1998, Wills et. al. 1999) give an equilibrium volume of the d phase Is 35% lower than experiment This is the largest discrepancy ever known in DFT based calculations. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Problems with LDA n n n LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et. al. ) If one treats the f electrons as part of the core LDA overestimates the volume by 30% LDA predicts correctly the volume of the a phase of Pu, using full potential LMTO (Soderlind and Wills). This is usually taken as an indication that a Pu is a weakly correlated system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams Nature 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Lda vs Exp Spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu Spectra DMFT(Savrasov) EXP (Arko et. al) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Case study Fe and Ni n n n Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures c has a Curie Weiss law with a moment much larger than the ordered moment. Magnetic anisotropy THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK PRL 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Iron and Nickel: magnetic properties (Lichtenstein, Katsenelson, GK PRL 01) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ni and Fe: theory vs exp n n n m( T=. 9 Tc)/ m. B ordered moment Fe 1. 5 ( theory) Ni. 3 (theory) meff / m. B 1. 55 (expt). 35 (expt) high T moment Fe 3. 1 (theory) 3. 12 (expt) Ni 1. 5 (theory) 1. 62 (expt) Curie Temperature Tc n n Fe 1900 Ni 700 ( theory) (theory) 1043(expt) 631 (expt) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outlook q The Strong Correlation Problem: How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR q. Strategy: advancing our understanding scale by scale q. Generalized cluster methods to capture longer range magnetic correlations q. New structures in k space. Cellular DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ni low T puzzles n n Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis(100) for Nickel. (instead of the correct one (111) LDA Fermi surface has features which are not seen in De. Haas Van Alphen ( Lonzarich) Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results ) I. Yang S. Savrasov and G. Kotliar PRL 2001 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Fe and Ni n n n Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction. 3 ev Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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DMFT: References Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W. Krauth, E. Lange, G. Moeller, Y. Motome, G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, X. Y. Zhang Other work: A. Brandt, W. Nolting, R. Bulla, M. Jarrell, D. Logan, J. Freericks, T. Prushke, W. Metzner, F. Gebhardt, A. Lichtenstein, M. Fleck D. Vollhardt ………………. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Reviews of DMFT n n Prushke T. Jarrell M. and Freericks J. Adv. Phys. 44, 187 (1995) A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68, 13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Challenges n n n The photoemission in cuprates has a strong momentum dependence Strong Magnetic Correlations (no orbital degeneracy) Single Site DMFT does not capture these effects THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Cuprates: Photoemission – Transfer of Spectral Weight with a) temperature and b) doping THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT q. Spin Orbital Ordered States q. Longer range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar, ) q. Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

DMFT q. Formulation as an electronic structure method (Chitra and Kotliar) q. Density vs Local Spectral Function q. Extensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar), Surfaces (Nolting), Stripes (Fleck Lichtenstein and Oles) q. Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and. RUTGERS Lichtenstein) THE STATE UNIVERSITY OF NEW JERSEY

Cuprates: Photoemission – Transfer of Spectral Weight with a) temperature and b) doping THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivity: Li. V 2 O 4 Takagi et. al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivities: Doped Hubbard Model (Prushke and Jarrell 1993) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivities: Doped Hubbard Model G. Palsson 1998 IPT NCA THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failure of the “Standard Model”: Cuprates Anomalous Resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Specific Heat Titanates THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre et

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Standard Model Odd # electrons -> metal Even # electrons -> insulator q. Theoretical foundation: Sommerfeld, Bloch and Landau q. Computational tools DFT in LDA q. Transport Properties, Boltzman equation , low temperature dependence of transport coefficients Typical Mott values of the resistivity 200 m. Ohmcm Residual instabilites SDW, CDW, SC THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failure of the “Standard Model”: Cuprates Anomalous Resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Solving the DMFT equations • Wide variety of computational tools (QMC, NRG, ED…. ) • Analytical Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT q. Formulation as an electronic structure method (Chitra and Kotliar) q. Density vs Local Spectral Function q. Extensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar), Surfaces (Nolting), Stripes (Fleck Lichtenstein and Oles) q. Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and. RUTGERS Lichtenstein) THE STATE UNIVERSITY OF NEW JERSEY

DMFT: Methods of Solution THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT q. Spin Orbital Ordered States q. Longer range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar, ) q. Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=. 9 ev, T/Tc=. 8) (Lichtenstein, Katsenelson, GK prl 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Photoemission and T/Tc=. 8 Spin Autocorrelation: Ni (U=3, J=. 9 ev) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strongly Correlated Electrons q. Competing Interaction q. Low T, Several Phases Close in Energy q. Complex Phase Diagrams q. Extreme Sensitivity to Changes in External Parameters q. Need for Quantitative Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failure of the Standard Model: Anomalous Spectral Weight Transfer Optical Conductivity o of Fe. Si for T=, 20, 250 200 and 250 K from Schlesinger et. al (1993) Neff depends on T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Hubbard model q. U/t q. Doping d or chemical potential q. Frustration (t’/t) q. T temperature Mott transition as a function of doping, pressure RUTGERS temperature etc. THE STATE UNIVERSITY OF NEW JERSEY

Landau Functional G. Kotliar EPJB (1999) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA functional Conjugate field, VKS(r) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Minimize LDA functional Kohn Sham eigenvalues, auxiliary THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

A time-honored example: Mott transition in V 2 O 3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Evolution at T=0 half filling full frustration X. Zhang M. Rozenberg G. Kotliar (PRL 1993) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Parallel development: Fujimori et. al THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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Pu: Complex Phase Diagram (J. Smith LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS