Theoretical Treatments of Correlation Effects Gabriel Kotliar Physics

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

What can theory contribute to materials research ? Summary n n 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 d Pu. 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/0004455 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 Sx G. Czek et. al. J. Mag. Mat. 3, 58 (1976) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Theoretical Approach n n 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 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

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 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, press ure… 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 Sx Miyasaka and Tagaki (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

Ising character of Mott endpoint n n n Singular part of the Weiss field is proportional to h a Max{ (p-pc) (T- Tc)}1/d d=3 in mean field and 5 in 3 d h 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 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

Delocalization Localization across the actinide series 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, when 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

LDA+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 of viewed as parameters n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

effective action construction ( Fukuda, Valiev and Fernando , Chitra and GK). n n 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 n n n V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 -7367 (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 n n 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 d 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 F 0 only. Generalization Lichtenstein et. al in The context of LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS