THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights into real materials : DMFT at work. From theoretical solid state physics to materials science.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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)