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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline 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 Copper Oxides. High Temperature Superconductivity. Uranium and Cerium Based Compounds. Heavy Fermion Systems. (LaSr)MnO3 Colossal Magnetoresistence.

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model High densities, electron as a wave, band theory, k- space 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 Hall Coefficient Resistivity Thermopower Specific Heat Susceptibility Predicts low temperature dependence of thermodynamics and transport

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 Array of hydrogen atoms is insulating if a>>a B e_ e_ e_ e_ Superexchange Think in real space, atoms High T : local moments Low T: spin orbital order

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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….) 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 S Lefebvre et al. cond-mat/

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

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

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Review of DMFT, technical tools for solving DMFT eqs.., applications, references…… 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  Low temperatures several competing phases. Their relative stability depends on chemistry and crystal structure  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 S x

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT  The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase  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 NiS 2-x Se x 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 S x

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT 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 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 The heavy, D (or F) electrons are localized,treat by DMFT. 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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Fukuda, Valiev and Fernando, Chitra and GK ). 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.  DFT  (r)] Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional 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 the atomic limit. No explicit expression exists. DFT is useful because good approximations to the exact density functional  DFT  (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  Sum of local 2PI graphs with local U matrix and local G

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF 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  phase

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Problems with LDA 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  phase of Pu, using full potential LMTO (Soderlind and Wills). This is usually taken as an indication that  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 Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures  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  ( T=.9 Tc)/   ordered moment Fe 1.5 ( theory) 1.55 (expt) Ni.3 (theory).35 (expt)  eff    high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature T c Fe 1900 ( theory) 1043(expt) Ni 700 (theory) 631 (expt)

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ni low T puzzles 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 DeHaas Van Alphen ( Lonzarich) Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results ) I. Yang S. Savrasov and G. Kotliar PRL2001

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe and Ni 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

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

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

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 Anomalous Resistivity:LiV 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 S Lefebvre et al. cond-mat/

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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  Formulation as an electronic structure method (Chitra and Kotliar)  Density vs Local Spectral Function  Extensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar),Surfaces (Nolting),Stripes (Fleck Lichtenstein and Oles)  Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and Lichtenstein)

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,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  Competing Interaction  Low T, Several Phases Close in Energy  Complex Phase Diagrams  Extreme Sensitivity to Changes in External Parameters  Need for Quantitative Methods

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

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

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

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

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

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 X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Spectral Evolution at T=0 half filling full frustration

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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