Introduction to Strongly Correlated Electron Materials Dynamical Mean

Introduction to Strongly Correlated Electron Materials, Dynamical Mean Field Theory (DMFT) and its extensions. Application to the Mott Transition. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University • Physics of Condensed-Matter Systems. Princeton Center for Complex Materials Princeton. July 18 -21 (2005).

Summary n n Strongly Correlated Electron Systems require a new starting point or (non. Gaussian) reference system for their description. DMFT provides such a reference frame, mapping the full many body problem on the lattice to a much simpler system, a quantum impurity model in a self consistent medium. DMFT a first stab at a problem. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Summary Application. Pressure and temperature driven Mott transition. Universal aspects of the Mott transition in transition metal oxides. n Three peak structure in the one particle density of states. QP and Hubbard bands. Mott transition is driven by transfer of spectral weight [non rigid band picture ]. n Low energy quasiparticle coherence scale. Coherence-incoherent crossover. Place where gap closure occurs differs from the place where coherence disappears. Uc 1 vs Uc 2. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Summary Zeroth order picture to confront with experiments in a wide range of materials. n Realistic extensions. Interface with band theory. Illustrate with the physics of actinides. n Plaquette as a reference frame. Cluster DMFT. Superconductivity as a result of proximity to a Mott insulator singlet state [Anderson’s RVB picture ] n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Some References n n n Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP 68 , 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) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Weakly correlated electrons: band theory. n n Fermi Liquid Theory. Simple conceptual picture of the ground state, excitation spectra, transport properties of many systems (simple metals, semiconductors, …. ). In a certain low energy regime, adiabatic Continuity to a Reference Systen of Free Fermions with renormalized parameters. Rigid bands , optical transitions , thermodynamics, transport……… 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. Filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals. Kinetic Boltzman equations for QP. scattering off phonons or disorder, ee. int etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Quantitative Tools of Electronic Static mean field theory. Derived from a functional which gives the total energy. Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction. RUTGERS GW. THE STATE UNIVERSITY OF NEW JERSEY

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 Anderson superexchange. spin-orbital order , RVB. 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

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

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. RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

Model Hamiltonians: Hubbard model q. U/t q. Doping d or chemical potential q. Frustration (t’/t) q. T temperature THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strongly correlated systems are usually treated with model Hamiltonians n n Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom. In practice other methods (eg constrained LDA , GW, etc. are used) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

One Particle Spectral Function and An Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k f(w) A(w, K) M 2 n Probability of absorbing an electron and transfering energy w=Ei-Ef, and momentum k (1 -f(w)) A(w K ) M 2 n Theory. Compute one particle greens function and use spectral function. e n n n e THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Photoemission and the Theory of Electronic Structure Local Spectral Function Limiting case itinerant electrons Limiting case localized electrons Hubbard bands THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strong Correlation effects appear in 3 d- 4 f (and sometimes 5 f) systems. Because their wave functions are more localized. Many compounds. Also p electron in organic materials with large volumes can be strongly correlated. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Breakdown of the Standard Model. Strong Correlation Anomalies cannot be understood within the Breakdown of standard model of solids. Metallic “resistivities beyond the Mott limit. THE(2000) STATE UNIVERSITY OF NEW JERSEY C. Urano et. al. PRL 85, 1052 RUTGERS

Failure of the Standard Model: Anomalous Spectral Weight Transfer as a function of T. Optical Conductivity Schlesinger et. al (1993) Neff depends on T THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Correlated Materials do big things n n Huge resistivity changes. Mott transition. 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). Gigantic Volume Collapses. Lanthanide and actinides. 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

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 (say DFT-GW) in predicting physical properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n n n Basic competition between kinetic energy and Coulomb interactions. One needs a tool that treats quasiparticle bands and Hubbard bands on the same footing to contain the band atomic limit. The approach should allow to incorporate material specific information. When the neither the band or the atomic description applies, a new reference point for thinking about correlated electrons is needed. DMFT! THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Limit of large lattice coordination Metzner Vollhardt, 89 Neglect k dependence of self energy Muller. Hartmann 89 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT mapping (Georges Kotliar 1992) Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Single site DMFT cavity construction: Weiss field Semicircular density of states. Behte lattice. 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)

DMFT mapping (Georges Kotliar 1992) Notice that if the self energy is local it is the self energy of an Anderson impurity model. Determine the bath of the impurity model from: 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

Main Omission of this Course n n n Techniques for solving quantitatively the Anderson Impurity Model. G[G 0]See Reviews. Qualitative behavior of the solution of the Anderson Impurity Model. Kondo Physics. Extension to describe ordered phases. Superconductivity. Antiferromagnetism. Etc… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Medium of free electrons : impurity model. Solve for the medium using Self Consistency THEG. STATE UNIVERSITY OF NEW G. JERSEY G. . Kotliar, S. Savrasov, Palsson and Biroli, Phys. Rev. Lett. 87, RUTGERS 186401 (2001)

Extension to clusters. Cellular DMFT. C-DMFT. G. Kotliar, S. Y. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) is the hopping expressed in the superlattice notations. • Other cluster extensions (DCA, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Testing CDMFT (G. . Kotliar, S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M. Capone M. Civelli V Kancharla C. Castellani and GK PR B 69, 195105 (2004) ] U/t=4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Comments on DMFT. n n Review of DMFT, technical tools for solving DMFT eqs. A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68, 13 (1996)] CDMFT , instead of studying finite systems with open or periodic boundary conditions, study a system in a medium. Connection with DMRG, infer the density matrix by using a Gaussian anzats, and the periodicity of the system. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT as an approximation t THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Strongly Correlated Electrons and DMFT. n The challenge (besides learning to solve the DMFT equations more accurately or more explicitly) is to identify which strong correlation phenomena can be capture from a local DMFT perspective using sites, linkes, plaquettes, etc as reference systems, and which aspects involve non local and non Gaussian fluctuations. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

V 2 O 3 under pressure or THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ni. Se 2 -x. Sx 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

Pressure Driven Mott transiti How does the electron go from the localized to the itinerant limit ? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

M. Rozenberg et. al. Phys. Rev. Lett. 75, 105 (1995) T/W Phase diagram of a Hubbard model with partial frustration at STATE UNIVERSITY OF NEW JERSEY the Mott transition in single site integer THE filling. Thinking about RUTGERS DMFT. High temperature universality

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

Spectral Evolution at T=0 half filling full frustration X. Zhang M. Rozenberg G. Kotliar (PRL 1993) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Qualitative single site DMFT predictions. n n Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight. 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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Qualitative single site DMFT predictions: Optics n n Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight. Consequences for optics. 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 transfer of optical spectral weight, Ni. Se. S. [Miyasaka and Takagi 2000] 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

Anomalous Resistivity and Mott transition Ni Se 2 -x Sx Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Single site DMFT and kappa organics THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ising critical endpoint! In V 2 O 3 P. Limelette et. al. Science 302, 89 (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Searching for a quasiparticle peak 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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Schematic DMFT phase Implications for transport. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivity and Mott transition Ni Se 2 -x Sx Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phase Diagram k Organics THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Transport in k organics: hysteresis. Limelette et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ising endpoint finally found 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 nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

V 2 -x. Crx O 3 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ising critical endpoint! In VCr 2 O 3 Limelette et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conclusion. n n An electronic model accounts for all the qualitative features of the finite temperature of a frustrated system at integer occupancy. The electronic degrees of freedom rather than the lattice drives the transition. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conclusion n Single site DMFT describes the main features of the experiments at high temperatures using a simple model. Made non trivial predictions. Finite temperature conclusions are robust. At low temperatures clusters will bring refinements of this picture. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS