Electronic Structure Near the Mott transition Gabriel Kotliar

Electronic Structure Near the Mott transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

Outline n n Introduction to the strong correlation problem and to the Mott transition Some dynamical mean field ideas Applications to the Mott transition problem: some insights from studies of model Hamiltonians. n Towards an electronic structure method: applications to materials. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The electron in a solid: wave picture Momentum Space , bands, k in Brillouin zone is good quantum number. Maximum metallic resistivity 200 mohm cm Landau Fermi liquid theory interactions renormalize away at low energy, simple band picture in effective field holds. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Standard Model of Solids n Qualitative predictions: low temperature dependence of thermodynamics and transport Optical response, transitions between bands. Qualitative predictions. Filled bands-Insulators, Unfilled bands metals. Odd number of electrons metallicity. Quantitative tools: DFT, LDA, GGA, total energies, good starting point for spectra, GW, and transport THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The electron in a solid: particle picture. n Ni. O, Mn. O, …Array of atoms is insulating if a>>a. B. Mott: correlations localize the electron 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, work in 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 motion of H+ forms the lower Hubbard band • H H- H H motion of H_ forms the upper Hubbard band • Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic RUTGERS Physics. THE STATE UNIVERSITY OF NEW JERSEY

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

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 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). Huge volume collapses, Ce, Pu…… 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

The Mott transition Electronically driven MIT. n Forces to face directly the localization delocalization problem. n Relevant to many systems, eg V 2 O 3 n Techniques applicable to a very broad range or problems. n 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

Universal and non universal features. Top to bottom approach to correlated materials. Some aspects at high temperatures, depend n n weakly on the material (and on the model). Low temperature phase diagram, is very sensitive to details, in experiment (and in theory). 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

Phase Diagrams : V 2 O 3, Ni Se 2 -x Sx Mc Whan et. Al 1971, . Czek et. al. J. Mag. Mat. 3, 58 (1976), THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n Introduction to the strong correlation problem and to the Mott transition. DMFT ideas Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Ni. O, Pu, Fe, Ni, La. Sr. Ti. O 3, ………. Outlook 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

Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89 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)

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

view. R. Chitra and G. Kotliar Phys Rev. B. n (2000). 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Example: DMFT for lattice model (e. g. single band Hubbard). Muller Hartman 89, Chitra and Kotliar 99. 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

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, GW+EDMFT or “GWU”…. . n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V 2 O 3 from optics. M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) 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. Three peak structure. X. Zhang M. Rozenberg G. Kotliar (PRL 1993) 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

Insights from DMFT n n Three peak structure of the density of states. In the strongly correlated metallic regime the Hubbard bands are well formed. 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

n What about experiments? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Parallel development: Fujimori et. al 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 transfer of spectral weight in v 2 O 3 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. Doniach 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 2000] 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

Recent exps. Moo et. al. (2003) Theory Held et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

• Transport in 2 d organics. Limlet et. al. 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

Conclusions: generic aspects n n n Three peak structure, quasiparticles and Hubbard bands. Non local transfer of spectral weight. Large resistivities. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights from DMFT. n n 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 pictures are needed as synthesized in DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Pu, Fe, Ni, Ce, La. Sr. Ti. O 3, Ni. O, Mn. O, Cr. O 2, K 3 C 60, 2 d and quasi-1 d organics, magnetic semiconductors, Sr. Ru. O 4, V 2 O 3…………. Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Interface DMFT with electronic structure. n q q Derive model Hamiltonians, solve by DMFT (or cluster extensions). Full many body aproach, treat light electrons by GW or screened HF, heavy electrons by DMFT. Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

view. R. Chitra and G. Kotliar Phys Rev. B. n (2000). 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. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Spectral Density Functional : effective action construction (Chitra and GK). n n Introduce local orbitals, ca. R(r-R)orbitals, 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)] Approximate functional using DMFT insights. 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: Savrasov GK and Abrahams (Nature 2001) Dai Savrasov GK Migliori Letbetter and Abrahams (Science 2003) 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 2003) Paramagnetic Mott insulators. Ni. O Mn. O, Savrasov and RUTGERS GK( PRL 2002)………… THE STATE UNIVERSITY OF NEW JERSEY

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

Physics of Pu 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 Alternative approach Wills et. al. (5 f)4 core+ 1 f(5 f)in conduction band. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Shear anisotropy fcc Pu (GPa) n C’=(C 11 -C 12)/2 n C 44= 33. 59 n n = 4. 78 C 44/C’ ~ 8 Largest shear anisotropy in any element! LDA Calculations (Bouchet et. al. ) C’= -48 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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 (Savrasov and Kotliar PRL 2002) Solid circles – theory, open circles – exp. (Roy et. al, 1976) DMFT

Phases of Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Dai et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Epsilon Plutonium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n Introduction to the strong correlation problem. Essentials of DMFT Applications to the Mott transition problem: some insights from studies of models. Towards an electronic structure method: applications to materials: Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outlook n n Local approach to strongly correlated electrons. Many extensions, make the approach suitable for getting insights and quantitative results in correlated materials. 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 to the electronic structure of correlated materials. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outlook n n n Systematic improvements, short range correlations, cluster methods, improved mean fields. Improved interfaces with electronic structure. Exploration of complex strongly correlated materials. Correlation effects on surfaces, large molecules, systems out of equilibrium, illumination, finite currents, aeging. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Acknowledgements: Development of DMFT Collaborators: V. Anisimov, G. Biroli, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, K. Haujle, W. Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, O. Parcollet , G. Palsson, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X. Y. Zhang Support: NSF DMR 0096462 Support: Instrumentation. NSF DMR-0116068 Work on Fe and Ni: ONR 4 -2650 Work on Pu: DOE DE-FG 02 -99 ER 45761 and LANL subcontract No. 03737 -001 -02 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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Expts’ Wong et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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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, 36783681 (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

DMFT plus other methods. n n DMFT+ LDA , 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 and G. Kotliar, funcional formulation for full self consistent implementation. Savasov Kotliar and Abrahams. Application to delta Pu Nature (2001) Combining EDMFT with GW. Ping Sun and Phys. Rev. B 66, 085120 (2002). G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259 -301. cond-mat/0208241 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

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

QP in V 2 O 3 was recently found Mo et. al 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

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

Anomalous Spectral Weight Transfer: Optics Below energy Appreciable. 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

Comments on LDA+DMFT • • Static limit of the LDA+DMFT functional , with F= FHF 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

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

view. R. Chitra and G. Kotliar Phys Rev. B. n (2000). 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

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

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