Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University ISSP-Kashiwa 2001 Tokyo 1 st -5 th October

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS the 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. New insights and new techniques from the solution of the Mott transition problem within dynamical mean field of simple model Hamiltonians Use the ideas and concepts that resulted from this development to give physical insights into real materials. Steps taken to turn the technology developed to solve the toy models into a practical electronic structure method.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Background: DMFT study of the Mott transition in a toy model. Behavior of the compressibility near the Mott transition endpoint. DMFT as an electronic structure method. From Lda to LDA+U to LDA+ DMFT. DMFT results for delta Pu, and some qualitative insights into the “Mott transition across the actinide series” Fe and Ni, a new look at the classic itinerant ferromagnets

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Goal of the talk Describe some recent steps taken to make DMFT into an electronic structure tool. model Hamiltonian review see A. Georges talk in this workshop and consult reviews: 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 Outline: Choice of Basis. Realistic self consistency condition Brief Comment on Impurity Solvers Integration with LDA. Effective action formulation. Comparison with LDA and LDA+U Some examples in real materials, transition metals and actinides.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Collaborators, Colleagues, Support for realistic work…………. S. Lichtenstein (Nijmeigen), E Abrahams (Rutgers) G. Biroli (Rutgers), R. Chitra (Rutgers- Jussieux), V. Udovenko (Rutgers), S. Savrasov (Rutgers-NJIT) G. Palsson, I. Yang (Rutgers) NSF, DOE and ONR

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field

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. Strategy, look electronic structure problems where this physics is absolutely essential, Fe, Ni, Pu …………….

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elements of the Dynamical Mean Field Construction and C-DMFT. Definition of the local degrees of freedom Expression of the Weiss field in terms of the local variables (I.e. the self consistency condition) Expression of the lattice self energy in terms of the cluster self energy.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cellular DMFT : Basis selection. Exact spectra is basis independent DMFT results are not.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice action

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elimination of the medium variables

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Determination of the effective medium.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Connection between cluster and lattice self energy. The estimation of the lattice self energy in terms of the cluster energy has to be done using additional information Ex. Translation invariance C-DMFT is manifestly causal: causal impurity solvers result in causal self energies and Green functions ( GK S. Savrasov G. Palsson and G. Biroli ) Improved estimators for the lattice self energy are available (Biroli and Kotliar) In simple cases C-DMFT converges faster than other causal cluster schemes.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Convergence of CDMFT, test in a soluble problem (G. Biroli and G. Kotliar)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT self consistency loop

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic implementation of the self consistency condition H and S, do not commute Need to do k sum for each frequency DMFT implementation of Lambin Vigneron tetrahedron integration V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, (1997). Transport Coeff (G. Palsson V. Udovenko and G. Kotliar)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of computational tools (QMC, NRG,ED….) Semi-analytical Methods

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT+QMC (A. Lichtenstein, M. Rozenberg)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the impurity 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 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 Recent QMC phase diagram of the frustrated Half filled Hubbard model with semicircular DOS ( Joo and Udovenko).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Case study: IPT half filled Hubbard one band (Uc1) exact = 2.1 (Exact diag, Rozenberg, Kajueter, Kotliar PRB 1996), (Uc1) IPT =2.4 (Uc2) exact =2.97+_.05(Projective self consistent method, Moeller Si Rozenberg Kotliar Fisher PRL 1995 ) (Uc 2 ) IPT =3.3 (T MIT ) exact =.026+_.004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (T MIT ) IPT =.5 (U MIT ) exact = (QMC Rozenberg Chitra and Kotliar PRL 1991), (U MIT ) IPT =2.5 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 Compressibility near a Mott transition Interaction driven Mott transition Brinkman Rice.  ~ (Uc –U) Doping driven Mott transition (Gutzwiller, Brinkman Rice, Slave Boson method).  is non singular Numerical simulations T=0 QMC,.  diverges As 1/  (Furukawa and Imada)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition as a bifurcation At different points in the phase diagram, different behaviors.  vanishes at Uc2 (interaction driven Mott transition) At zero temperature  is non singular, at the doping driven Mott transition Behavior at U MIT T MIT ?

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition as a bifurcation in effective action Zero mode with S=0 and p=0, couples generically Divergent compressibility

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative phase diagram in the U, T,  plane ( Murthy Rozenberg and Kotliar 2001 )

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QMC calculationof n vs  ( Murthy Rozenberg and Kotliar 2001, 2 band model, U=3.0 )

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QMC n vs  ( Murthy Rozenberg and Kotliar 2001, 2 band, low T

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Compresibility vs T

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 functional Conjugate field, V KS (r)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Minimize LDA functional

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+U functional

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double counting term (Lichtenstein et.al)

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 Comments on LDA+DMFT Static limit of the LDA+DMFT functional, with  =  HF 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 Connection with atomic limit Weiss field

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic DMFT loop

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 G.Kotliar, funcional formulation for full self consistent implementation of a spectral density functional( cond- mat 2001)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional Approach The functional approach offers a direct connection to the atomic energies. One is free to add terms which vanish quadratically at the saddle point. Allows us to study states away from the saddle points, All the qualitative features of the phase diagram, are simple consequences of the non analytic nature of the functional. Mott transitions and bifurcations of the functional.

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

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 Pu: Anomalous thermal expansion (J. Smith LANL)

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% Notice however that LDA predicts correctly the volume of the  phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that  Pu is a weakly correlated system

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conventional viewpoint Alpha Pu is a simple metal, it can be described with LDA + correction. In contrast delta Pu is strongly correlated. Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized. Same situation in LDA + U (Savrasov andGK Bouchet et. al. [Bouchet’s talk]).Delta Pu has U=4,Alpha Pu has U =0.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Problems with the conventional viewpoint of Pu 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. Only the structural and elastic properties are completely different.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Specific Heat

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity J. Smith LANL

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS MAGNETIC SUSCEPTIBILITY

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field View of Pu ( Savrasov Kotliar and Abrahams, Nature 2001) 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. This result resolves one of the basic paradoxes in the physics of Pu.

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 PU: ALPHA AND DELTA

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 Main challenge, finite T properties (Lichtenstein’s talk).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS However not everything in low T phase is OK as far as LDA goes.. Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis for Nickel.(instead of 111) LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich) Use LDA+ U to tackle these refined issues, (WE cannot be resolved with DMFT, compare parameters with Lichtenstein’s )

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Some Earlier Work: Kondorskii and E Straube Sov Phys. JETP 36, 188 (1973) G. H Dallderop P J Kelly M Schuurmans Phys. Rev. B 41, (1990) Trygg, Johansson Eriksson and Wills Phys. Rev. Lett (1995) Schneider M Erickson and Jansen J. Appl Phys (1997) I Solovyev, Lichenstein Terakura Phys. Rev. Lett 80, 5758 (LDA+U +SO Coupling)……. Present work : Imseok Yang, S Savrasov and GK

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Origin of Magnetic Anisotropy Spin orbit coupling L.S L is a variable which is sensitive to correlations, a reminder of the atomic physics Crystal fields quench L, interactions enhance it, T2g levels carry moment, eg levels do not any redistribution of these no matter how small will affect L. Both J and U matter !

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic anisotropy of Fe and Ni LDA+ U

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Surprise correct Ni Fermi Surface!

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 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 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 Palsson and GK ) 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, 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 Ni moment

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Fe moment

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic anisotropy vs U, J=.95 Ni 1 3

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Magnetic anisotropy Fe J=.8