Dynamical Mean Field Theory or Metallic Plutonium. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. Collaborators: S. Savrasov (NJIT) and Xi Dai (Rutgers). IWOSMA Berkeley October 2002.
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Dynamical Mean Field Theory or Metallic Plutonium Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Collaborators: S. Savrasov (NJIT) and Xi Dai (Rutgers) IWOSMA Berkeley October 2002
Evolution of the electronic structure between the atomic limit and the band limit in an open shell situation. The “”in between regime” is ubiquitous central them in strongly correlated systems, gives rise to interesting physics. Example Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] Revisit the problem using a new insights and new techniques from the solution of the Mott transition problem within dynamical mean field theory in the model Hamiltonian context. Use the ideas and concepts that resulted from this development to give physical qualitative insights into real materials. Turn the technology developed to solve simple models into a practical quantitative electronic structure method . Mott Phenomena THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Connection between local spectra and cohesive energy using Anderson impurity models foreshadowed by J. Allen and R. Martin PRL 49, 1106 (1982) in the context of KVC for cerium. Identificaton of Kondo resonance n Ce , PRB 28, 5347 (1983). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mott transition in the actinide series (Smith Kmetko phase diagram) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Small amounts of Ga stabilize the d phase (A. Lawson LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
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 phaseIs 35% lower than experiment This is the largest discrepancy ever known in DFT based calculations. Plutonium Puzzles THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
LSDA predicts magnetic long range (Solovyev et.al.) Experimentally d Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% DFT in GGA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that a Pu is a weakly correlated system DFT Studies THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pu Specific Heat THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anomalous Resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pu is NOT MAGNETIC THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Specific heat and susceptibility. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
U/W is not so different in alpha and delta The specific heat of delta Pu, is only twice as big as that of alpha Pu. The susceptibility of alpha Pu is in fact larger than that of delta Pu. The resistivity of alpha Pu is comparable to that of delta Pu. Problems with the conventional viewpoint of a Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Local approximation (Treglia and Ducastelle PRB 21,3729), local self energy, as in CPA. Exact the limit defined by Metzner and Vollhardt prl 62,324(1989) inifinite. Mean field approach to many body systems, maps lattice model onto a quantum impurity model (e.g. Anderson impurity model )in a self consistent medium for which powerful theoretical methods exist. (A. Georges and G. Kotliar prb45,6479 (1992). Dynamical Mean Field Theory(DMFT)Review: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000), (2001). • Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a. • Example, density in DFT theory. (Fukuda et. al.) • When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. • The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. • It is useful to introduce a Lagrange multiplier l conjugate to a, G [a, l ]. • It gives as a byproduct a additional lattice information. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation to the functional. Example: DMFT for lattice model (e.g. single band Hubbard). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions Outline 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
Spectral Evolution at T=0 half filling full frustration X.Zhang M. Rozenberg G. Kotliar (PRL 1993) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Phase Diagrams :V2O3, Ni Se2-x SxMc Whan et. Al 1971,. Czek et. al. J. Mag. Mag. Mat. 3, 58 (1976), THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Mott transition in layered organic conductors S Lefebvre et al. Ito et.al, Kanoda’s talk Bourbonnais talk Magnetic Frustration THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Cerium THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Coexistence regions between localized and delocalized spectral functions. k diverges at generic Mott endpoints Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL (2002). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Ultrasound study of Fournier et. al. (2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Divergence of the compressibility at the Mott transition endpoint. Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region. Slow variation of the volume as a function of pressure in the liquid phase Elastic anomalies, more pronounced with orbital degeneracy. Minimum of the melting point THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Minimum in melting curve and divergence of the compressibility at the Mott endpoint THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Cerium THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Derive model Hamiltonians, solve by DMFT (or cluster extensions). Total energy? Full many body aproach, treat light electrons by GW or screened HF, heavy electrons by DMFT [E-DMFT frequency dependent interactionsGK and S. Savrasov, P.Sun and GK cond-matt 0205522] Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT) Interface DMFT with electronic structure. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
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 (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters LDA+DMFT approximate functional THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
LDA+DMFT-outer loop relax Edc U DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Outer loop relax Edc G0 Impurity Solver Imp. Solver: Hartree-Fock G,S U SCC DMFT LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Static limit of the LDA+DMFT functional , with FatomFHF reduces to the LDA+U functional of Anisimov Andersen and Zaanen. Crude approximation. Reasonable in ordered Mott insulators. Total energy in DMFT can be approximated by LDA+U with an effective U . Extra screening processes in DMFT produce smaller Ueff. ULDA+U < UDMFT LDA+DMFT and LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
QP bands in ruthenides: A. Liebsch et al (PRL 2000) N phase of Pu: S. Savrasov G. Kotliar and E. Abrahams (Nature 2001) MIT in V2O3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein M. Katsenelson and G. Kotliar et al PRL (2001) J-G transition in Ce: K. Held A. Mc Mahan R. Scalettar (PRL 2000); M. Zolfl T. et al PRL (2000). 3d doped Mott insulator La1-xSrxTiO3 Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2002) ……………….. Very Partial list of application of realistic DMFT to materials THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsenelson PRB (1998). Reviews:Kotliar, Savrasov, in New Theoretical approaches to strongly correlated systems, Edited by A. Tsvelik, Kluwer Publishers, (2001). Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar (2002) LDA+DMFT References THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Introduce local orbitals, caR(r-R), and local GF G(R,R)(i w) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for r(r) and G and performing a Legendre transformation, G[r(r),G(R,R)(iw)] Spectral Density Functional : effective action construction THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
LDA+DMFT Self-Consistency loop E U DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Static limit of the LDA+DMFT functional , with F= FHF reduces to LDA+U Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing. Luttinger theorem is obeyed. Functional formulation is essential for computations of total energies, opens the way to phonon calculations. Comments on LDA+DMFT 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, 7359-7367 (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov G.Kotliar funcional formulation for full self consistent implementation of a spectral density functional. Application to Pu S. Savrasov G. Kotliar and E. Abrahams (Nature 2001). References THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Long range Coulomb interactios, E-DMFT. R. Chitra and G. Kotliar Combining E-DMFT and GW, GW-U , G. Kotliar and S. Savrasov Implementation of E-DMFT , GW at the model level. P Sun and G. Kotliar. Also S. Biermann et. al. References THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Introduction: some Pu puzzles. DMFT , qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions Outline THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Snapshots of the f electron Dominant configuration:(5f)5 Naïve view Lz=-3,-2,-1,0,1 ML=-5 mB S=5/2 Ms=5 mB Mtot=0 What is the dominant atomic configuration? Local moment? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
LDA+U bands. (Savrasov GK ,PRL 2000). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
L=5, S=5/2, J=5/2, Mtot=Ms=mB gJ =.7 mB Crystal fields G7 +G8 GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1 This bit is quenched by Kondo effect of spd electrons [ DMFT treatment] Experimental consequence: neutrons large magnetic field induced form factor (G. Lander). Magnetic moment THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Multiorbital situation and several atoms per unit cell considerably increase the size of the space H (of heavy electrons). QMC scales as [N(N-1)/2]^3 N dimension of H Fast interpolation schemes (Slave Boson at low frequency, Roth method at high frequency, + 1st mode coupling correction), match at intermediate frequencies. (Savrasov et.al 2001) Technical details THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Atomic sphere approximation. Ignore crystal field splittings in the self energies. Fully relativistic non perturbative treatment of the spin orbit interactions. Technical details THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Pu: DMFT total energy vs Volume (Savrasov 00) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Qualitative explanation of negative thermal expansion Sensitivity to impurities which easily raise the energy of the a -like minimum. Double well structure and d Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS
Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Is the natural consequence of the model Hamiltonian phase diagram once electronic structure is about to vary. Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS