Theoretical Treatments of Correlation Effects Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Workshop on Chemical Physics of Emerging Materials Schloss Rinberg May 29 th 2001

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What can theory contribute to materials research ? Some universal aspects can be gleaned from simple models. Example, recent DMFT study of the Mott transition endpoint. Non universal physics requires detailed modeling. Case study Recent LDA+DMFT study of  Pu. Summary

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Why 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.

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 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 Kuwamoto Honig and Appell PRB (1980)

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 Theoretical Approach Mean field approach to quantum many body systems, constructing equivalent impurity models embedded in a bath to be determined self consistently. Use exact numerical techniques as well as semianalytical approaches to study this problem. (DMFT). Exact in infinite dimensions (Metzner and Vollhardt ), can be improved systematically using cluster methods (DCA, CDMFT). Study simple model Hamiltonians (such as the one band model on simple lattices) Understand the results physically in terms of a Landau theory :certain high temperature aspects are independent of the details of the model and the approximations used. Other results are approximate, and very sensitive on solid state aspects.

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 Schematic DMFT phase diagram one band Hubbard model (half filling, semicircular DOS, partial frustration) Rozenberg et.al PRL (1995)

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 an Ising Mott endpoint (Kotliar Lange and Rozenberg PRL 84, 5180 (2000))

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 Miyasaka and Tagaki (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 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 specific heat.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition endpoint Rapid variation has been observed in optical measurements in vanadium oxide and nises mixtures 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. Unify concepts from different theoretical approaches, condensation of d and onset of coherence.

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 Delocalization Localization across the actinide series

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, 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 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 effective action construction ( Fukuda, Valiev and Fernando, Chitra and GK). Select a set of local orbitals. Define a frequency dependent, local Greens function by projecting onto the local orbitals. The exact free energy can be expressed as a functional of the local Greens function and of the density A useful approximation to the exact functional road to total energy calculations.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT 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 G. Kotliar and E. Abrahams full self consistent implementation ( Nature, 2001)

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu: DMFT total energy vs Volume (S. Savrasov )

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Some universal aspects can be gleaned from simple models. Recent DMFT study of the Mott transition endpoint. Many more simple qualitative pictures of little corners in the space of all materials, are still to be found. Non universal physics requires detailed modeling. Recent LDA+DMFT study of  Pu. New developments in many body and electronic structure methods, predictions of new compounds? More interactions with chemical physics and material science.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mean-Field : Classical vs Quantum Classical case Quantum case

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double counting correction Simplest case F0 only. Generalization Lichtenstein et.al in The context of LDA+U