The Mott transition in f electron systems Pu

The Mott transition in f electron systems, Pu, a dynamical mean field perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Collaborators: S. Savrasov (NJIT) ASR 2002 Tokai Japan November 12 -24 2002

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

Outline n n n Introduction: some Pu puzzles. Introduction to DMFT. Some qualitative insights from model Hamiltonian studies. Towards and understanding of elemental Pu. Conclusions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Plutonium Puzzles 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

DFT Studies LSDA predicts magnetic long range (Solovyev et. al. ) Experimentally d Pu is not magnetic. n If one treats the f electrons as part of the core LDA overestimates the volume by 30% n 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 n U/W is not so different in alpha and delta n 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

Plutonium puzzles. n n How to think about the alpha and delta phases and compute their physical properties? Why does delta have a negative thermal expansion? Why do minute amount of impurities stablize delta? Where does epsilon fit? Why is it smaller than delta? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n Introduction: some Pu puzzles. Introduction to DMFT. Some qualitative insights from model Hamiltonian studies. Towards and understanding of elemental Pu. Conclusions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mean-Field : Classical vs Quantum Classical case Phys. Rev. B 45, 6497 Quantum case THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A. Georges, G. Kotliar (1992)

Extensions of DMFT. Renormalizing the quartic term in the local impurity action. EDMFT. n Taking several sites (clusters) as local entity. CDMFT n Combining DMFT with other methods. LDA+DMFT, GWU. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT: effective action point of view. R. Chitra and G. Kotliar Phys. Rev. B 62, 12715 (2000), B 63, 115110 (2001) n n Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a. Construct approximations to the functional G to perform practical calculations. Example: Density functional theory (Fukuda et. al. ), density, LDA, GGA. Example: model DMFT. Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation the functional keeping 2 PI graphs THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Density Functionals for Electronic Structure Calculations, Sergej Savrasov and Gabriel Kotliar, cond-mat/0106308 n n Effective action construction. Introduce local orbitals, ca. R(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)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Construct approximate functional which gives the LDA+DMFT equations. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 -7367 (1997). 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 (Gunnarson and Anisimov, Mc. Mahan et. al. Hybertsen et. al) of viewed as parameters n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outer loop relax Edc G 0 DMFT LDA+U Impurity Imp. Solver: Solver Hartree-Fock SCC THE STATE UNIVERSITY OF NEW JERSEY RUTGERS G, S U

Review A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68, 1 (1996) 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 Phys. B 15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar condmat 0211076(2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n Introduction: some Pu puzzles. Introduction to DMFT. Some qualitative insights from model Hamiltonian studies. Towards and understanding of elemental Pu. Conclusions 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

Cerium THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Evolution at T=0 , half filling full frustration, Hubbard bands in the metal, transfer of spectral weight. X. Zhang M. Rozenberg G. Kotliar (PRL 1993) A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL (2002). n n Coexistence regions between localized and delocalized spectral functions. k diverges at generic Mott endpoints 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

Generalized phase diagram T Structure, bands, orbital s U/W 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

Outline n n n 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

What is the dominant atomic configuration? Local moment? n n n Snapshots of the f electron Dominant configuration: (5 f)5 Naïve view Lz=-3, -2, -1, 0, 1 ML=-5 m. B S=5/2 Ms=5 m. B Mtot=0 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

GGA+U bands. Savrasov Kotliar , Phys. Rev. Lett. 84, 3670 -3673, (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

How is the Magnetic moment quenched. n n n L=5, S=5/2, J=5/2, Mtot=Ms=m. B g. J =. 7 m. B Crystal fields G 7 +G 8 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). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu: DMFT total energy vs Volume S. Savrasov, G. Kotliar, and E. Abrahams, Nature 410, 793 (2001), THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Double well structure and d Pu Qualitative explanation of negative thermal expansion THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Dynamical Mean Field View of Pu (Savrasov Kotliar and Abrahams, Nature 2001) n Delta and Alpha Pu are both strongly n 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comments on the HF static limit for Pu n n Describes only the Hubbard bands. No QP states. Single well structure in the E vs V curve. (Savrasov and Kotliar PRL). Same if one uses a Hubbard one impurity solver. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Lda vs Exp Spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Summary Spectra Method LDA+U DMFT E vs V

The delta –epsilon transition n The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? A functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Effects of structure. GGA+DMFT_Hubbard 1 imp. solver Ee-Ed=350 K GGA gives Ee-Ed= -6000 K THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonon freq (THz) vs q in delta Pu (S. Savrasov) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonon frequency (Thz ) vs q in epsilon Pu. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Epsilon plutonium. n n Compute the energy of the most unstable frozen mode. Transverse mode at ( 0, pi) with polarization (0, 1, -1). Extrapolate the form of the quartic interaction to the whole Brillouin zone. Carry out a self consistent Born approximation to obtain the restabilize phones. Recompute the entropy difference between delta and epsilon. Estimate the critical temperatures: 500 -700 K , depending on the detials of the extrapolation. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonon entropy drives the epsilon delta phase transition n n Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonons epsilon THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n 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 into other phases. Conclusions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conclusions n n n DMFT produces non magnetic state, around a fluctuating (5 f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve. Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon). Calculations can be refined. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conclusions n n n Phonons matter. Role of electronic entropy. In the making, new generation of DMFT programs, QMC with multiplets, full potential DMFT, frequency dependent U’s, multiplet effects , combination of DMFT with GW Other materials, Cerium and Yterbium compounds………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Acknowledgements: Development of DMFT Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, D. Fisher, A. Georges, H. Kajueter, W. Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, S. Pankov, M. Rozenberg, S. Murthy , S. Savrasov, Q. Si, V. Udovenko, X. Y. Zhang Funding: National Science Foundation. Department of Energy and LANL. Office of Naval Research. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Technical details n n n 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, + 1 st mode coupling correction), match at intermediate frequencies. (Savrasov et. al 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Technical details n Atomic sphere approximation. n Ignore crystal field splittings in the self energies. n Fully relativistic non perturbative treatment of the spin orbit interactions. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Solving the DMFT equations • Wide variety of computational (QMC, ED…. )Analytical Methods • Extension to ordered states. tools Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68, 13 (1996)] n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Temperature stabilizes a very anharmonic phonon mode THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LSDA+DMFT functional F Sum of local 2 PI graphs with local U matrix and local G THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

E-DMFT references n n n H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph. D thesis (1995)). Q. Si and J L Smith PRL 77 (1996)3391. R. Chitra and G. Kotliar Phys. Rev. Lett 84, 3678 -3681 (2000 ) Y. Motome and G. Kotliar. PRB 62, 12800 (2000) R. Chitra and G. Kotliar Phys. Rev. B 63, 115110 (2001) S. Pankov and G. Kotliar PRB 66, 045117 (2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT Impurity cavity construction THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Cluster extensions of DMFT n n Two impurity method. [A. Georges and G. Kotliar (1995 unpublished ) and RMP 68, 13 (1996) , A. Schiller PRL 75, 113 (1995)] M. Jarrell et al Dynamical Cluster Approximation [Phys. Rev. B 7475 1998] Periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000). G. Kotliar S. Savrasov G. Palsson and G. Biroli Cellular DMFT [PRL 87, 186401 2001] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

C-DMFT C: DMFT The lattice self energy is inferred from the cluster self energy. Alternative approaches DCA (Jarrell et. al. ) Periodic clusters (Lichtenstein and Katsnelson) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMF T vs Nc=1 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT MODELS. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mean-Field : Classical vs Quantum Classical case Phys. Rev. B 45, 6497 Quantum case THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A. Georges, G. Kotliar (1992)

Example: Single site DMFT, functional formulation Local self energy (Muller Hartman 89) n Express in terms of Weiss field (G. Kotliar EPJB 99) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT Impurity cavity construction 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

Case study: IPT half filled Hubbard one band n n n (Uc 1)exact = 2. 2+_. 2 (Exact diag, Rozenberg, Kajueter, Kotliar PRB 1996) , confirmed by Noack and Gebhardt (1999) (Uc 1)IPT =2. 6 (Uc 2)exact =2. 97+_. 05(Projective self consistent method, Moeller Si Rozenberg Kotliar Fisher PRL 1995 ), (Confirmed by R. Bulla 1999) (Uc 2)IPT =3. 3 (TMIT ) exact =. 026+_. 004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =. 045 (UMIT )exact =2. 38 +-. 03 (QMC Rozenberg Chitra and Kotliar PRL 1999), (UMIT )IPT =2. 5 (Confirmed by Bulla 2001) For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Density Functional n n n 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 GDFT[r(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

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

E-DMFT+GW effective action G= D= THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

E-DMFT +GW P. Sun and G. Kotliar Phys. Rev. B 2002 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA+DMFT and LDA+U Static limit of the LDA+DMFT functional , • with Fatom FHF reduces to the LDA+U functional of Anisimov Andersen and Zaanen. • Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems. • 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 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

Interface DMFT with electronic structure. Derive model Hamiltonians, solve by DMFT (or cluster extensions). Total energy? q Full many body aproach, treat light electrons by GW or screened HF, heavy electrons by DMFT [E-DMFT frequency dependent interactions. GK and S. Savrasov, P. Sun and GK cond-matt 0205522] q Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT) n 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 G 0 DMFT LDA+U Impurity Imp. Solver: Solver Hartree-Fock SCC THE STATE UNIVERSITY OF NEW JERSEY RUTGERS G, S U

LDA+DMFT and LDA+U Static limit of the LDA+DMFT functional , • with Fatom FHF reduces to the LDA+U functional of Anisimov Andersen and Zaanen. • Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems. • Total energy in DMFT can be approximated by LDA+U with an effective U. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Very Partial list of application of realistic DMFT to materials n n n n 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 V 2 O 3: 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). 3 d doped Mott insulator La 1 -x. Srx. Ti. O 3 Anisimov et. al 1997, Nekrasov et. al. 1999, Udovenko et. al 2002) ………………. . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA+DMFT References Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997). n 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 Phys. B 15, 2611 (2001). A. Lichtenstein M. Katsnelson and G. Kotliar (2002) n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

LDA+DMFT functional F Sum of local 2 PI graphs with local U matrix and local G THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comments on LDA+DMFT • • 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

References n n n 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). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Debye temperatures THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

References n n 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Wilson and Kadowaki Woods Ratio THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Dependence on structure Expt: Ve-Vd=. 54 A Theory: Ve-Vd=. 35 A THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Dynamical Mean Field Theory(DMFT) Review: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68, 1 (1996) Local approximation (Treglia and Ducastelle PRB n n n 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 prb 45, 6479 (1992)). See also M. Jarrell (PRL 1992). Connect local spectra and lattice total energy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n n Correlation betwee the Minimum of the melting point and the Mott transition endpoint. 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Localization delocalization transition and f electrons. n n n Mott phenomena. 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)] 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT and f electrons. n n n These views of the localization delocalization transition are not orthogonal and were incorporated into a more complete Dynamical Mean Field picture of the Mott transition. G. Kotliar, EPJ-B, 11, (1999), 27. A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68, 1 (1996). Moeller Q. Si G. Kotliar M. Rozenberg and D. S Fisher, PRL 74 (1995) 2082. DMFT: Powerful new tool for studying f electrons. Qualitative insights into complex materials. Turn technology developed to solve models into practical quantitative electronic structure method , to study eg. PU. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Evolution of the Spectral Function with Temperature near Mott endpoint. Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

view. R. Chitra and G. Kotliar Phys Rev. B. n (2000), (2001). Identify observable, A. Construct an exact functional of n n n <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

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

Mott transition in layered organic conductors et al. Ito et. al, Kanoda’s talk Bourbonnais talk Magnetic Frustrati on THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre

Ultrasound study of Fournier et. al. (2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comparaison with LDA+U THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

view. R. Chitra and G. Kotliar Phys Rev. B. (2000), Phys. Rev. B 63, 115110 n Identify observable, A. Construct an exact functional of (2001) <A>=a, G [a] which is stationary at the physical value of a. n n n 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Example: DMFT for lattice model (e. g. single band Hubbard). n n n Observable: Local Greens function Gii (w). Exact functional G [Gii (w) ]. DMFT Approximation to the functional. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Wilson and Kadowaki Woods Ratio THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comparaison with the Hartree Fock static limit: GGA+U. Volume, total energies are OK much better than LDA, but no double minima Ee-Ed=350 K THE STATE UNIVERSITY OF NEW JERSEY RUTGERS