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## Electronic Structure Near the Mott transition

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**Electronic Structure Near the Mott transition**Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University**Outline**• 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. • Towards an electronic structure method: applications to materials. 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**• 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.**• NiO, MnO, …Array of atoms is insulating if a>>aB. Mott: correlations localize the electron e_ 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**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. bandsNiO, CoO MnO….) Low densities, electron behaves as a particle,use atomic physics, work in real space. H H H+ H H H motion of H+ forms the lower Hubbard band H H 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 Physics. 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**Correlated Materials do big things**• Huge resistivity changes V2O3. • Copper Oxides. .(La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 . • Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000 • (La1-xSrx)MnO3 Colossal Magneto-resistance. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Strongly Correlated Materials.**• Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999). • Huge volume collapses, Ce, Pu…… • Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**The Mott transition**• Electronically driven MIT. • Forces to face directly the localization delocalization problem. • Relevant to many systems, eg V2O3 • Techniques applicable to a very broad range or problems. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Mott transition in V2O3 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 weakly on the material (and on the model). • Low temperature phase diagram, is very sensitive to details, in experiment (and in the theory). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Mott transition in layered organic conductors S**Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Failure of the Standard Model: NiSe2-xSx**Miyasaka and Takagi (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Phase Diagrams :V2O3, Ni Se2-x SxMc Whan et. Al 1971,. Czek**et. al. J. Mag. Mag. Mat. 3, 58 (1976), THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• 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: NiO, Pu, Fe, Ni, LaSrTiO3, ………. • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Hubbard model**• U/t • Doping d or chemical potential • Frustration (t’/t) • T temperature Mott transition as a function of doping, pressure temperature etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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 Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**DMFT: Effective Action point of view.R. Chitra and G.**Kotliar Phys Rev. B.(2000). • Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a. • Example, density in DFT theory. (Fukuda et. al.) • When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. • The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Example: DMFT for lattice model (e.g. single band**Hubbard).Muller Hartman 89, Chitra and Kotliar 99. • 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. • Taking several sites (clusters) as local entity. CDMFT • Combining DMFT with other methods. LDA+DMFT, GW+EDMFT or “GWU”….. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Outline**• 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 V2O3 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**• 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**• The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase • Control parameters: doping, temperature,pressure… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**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 V2O3 under pressure or chemical**substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Anomalous transfer of optical spectral weight V2O3**: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 v2O3**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Anomalous Spectral Weight Transfer: Optics**Below energy ApreciableT dependence found. Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994), Rozenberg et.al. PRB 54, 8452, (1996). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**ARPES measurements on NiS2-xSexMatsuura 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, NiSeS.**[Miyasaka and Takagi 2000] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Anomalous Resistivity and Mott transition Ni Se2-x Sx**Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Recent exps. Moo et. al. (2003)Theory Held et. al.**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Transport in 2d organics. Limlet et. al.**THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Strong correlation anomalies**• 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**• 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**. • 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**• 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, LaSrTiO3, NiO,MnO,CrO2,K3C60,2d and quasi-1d organics, magnetic semiconductors,SrRuO4,V2O3…………. • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Interface DMFT with electronic structure.**• 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**DMFT: Effective Action point of view.R. Chitra and G.**Kotliar Phys Rev. B.(2000). • Identify observable, A. Construct an exact functional of <A>=a, G [a] which is stationary at the physical value of a. • Example, density in DFT theory. (Fukuda et. al.) • When a is local, it gives an exact mapping onto a local problem, defines a Weiss field. • The method is useful when practical and accurate approximations to the exact functional exist. Example: LDA, GGA, in DFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS**Spectral Density Functional : effective action construction**(Chitra and GK). • Introduce local orbitals, caR(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 • 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 V2O3: 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). • 3d doped Mott insulator La1-xSrxTiO3 (Anisimov et.al 1997, Nekrasov et.al. 1999, Udovenko et.al 2003) • Paramagnetic Mott insulators. NiO MnO, Savrasov and GK( PRL 2002)…………………… 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