Dynamical Mean Field Theory of the Mott Transition

Dynamical Mean Field Theory of the Mott Transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Jerusalem Winter School January 2002

OUTLINE OF THE COURSE n n Motivation. Electronic structure of correlated materials, limiting cases and open problems. The standard model of solids and its failures. Introduction to the Dynamical Mean Field Theory (DMFT). Cavity construction. Statistical Mechanical Analogies. Lattice Models and Quantum Impurity models. Functional derivation. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n The limit of large lattice coordination. Ordered phases. Correlation functions. Techniques for solving the Dynamical Mean Field Equations. [ Trieste School June 17 -22 2002] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n n The Mott transition. Early ideas. Brinkman Rice. Hubbard. Slater. Analysis of the DMFT equations: existence of a Mott transition. The Mott transition within DMFT. Overview of some important results of DMFT studies of the Hubbard Model. Electronic Structure of Correlated Materials. Canonical Phase diagram of a fully frustrated Hubbard model. Universal and non universal aspects of the physics of strongly correlated materials. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n n Analysis of the DMFT equations. Existence of a Mott transition. Analysis from large U and small U. The destruction of the metallic phase. Landau analysis. Uc 1. Uc 2. The Mott transition endpoint. A new look at experiments. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline The electronic structure of real materials. Examples of problems where DMFT gives new insights, and quantitative understanding: itinerant ferromagnetism, Fe, Ni. Volume collapse transitions, actinide physics. Doping driven Mott transition titanites. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Outline n New directions, beyond single site DMFT. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Realistic Theories of Correlated Materials ITP, Santa-Barbara workshop July 29 – December 16 (2002) O. K. Andesen, A. Georges, G. Kotliar, and A. Lichtenstein Contact: kotliar@physics. rutgers. edu Conference: November 25 -29, (2002)

The promise of Strongly Correlated Materials n n n Copper Oxides. High Temperature Superconductivity. Uranium and Cerium Based Compounds. Heavy Fermion Systems. (La. Sr)Mn. O 3 Colossal Magnetoresistence. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The Promise of Strongly Correlated Materials. n High Temperature Superconductivity in doped filled Bucky Balls (B. Battlog et. al Science) Thermoelectric response in Ce. Fe 4 P 12 (H. Sato et al. cond-mat 0010017). Large Ultrafast Optical Nonlinearities Sr 2 Cu. O 3 (T Ogasawara et. al cond-mat 000286) n Theory will play an important role in optimizing their physical properties. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

How to think about the electron in a solid? Drude Sommerfeld Bloch, Periodic potential Bands, k in Brillouin zone Maximum metallic resistivity 200 mohm cm THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Standard Model High densities, electron as a wave, band theory, kspace Landau: Interactions Renormalize Away One particle excitations: quasi-particle bands Density Functional Theory in Kohn Sham Formulation, successful computational tool for total energy, and starting point For perturbative calculation of spectra, Si Au, Li, Na ………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Standard Model : Metals Predicts low temperature dependence of thermodynamics and transport Hall Coefficient Resistivity Thermopower Specific Heat Susceptibility THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Quantitative Tools : Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, GW, transport. ………… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

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…. ) Rich structure of Magnetic and Orbital Ordering at low T 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 which are not close to the well understood limits (localized or itinerant). • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock works well • Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455 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

Failure of the standard model : Anomalous Resistivity: Li. V 2 O 4 Takagi et. al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failure of the Standard Model: Anomalous Spectral Weight Transfer Optical Conductivity 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

Strong Correlation Problem n n n Large number of compounds (d, f, p…. ). Departure from the standard model. Hamiltonian is known. Identify the relevant degrees of freedom at a given scale. Treat the itinerant and localized aspect of the electron The Mott transition, head on confrontation with this issue Dynamical Mean Field Theory simplest approach interpolating between that bands and atoms THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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 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

Comments on DMFT n n Exact in both atomic and band limits Weiss field is a function Multiple energy scales in a correlated electron problem, non linear coupling between them. Frezes spatial fluctuations but treats quantum fluctuations exactly, local view of the quantum many body problem. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Example: semicircular DOS 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 tools (QMC, NRG, ED…. ) • Analytical Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992) Mean-Field : Classical vs Quantum case Classical case THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Single site DMFT, functional formulation n n Express in terms of Weiss field (semicircular. DOS) The Mott transition as bifurcation point in functionals o. G[G] or F[D], (G. Kotliar EPJB 99) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT for lattice hamiltonians k independent S k dependent G, Local Approximation Treglia et. al 1980 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

How to compute S ? View locally the lattice problem as a (multiorbital) Anderson impurity model The local site is now embedded in a medium characterized by THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

How to determine the medium n Use the impurity model to compute S and the impurity local Greens function. Require that impurity local Greens function equal to the lattice local Greens function. Weiss field THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Response functions THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Evaluation of the Free energy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Solving the DMFT equations • Wide variety of computational tools (QMC, NRG, ED…. ) • Analytical Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

tools for solving DMFT eqs. . , applications, references…… n A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68, 13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Mott transition: Early ideas. Half filling. Evolution of the one electron spectra [physical quantity measured in photoemission and BIS] as a function of control parameters. ( U/t, pressure, temperature ) n Hubbard, begin in paramagnetic insulator. As U/t is reduced Hubbard bands merge. Gap closure. Mathematical description, closure of equations of motion, starting from atoms (I. e. large U). Incoherent motion, no fermi surface. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition: early ideas. Brinkman and Rice. Gutzwiller. Begin in paramagnetic metallic state, as U/t approaches a critical value the effective mass diverges. Luttinger fermi surface. n Mathematical description, variational wave function, slave bosons, quantum coherence and double occupancy. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Slave bosons: mean field +fluctuations n n n Fluctuations of the slave bosons around the saddle point gives rise to Hubbard bands. Starting from the insulating side, in a paramagnetic state, the gap closes at the same U, where Z vanishes. No satisfactory treatement of finite temperature properties. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott vs Slater Mott: insulators in the absence of magnetic long range order. e. g. Vanadium Oxide Nickel Oxide. Mott transition in the paramagnetic state. • Slater: insulating behavior as a consequence of antiferromagnetic long range order. Double the unit cell to convert a Mott insulator into a band insulator. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Local view of the spectral function Partition function of the Anderson impurity model : gas of kinks [Anderson and Yuval] Metallic state, proliferation of kinks. Insulating state. Kinks are confined. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Local view of the spectral function. n n Consistent treatement of quasiparticles and collective modes. Kinky paths, with may spin fluctuations: low energy resonance [Abrikosov Suhl Resonance] Confined kinks, straight paths, Hubbard bands. [control the insulator partition function] Strongly correlated metal has both. 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

Destruction of the metal The gap is well formed at Uc 2, when the metal is destroyed. Hubbard bands are well formed in the metal. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

Destruction of the insulator n n n Continue the insulating solution below Uc 2. Coexistence of two solutions between Uc 1 and Uc 2 Mott Hubbard gap vanishes linearly at Uc 1. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Recent calculation of the phase diagram of the frustrated Half filled Hubbard model with semicircular DOS (QMC Joo and Udovenko PRB 2001). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Case study: IPT half filled Hubbard one band n n (Uc 1)exact = 2. 1 (Exact diag, Rozenberg, Kajueter, Kotliar 1995) , (Uc 1)IPT =2. 4 (Uc 2)exact =2. 95 (Projective self consistent method, Moeller Si Rozenberg Kotliar PRL 1995 ) (Uc 2)IPT =3. 3 (TMIT ) exact =. 026+_. 004 (QMC Rozenberg Chitra and Kotliar PRL 1999), (TMIT )IPT =. 5 (UMIT )exact =2. 38 +-. 03 (QMC Rozenberg Chitra and Kotliar PRL 1991), (UMIT )IPT =2. 5 For realistic studies errors due to other sources (for example the value of U, are at least of the same order of magnitude). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Schematic DMFT phase diagram Hubbard model (partial frustration) Rozenberg et. al. PRL (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Kuwamoto Honig and Appell PRB (1980) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phase Diag: Ni Se 2 -x Sx THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights from DMFT q Low temperatures several competing phases. Their relative stability depends on chemistry and crystal structure q. High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT 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 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 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

Anomalous Resistivity and Mott transition Ni Se 2 -x Sx THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous resisitivity near Mott transition. 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 spectral weight in v 2 O 3 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 spectral weight in heavy fermions [Rozenberg etal] 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. [Non local in frequency] Real and momentum space. – THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivity: Li. V 2 O 4 Takagi et. al. PRL 2000 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre et

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Standard Model Odd # electrons -> metal Even # electrons -> insulator q. Theoretical foundation: Sommerfeld, Bloch and Landau q. Computational tools DFT in LDA q. Transport Properties, Boltzman equation , low temperature dependence of transport coefficients Typical Mott values of the resistivity 200 m. Ohmcm Residual instabilites SDW, CDW, SC THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Failure of the “Standard Model”: Cuprates Anomalous Resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT q. Formulation as an electronic structure method (Chitra and Kotliar) q. Density vs Local Spectral Function q. Extensions to treat strong spatial inhomogeneities. Anderson Localization (Dobrosavlevic and Kotliar), Surfaces (Nolting), Stripes (Fleck Lichtenstein and Oles) q. Practical Implementation (Anisimov and Kotliar, Savrasov, Katsenelson and. RUTGERS Lichtenstein) THE STATE UNIVERSITY OF NEW JERSEY

DMFT q. Spin Orbital Ordered States q. Longer range interactions Coulomb, interactions, Random Exchange (Sachdev and Ye, Parcollet and Georges, Kajueter and Kotliar, Si and Smith, Chitra and Kotliar, ) q. Short range magnetic correlations. Cluster Schemes. (Ingersent and Schiller, Georges and Kotliar, cluster expansion in real space, momentum space cluster DCA Jarrell RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

Strongly Correlated Electrons q. Competing Interaction q. Low T, Several Phases Close in Energy q. Complex Phase Diagrams q. Extreme Sensitivity to Changes in External Parameters q. Need for Quantitative Methods THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Landau Functional G. Kotliar EPJB (1999) 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

Ising character of the transfer of spectral weight Ising –like dependence of the photo-emission intensity and the optical spectral weight near the Mott transition endpoint 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