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Introduction to Strongly Correlated Electron Materials and to Dynamical Mean Field Theory (DMFT).

Introduction to Strongly Correlated Electron Materials and to Dynamical Mean Field Theory (DMFT).

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## Introduction to Strongly Correlated Electron Materials and to Dynamical Mean Field Theory (DMFT).

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**Introduction to Strongly Correlated Electron Materials and**to Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University • Workshop on Quantum Materials • Heron Island Resort • New Queensland Australia • 1-4 June 2005**Outline**• Introduction to strongly correlated electrons. • Introduction to Dynamical Mean Field Theory (DMFT) • First Application. The Mott transition problem. Theory and experiments. • More realistic calculations. LDA +DMFT. Pu Am and the Mott transition across the actinide series. Pu and Am • Cluster Extensions. Application to Cuprate Superconductors. • Conclusions. Current developments and future directions.**Electrons in a Solid:the Standard Model**Band Theory: electrons as waves. Landau Fermi Liquid Theory. Rigid bands , optical transitions , thermodynamics, transport……… • Quantitative Tools. Density Functional Theory+Perturbation Theory.**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)**The success of the standard model does NOT extend to**strongly correlated systems . Anomalies cannot be understood within a RIGID BAND PICTURE,e.g. very resistive metals C. Urano et. al. PRL 85, 1052 (2000)**Strong Correlation Anomalies : temperature dependence of**the integrated optical weight up to high frequency. Violations of low energy optical sum rule. Breakdown of rigid band picture.**Breakdown of standard model**• Large metallic resistivities exceeding the Mott limit. Maximum metallic resistivity 200 mohm cm • Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics. • The quantitative tools of the standard model fail, e.g. alpha gamma transition in Cerium, Mott transition in oxides, actinides etc…**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.**Strongly Correlated Materials.**• Large thermoelectric response in NaCo2-xCuxO4 • Huge volume collapses, Ce, Pu…… • Large and ultrafast optical nonlinearities Sr2CuO3 • Large Coexistence of Ferroelectricity and Ferromagnetism (multiferroics) YMnO3.**Localization vs Delocalization Strong Correlation Problem**• Many interesting compounds do not fit within the “Standard Model”. • Tend to have elements with partially filled d and f shells. Competition between kinetic and Coulomb interactions. • Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). • Non perturbative problem. • Require a framework that combines both atomic physics and band theory. DMFT.**Two paths for the calculation of electronic structure of**materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc.**MODEL HAMILTONIAN AND OBSERVABLES**Parameters: U/t , T, carrier concentration, frustration : Local Spectral Function Limiting case itinerant electrons Limiting case localized electrons Hubbard bands**Limit of large lattice coordination**Metzner Vollhardt, 89 Muller-Hartmann 89**Mean-Field Classical vs Quantum**Classical case Quantum case A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992) Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)**Mean-Field Quantum Case**H=Ho +Hm +Hm0 Determine the parameters of the mediu t’ so as to get translation invariance on the average. A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)**DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45,**6479 (1992).First happy marriage of atomic and band physics. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)**Pressure Driven Mott transition**R. Mckenzie, Science 278, 820-821 (1997). How does the electron go from the localized to the itinerant limit ?**M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)**T/W Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site DMFT. High temperature universality**M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995)**T/W Phase diagram of a Hubbard model with partial frustration at integer filling. Thinking about the Mott transition in single site DMFT. High temperature universality**Evolution of the Spectral Function with Temperature**Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)**V2O3:Anomalous transfer of spectral weight**Th. Pruschke and D. L. Cox and M. Jarrell, Europhysics Lett. , 21 (1993), 593 M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)**Transfer of optical spectral weight**M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)**Anomalous transfer of optical spectral weight, NiSeS.**[Miyasaka and Takagi 2000]**Anomalous Resistivity and Mott transition Ni Se2-x Sx**Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator. M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)**modeled to triangular lattice**modeled to triangular lattice t t t’ t’ k-(ET)2X are across Mott transition ET = Insulating anion layer X-1 conducting ET layer [(ET)2]+1**Single site DMFT and kappa organics Merino and McKenzie PRB**61, 7996 (2000)and 62 16442 (2000)**Ising critical endpoint! In V2O3 P. Limelette et.al.**Science 302, 89 (2003)**ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B**58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)Mo et al., Phys. Rev.Lett. 90, 186403 (2003). .**Conclusions.**• Three peak structure, quasiparticles and Hubbard bands. • Non local transfer of spectral weight. • Large metallic resistivities. • The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase. • Coherent and incoherence crossover. Real and momentum space. • Theory and experiments begin to agree on a broad picture.**Collaborators References**• Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996). • Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005). • Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)**Extensions of Single SiteDMFT and its applications to**correlated materials. • More realistic calculations. LDA +DMFT. Pu Am and the Mott transition across the actinide series. • Cluster Extensions. Application to Cuprate Superconductors. • Conclusions. Current developments and future directions. • Introduction to strongly correlated electrons. • Introduction to Dynamical Mean Field Theory (DMFT) • First Application. The Mott transition problem. Theory and experiments.**Two paths for calculation of electronic structure of**strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. DMFT ideas can be used in both cases.**Dynamical Mean Field Theory**• Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.[A. Georges and GK Phys. Rev. B 45, 6497, 1992]. • Merge atomic physics and band theory. Atom in a medium. Weiss field. = Quantum impurity model. Solid in a frequency dependent potential. • Incorporate band structure and orbital degeneracy to achive a realistic description of materials. LDA +DMFT. Realistic combination with band theory: LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). • .**LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin**and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). • The light, sp (or spd) electrons are extended, well described by LDA .The heavy, d (or f) electrons are localized treat by DMFT. Use Khon Sham Hamiltonian after substracting the average energy already contained in LDA. • Add to the substracted Kohn Sham Hamiltonian a frequency dependent self energy, treat with DMFT. In this method U is either a parameter or is estimated from constrained LDA • Describes the excitation spectra of many strongly correlated solids. .**Spectral Density Functional**• Determine the self energy , the density and the structure of the solid self consistently. By extremizing a functional of these quantities. (Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, PRB 2005). Coupling of electronic degrees of freedom to structural degrees of freedom. Full implementation for Pu. Savrasov and Kotliar Nature 2001. • Under development. Functional of G and W, self consistent determination of the Coulomb interaction and the Greens fu**LDA+DMFT Self-Consistency loop**Edc U DMFT**Pu in the periodic table**actinides**Mott Transition in the Actinide Series**Lashley et.al.**Pu phases: A. Lawson Los Alamos Science 26, (2000)**LDA underestimates the volume of fcc Pu by 30%. Within LDA fcc Pu has a negative shear modulus. LSDA predicts d Pu to be magnetic with a 5 ub moment. Experimentally it is not. Treating f electrons as core overestimates the volume by 30 %**Pu is not MAGNETIC, alpha and delta have comparable**susceptibility and specifi heat.**Total Energy as a function of volume for Pu W (ev) vs**(a.u. 27.2 ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein Savrasov and Kotliar (2004)**Double well structure and d Pu**Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ]**DMFT and the Invar ModelA. Lawson et. al. LA UR 04-6008**(LANL)**Phonon Spectra**• Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. • Phonon spectra reveals instablities, via soft modes. • Phonon spectrum of Pu had not been measured.**Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science**vol 300, 953, 2003