Electronic Structure of Correlated Materials a DMFT Perspective

Electronic Structure of Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University and KITP Institute for Theoretical Physics UCSB Santa Barbara Brookhaven National Laboratory September 12 th 2002 Supported by the NSF DMR 0096462

Outline n n n The Mott transition problem and electronic structure. Dynamical Mean Field Theory Model Hamiltonian Studies of the Mott transition. Universal aspects. System specific studies of materials. LDA+DMFT. Some case studies. Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Weakly correlated electrons: band theory. n n Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors, …. ). A methods for performing quantitative calculations. (Density functional theory, in various approximations). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The electron in a solid: wave picture Momentum Space (Sommerfeld) Maximum metallic resistivity 200 mohm cm Standard model of solids (Bloch, Landau) Periodic potential, waves form bands , k in Brillouin zone. Interactions renormalize away. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Standard Model of Solids § § Qualitative predictions: low temperature dependence of thermodynamics and transport. Optical response, transition between the bands. Qualitative predictions: filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals. Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra, eg. GW. Kinetic equations yield transport coefficients. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Kohn Sham reference system Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Success story : Density Functional Linear Response Tremendous progress in ab initio modelling of lattice dynamics & electron-phonon interactions has been achieved (Review: Baroni et. al, Rev. Mod. Phys, 73, 515, 2001) (Savrasov, PRB 1996)

The electron in a solid: particle picture. n Array of hydrogen atoms is insulating if a>>a. B. Mott: correlations localize the electron e_ e_ e_ Superexchange Think in real space , solid collection of atoms High T : local moments, Low T spin-orbital order THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott : Correlations localize the electron Low densities, electron behaves as a particle, use atomic physics, real space One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bands. Ni. O, Co. O Mn. O…. ) H H H+ H H H- H H motion of H+ forms the lower Hubbard band motion of H_ forms the upper Hubbard band Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Photoemission spectroscopy. Measures density of states for (BIS)adding and (PES) removing electrons THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Localization vs Delocalization Strong Correlation Problem • A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock work well. • Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strong correlation anomalies n n n Metals with resistivities which exceed the Mott Ioffe Reggel limit. Transfer of spectral weight which is non local in frequency. Dramatic failure of DFT based approximations in predicting physical properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Correlated Materials do big things n n Huge resistivity changes V 2 O 3. Copper Oxides. . (La 2 -x Bax) Cu. O 4 High Temperature Superconductivity. 150 K in the Ca 2 Ba 2 Cu 3 Hg. O 8. Uranium and Cerium Based Compounds. Heavy Fermion Systems, Ce. Cu 6, m*/m=1000 (La 1 -x. Srx)Mn. O 3 Colossal Magneto-resistance. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strongly Correlated Materials. n n Large thermoelectric response in Ce. Fe 4 P 12 (H. Sato et al. cond-mat 0010017). Ando et. al. Na. Co 2 -x. Cux. O 4 Phys. Rev. B 60, 10580 (1999). Large and ultrafast optical nonlinearities Sr 2 Cu. O 3 (T Ogasawara et. a Phys. Rev. Lett. 85, 2204 (2000) ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in V 2 O 3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre et

Failure of the Standard and Model: Ni. Se. Miyasaka S 2 -x x (2000) Takagi 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)

Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT Impurity cavity construction THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

E-DMFT references n n H. Kajueter and G. Kotliar (unpublished and Kajuter’s Ph. D thesis). Q. Si and Smith PRL [analysis of quantum critical points] 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

Insights from DMFT q Low temperature Ordered phases. Stability depends on chemistry and crystal structure q. High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration) M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Kuwamoto Honig and Appell PRB (1980) M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre et

Phase Diag: Ni Se 2 -x Sx 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, 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 the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (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. Doniach and Watanabe Phys. Rev. B 57, 3829 (1998) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in V 2 O 3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous transfer of optical spectral weight V 2 O 3 : M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996). M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Spectral Weight Transfer: Optics Below energy Apreciable. T dependence found. Schlesinger et. al (Fe. Si) PRL 71 , 1748 , (1993) B Bucher et. al. Ce 2 Bi 4 Pt 3 PRL 72, 522 (1994), Rozenberg et. al. PRB 54, 8452, (1996). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

ARPES measurements on Ni. S 2 -x. Sex . Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous transfer of optical spectral weight, Ni. Se. S. [Miyasaka and Takagi] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivity and Mott transition Ni Se 2 -x Sx Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

Qualitative phase diagram in the U, T , m plane (two band Kotliar Murthy Rozenberg PRL (2002). n Coexistence regions between localized and delocalized spectral functions. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

QMC calculationof n vs m (Kotliar Murthy Rozenberg PRL 2002, 2 band, U=3. 0) k diverges at generic Mott endpoints RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre et

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights from DMFT n n n Mott transition as a bifurcation of an effective action Important role of the incoherent part of the spectral function at finite temperature Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Two roads for ab-initio calculation of electronic structure of strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n n n Realistic Calculations of the Electronic Structure of Correlated materials Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials. Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997) Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Combining LDA and 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, subtract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles or viewed as parameters n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Materials…… n n n Pu Fe, Ni, La 1 -x Srx Ti. O 3 Ni. O ……………. 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

Problems with density functional treatements of d Pu DFT in the LDA or GGA is a well established tool for the calculation of ground state properties. • Many studies (APW Freeman, Koelling 1972, ASA and FP-LMTO, Soderlind et. al 1990, Kollar et. al 1997, Boettger et. al 1998, Wills et. al. 1999) show • an equilibrium volume of the d phase Is 35% lower than experiment • This is the largest discrepancy ever known in DFT based calculations. • 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% • Weak correlation picture for alpha phase. •

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

Lda vs Exp Spectra (Joyce et. al. ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Pu n Alpha and delta are strongly correlated, The coexistence in the toy model, give rise to two distinct phases in the realistic calculation. In progress: phonon spectrum, epsilon phase…… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Lax. Sr 1 -x. O 3 n n Adding holes to a Mott insulator in three dimensions. Canonical example of a Brinkman Rice system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

(Tokura et. Al. 1993)A doped Mott insulator: Lax. Sr 1 -x. O 3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT calculation U near the Mott transition, Rozenberg et. al 94 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Hall Coefficient, electron like. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

La 1 -x. Srx. Ti. O 3 photoemission THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Evolution of spectra with doping U=4 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Optical conductivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Realistic Computation of Optical Properties : La 1 x. Srx. Ti. O 3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Case study Fe and Ni n n Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T Main puzzle: at high temperatures c has a Curie Weiss law with a moment larger than the ordered moment. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205 , 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Iron and Nickel: magnetic properties (Lichtenstein, Katsenelson, GK PRL 01) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ni and Fe: theory vs exp n meff / m. B high T moment Fe 3. 1 (theory) 3. 12 (expt) Ni 1. 5 (theory) 1. 62 (expt) Curie Temperature Tc n n Fe 1900 Ni 700 ( theory) (theory) 1043(expt) 631 (expt) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Fe and Ni Consistent picture of Fe (more localized) and Ni (more correlated) n Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction. 3 ev n Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=. 9 ev, T/Tc=. 8) (Lichtenstein, Katsenelson, Kotliar Phys Rev. Lett 87, 67205 , 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Photoemission and T/Tc=. 8 Spin Autocorrelation: Ni (U=3, J=. 9 ev) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failures of lda for ground state properties. Ni. O dielectric constant. LSDA: 35. 7 Exp: 5. 7 Lattice dynamics cannot be predicted: • Optical G-phonon in Mn. O within LSDA: 3. 04 THz, Experimentally: 7. 86 THz (Massidda, et. al, PRL 1999) • Bulk modulus for metallic Plutonium is one order of magnitude too large within LDA (214 GPa vs. 30 GPa) Also elastic constants are off. (Bouchet, et. al, J. Phys. C, 2001)

Functional approach allows computation of linear response. (S. Savrasov and GK 2002 Apply to Ni. O, canonical Mott insulator. U=8 ev, J=. 9 ev Simple Impurity solver Hubbard 1. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Results for Ni. O: Phonons Solid circles – theory, open circles – exp. (Roy et. al, 1976)

Ni. O U=8 ev, J=1 ev, Savrasov and GK (2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Summary n n n Introduction to strongly correlated electrons Dynamical Mean Field Theory Model Hamiltonian Studies. Universal aspects insights from DMFT System specific studies: LDA+DMFT Outlook 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 et al (Nature 2001) MIT in V 2 O 3: K. Held et al (PRL 2001) Magnetism of Fe, Ni: A. Lichtenstein et al PRL (2001) J-G transition in Ce: K. Held et al (PRL 2000); M. Zolfl 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

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, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, X. Y. Zhang Support: National Science Foundation. Work on Pu: Departament of Energy and LANL. Work on Fe and Ni: Office of Naval Research THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Challenges n n Short Range Magnetic Correlations without magnetic order. Single Site DMFT does not capture these effects THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu: Anomalous thermal expansion (J. Smith LANL) 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% Notice however that LDA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that a Pu is a weakly correlated system THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conventional viewpoint n n n 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 5 f electrons are treated as band like, while in Delta Pu, 4 5 f electrons are band-like while one 5 f electron is deloclized. Same situation in LDA + U (Savrasov and. GK 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 n n 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) n Delta and Alpha Pu are both strongly n 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. 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 n n n Archetypical itinerant ferromagnets LSDA predicts correct low T moment Band picture holds at low T But at high temperatures, they resemble more a collection of atoms with reduced moment. The crossover from low T to high T requires DMFT. 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. . n n n Magnetic anisotropy puzzle. LDA predicts the incorrect easy axis for Nickel. (instead of 111) LDA Fermi surface has features which are not seen in De. Haas Van Alphen ( Lonzarich) Use LDA+ U to tackle these refined issues, (WE cannot be resolved with DMFT, compare parameters with Lichtenstein’s ) RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

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, 11919 (1990) Trygg, Johansson Eriksson and Wills Phys. Rev. Lett. 75 2871 (1995) Schneider M Erickson and Jansen J. Appl Phys. 81 3869 (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 n n n 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, T 2 g 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

Conclusion n § 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

Resistivities THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Resistivities THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Li. VO 4 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Thermodynamics Li. VO 4 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Resistivity saturation THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

RUTGERS THE STATE UNIVERSITY OF NEW JERSEY other types of printers. cript printer, but not to Comment: was not saved a preview included in it. Preview: Creator: gnuplot Title: IPT NCA Anomalous Resistivities: Doped Hubbard Model G. Palsson 1998

Anomalous Resistivities: Doped Hubbard Model (QMC) Prushke and Jarrell 1993 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outlook n n Systematic improvements, short range correlations. Take a cluster of sites, include the effect of the rest in a G 0 (renormalization of the quadratic part of the effective action). What to take for G 0: 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 , 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 EDMFT 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 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 3

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conventional viewpoint n n n 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 5 f electrons are treated as band like, while in Delta Pu, 4 5 f electrons are band-like while one 5 f electron is deloclized. Same situation in LDA + U (Savrasov and. GK Bouchet et. al. [Bouchet’s talk]). Delta Pu has U=4, Alpha Pu has U =0. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Delocalization-Localization across the actinide series o o o f electrons in Th Pr U Np are itinerant. From Am on they are localized. Pu is at the boundary. Pu has a simple cubic fcc structure, the d phase which is easily stabilized over a wide region in the T, p phase diagram. The d phase is non magnetic. 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 d phase Is 35% lower than experiment This is one of the largest discrepancy ever known in DFT based calculations. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu: Anomalous thermal expansion (J. Smith LANL) 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% Notice however that LDA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills). This is usually taken as an indication that a Pu is a weakly correlated system THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conventional viewpoint n n n 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 5 f electrons are treated as band like, while in Delta Pu, 4 5 f electrons are band-like while one 5 f electron is deloclized. Same situation in LDA + U (Savrasov and. GK 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 n n 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) n Delta and Alpha Pu are both strongly n 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. This result resolves one of the basic paradoxes in the physics of Pu. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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 Removes inconsistencies and shortcomings of this approach. DMFT retain correlations effects in the absence of orbital ordering. • • 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

Anomalous Resistivities: Doped Hubbard Model (QMC) Prushke and Jarrell 1993 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

RUTGERS THE STATE UNIVERSITY OF NEW JERSEY other types of printers. cript printer, but not to Comment: was not saved a preview included in it. Preview: Creator: gnuplot Title: IPT NCA Anomalous Resistivities: Doped Hubbard Model G. Palsson 1998

DMFT: Methods of Solution THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA functional Conjugate field, VKS(r) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Minimize LDA functional Kohn Sham eigenvalues, auxiliary THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous transfer of spectral weight heavy fermions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous transfer of spectral weight THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous transfer of spectral weigth heavy fermions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

V 2 O 3 resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA+DMFT Self-Consistency loop E U DMFT 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

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

Ni and Fe: theory vs exp n n n m( T=. 9 Tc)/ m. B ordered moment Fe 1. 5 ( theory) Ni. 3 (theory) meff / m. B 1. 55 (expt). 35 (expt) high T moment Fe 3. 1 (theory) 3. 12 (expt) Ni 1. 5 (theory) 1. 62 (expt) Curie Temperature Tc n n Fe 1900 Ni 700 ( theory) (theory) 1043(expt) 631 (expt) 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

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 expanding around the atomic limit. 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. Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA functional Conjugate field, VKS(r) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failure of the Standard Model: Anomalous Spectral Weight Transfer Optical Conductivity o of Fe. Si for T=, 20, 250 200 and 250 K from Schlesinger et. al (1993) Neff depends on T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Degenerate Hubbard model q. U/t q. Doping d or chemical potential q. Frustration (t’/t) q. T temperature Mott transition as a function of doping, pressure RUTGERS temperature etc. THE STATE UNIVERSITY OF NEW JERSEY

Outlook q The Strong Correlation Problem: How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR q. Strategy: advancing our understanding scale by scale q. Generalized cluster methods to capture longer range magnetic correlations q. New structures in k space. Cellular DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Small amounts of Ga stabilize the d phase 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

res 85520219 st 9 v 9 t 9 es THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA functional Conjugate field, VKS(r) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Hubbard model q. U/t q. Doping d or chemical potential q. Frustration (t’/t) q. T temperature Mott transition as a function of doping, pressure RUTGERS temperature etc. THE STATE UNIVERSITY OF NEW JERSEY

Dynamical Mean Field Theory, cavity construction A. Georges G. Kotliar 92 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

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

Single site DMFT, functional formulation. Construct a functional of the observable , local Greens function. Express in terms of Weiss field, (semicircular. DOS) [G. Kotliar EBJB 99] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Compressibilty divergence : One band case (Kotliar Murthy and Rozenberg 2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and Kotliar). 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. GDFT[r(r)] n Introduce local orbitals, ca. R(r-R)orbitals, and local GF G(R, R)(i w) = n The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing G[r(r), G(R, R)(iw)] n A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and n Abrahams Nature 410, 793 (2001)) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Optical spectral weight THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Parallel development: Fujimori et. al THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

C-DMFT C: DMFT The lattice self energy is inferred from the cluster self energy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS