Dynamical Mean Field Theory from Model Hamiltonian Studies of the Mott Transition to Electronic Structure Calculations Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University 11 Conference on Recent Progress in Many Body Physics UMIST July 9-15 th 2001

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline What is DMFT, when is it useful and how is it done. What has been accomplished. Ex. model Hamiltonian studies of the finite temperature Mott transition. How to combine DMFT and band structure, formal aspects. Results for some real materials.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS References, Collaborators Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Finite T Mott endpoint: Kotliar Lange and Rozenberg PRL 84, 5180 (2000)) Realistic Calculations: S. Savrasov and GK cond-mat Application to Pu, S.Savrasov GK and E. Abrahams Nature 410, 793 (2001). Fe and Ni A. Lichtenstein M. Katsnelson and GK (PRL in press).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] Weiss field

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 Comments on the DMFT construction Exact in large dimensions [Metzner and Vollhardt 89] Trick to sum all LOCAL skeleton graphs, [Muller Hartman 89].  Can be used for susceptibilities, ordered states etc..  Non perturbative construction, works even when skeleton expansion fails.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Good method to study the Mott phenomena Evolution of the electronic structure between the atomic limit and the band limit. Basic solid state problem. Solved by band theory when the atoms have a closed shell. Mott’s problem: Open shell situation. The “”in between regime” is ubiquitous central them in strongly correlated systems. Some unorthodox examples Fe, Ni, Pu. Solution of this problem should lead to advances in electronic structure theory (LDA +DMFT)

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 Phase Diag: Ni Se 2-x S x G. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors Ito et al. (1986) Kanoda (1987) Lefebvre et al. (2001)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram one band Hubbard (half filling, semicircular DOS, role of partial frustration) Rozenberg et.al PRL (1995)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights into the Mott phenomena  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 an Ising Mott endpoint (Kotliar et.al.PRL 84, 5180 (2000))

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expt. Ni Se S Matsuura et. Al.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising character of Mott endpoint Singular part of the Weiss field is proportional to  Max{ (p-pc),(T- Tc)} 1/   in mean field and 5 in 3d  couples to all physical quantities which then exhibit a kink at the Mott endpoint. Resistivity, double occupancy,photoemission intensity, integrated optical spectral weight, etc. Divergence of the the compressibility,in particle hole asymmetric situations e.g. Furukawa and Imada

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase diagram 1 band model

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Divergent compressibility U=2.4

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Compressibility QMC two band model, U=3

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition endpoint Rapid variation has been observed in optical measurements in vanadium oxide (Thomas) and Ni mixtures(Miyasaka and Takgai) Experimental questions: width of the critical region. Ising exponents or classical exponents, validity of mean field theory Building of coherence in other strongly correlated electron systems. condensation of doubly occupied sites and onset of coherence.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optical Conductivty Miyasaka Takagi (2000)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights from DMFT: think in term of spectral functions, the density is not changing! Resistivity near the metal insulator endpoint ( Rozenberg et.al 1995) exceeds the Mott limit

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two Roads for calculations of the electronic structure of correlated materials Crystal Structure +atomic positions Correlation functions Total energies etc. Model Hamiltonian

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+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 of 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 from the atomic limit, but 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 Connection with atomic limit Weiss field

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional approach

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on LDA+DMFT Static limit of the functional reduces to LDA+U Removes inconsistencies of this approach, 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 LDA+DMFT References V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov and GK full self consistent implementation cond-mat Application to Pu, S.Savrasov GK and E. Abrahams Nature 410, 793 (2001)

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T

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 Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK cond-mat )

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 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 Case study in f electrons, Mott transition in the actinide series

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Delocalization-Localization across the actinide series o f electrons in Th Pr U Np are itinerant. From Am on they are localized. Pu is at the boundary. o Pu has a simple cubic fcc structure,the  phase which is easily stabilized over a wide region in the T,p phase diagram. o The  phase is non magnetic. an equilibrium volume of the  phase Is 35% lower than experiment o Many LDA, GGA studies ( Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give an equilibrium volume of the  phase Is 35% lower than experiment o This is one of the largest discrepancy ever known in DFT based calculations.

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, when 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

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 Conclusion The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights.  This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for simple elements

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest in a G0 (renormalization of the quadratic part of the effective action). What to take for G0: DCA (M. Jarrell et.al), CDMFT ( Savrasov GK Palsson and Biroli ) include the effects of the electrons to renormalize the quartic part of the action (spin spin, charge charge correlations) E. DMFT (Kajueter and GK, Si et.al)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Extensions of DMFT implemented on model systems, (e.g. Motome and GK ) carry over to more realistic framework. Better determination of Tcs. First principles approach: determination of the Hubbard parameters, and the double counting corrections long range coulomb interactions E- DMFT Improvement in the treatement of multiplet effects in the impurity solvers, phonon entropies, ………

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