When Band Theory Does Not Work and What One Can Do About It: Dynamical Mean Field Approach to Strongly Correlated Materials Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Princeton Materials Institute Nov

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Successes of band theory. Mott insulators, atomic perspective Strongly correlated materials, between atoms and bands. Brief Introduction to Dynamical Mean Field Theory. Results: qualitative insights into the electronic structure near a Mott transition: theory and recent experiments. Results: material specific studies. Plutonium Outlook.

References S. Savrasov and G. Kotliar Phys. Rev. Lett. 84, , (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410, 793 (2001).Phys. Rev. Lett. 84, , (2000)Nature 410, 793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003)..

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. Band Theory. The Standard Model of Solids and its successes.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Momentum Space (Sommerfeld) Standard model of solids Periodic potential, waves form bands, k in Brillouin zone The electron in a solid: wave picture Maximum metallic resistivity 200  ohm cm Landau: Interactions renormalize away

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Standard Model of Solids RIGID BAND PICTURE. Optical response, transitions between bands. Quantitative tools: DFT, LDA, GGA, total energies,good starting point for spectra, GW,and transport

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Density functional and Kohn Sham reference system Kohn Sham spectra, proved to be an excelent starting point for doing perturbatio theory in screened Coulomb interactions GW.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+GW: semiconducting gaps

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott insulators. The solid as a collection of atoms.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The electron in a solid: particle picture. NiO, MnO, …Array of atoms is insulating if a>>a B. Mott: correlations localize the electron e_ e_ e_ e_ Think in real space, solid collection of atoms High T : local moments, Low T spin-orbital order Superexchange

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. bandsNiO, CoO MnO….) Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. 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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The “in between regime” Strongly correlated materials.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 –GW or LDA+U or Hartree Fock work well. Need approach which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. New reference point, to replace the Kohn Sham system.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Correlated Materials do big things Mott transition.Huge resistivity changes pressure driven Mott transition V 2 O 3. Copper Oxides..(La 2-x Ba x ) CuO 4 High Temperature Superconductivity. 150 K in the Ca 2 Ba 2 Cu 3 HgO 8. Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu 6,m*/m=1000 (La 1-x Sr x )MnO 3 Colossal Magneto- resistance.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly Correlated Materials. Large thermoelectric response in CeFe 4 P 12 (H. Sato et al. cond-mat ). Ando et.al. NaCo 2- x Cu x O 4 Phys. Rev. B 60, (1999). Thermoelectric with a large number of carriers. Large and ultrafast optical nonlinearities Sr 2 CuO 3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) Huge volume changes (volume collapses) in Lanthanides and Actinides Ce, Pu.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Physical properties of correlated Electron Materials are based on different physical principles which lie outside the “standard model”, exciting perspectives for applications

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Breakdown of the Standard Model Pressure Driven Mott transition

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Breakdown of standard model Many Qualitative Failures Large metallic resistivities exceeding the Mott limit. [Anderson, Emery and Kivelson] Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics. [G. Sawatzki]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the standard model : Anomalous Resistivity :LiV 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 Schlesinger et.al (1993) Neff depends on T

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Breakdown of the standard model : Anomalous transfer of optical weight [D. Van der Marel A. Damascelli ]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Breakdown of the Standard Model The LDA+GW program fails badly, Qualitatively incorrect predictions. Incorrect phase diagrams. Physical Reason: The one electron spectra, contains both Hubbard Bands and Quasiparticle features.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Basic Difficulties Lack of a small parameter. Kinetic energy is comparable to Coulomb energies. Relevant degrees of freedom change their form in different energy scales, challenge for traditional RG methods. WANTED: a simple picture of the physical phenomena, and the physics underlying a given material. WANTED: a computational tool to replace LDA+GW

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated systems are usually treated with model Hamiltonians 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 are used)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory DMFT is a new reference frame to approach strongly correlated phenomena, and describes naturally, NON RIGID BAND picture, highly resistive states, treats quasiparticle excitations and Hubbard bands on the same footing.. Allowed progress in solution of model Hamiltonians, and in realistic (system specific) model of materials.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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. Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

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] 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. DMFT, build functionals of the LOCAL spectral function. Exact functionals exist. We also have good approximations! Extension to an ab initio method.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic applications of DMFT References: combinations of DMFT with band theory. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov and G.Kotliar and Abrahams funcional formulation for full self consistent Nature {410}, 793(2001). Reviews: Held et.al., Psi-k Newsletter \#{\bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and Kotliar cond-mat/ :

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

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How good is the LOCAL approximation?

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 CDMFT vs Nc=1

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS N vs mu in one dimension. Compare 2+8 vs exact Bethe Anzats, [M. Capone and M.Civelli]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Electronic structure of strongly correlated electrons. Insights from Dynamical Field Theory. Qualitative predictions and recent experiments.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition Electronically driven MIT. Forces to face directly the localization delocalization problem. Central issue in correlated electron systems. 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 V 2 O 3 under pressure or chemical substitution on V-site

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. cond-mat/ , Phys. Rev. Lett. 85, 5420 (2000)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pressure Driven Mott transition

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insight Phase diagram in the T, U plane of a frustrated ((the magnetic order is supressed)) correlated system at integer filling. At high temperatures, the phase diagram is generic, insensitive to microscopic details. At low temperatures, detail matters.

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 Insight, in the strongly correlated region the one particle density of states has a three peak structure Low energy Quasiparticle Peak plus Hubbard bands.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT has bridged the gap between band theory and atomic physics. Delocalized picture, it should resemble the density of states, (perhaps with some additional shifts and satellites). Localized picture. Two peaks at the ionization and affinity energy of the atom.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS One electron spectra near the Mott transition. Mott transition: transfer of spectral weight. X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

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 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 Some experiments.

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

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QP peak in V2O3 found Mo et.al PRL 90, (2003)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous metallic resistivities In the “ in between region “ anomalous resistivities are the rule rather than the exception.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Failure of the Standard Model: NiSe 2-x S x Miyasaka and Takagi (2000)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition (Rozenberg et. Al. ) Ni Se 2-x S x Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles )

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS More recent work, organics, Limelette et. al. PRL 91, (2003)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative single site DMFT predictions: Optics 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 heavy fermions Rozenberg Kajueter Kotliar (1996)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS More recent work

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 Ising critical endpoint! In V 2 O 3 P. Limelette et.al. Science Vol 302,89 (2003).

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Formations of structures in k space. Cluster dynamical mean field study. Parcollet Biroli and Kotliar Cond-Matt

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. cond-mat/ , Phys. Rev. Lett. 85, 5420 (2000)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the distribution in k space of the low energy spectral intensity as the Mott transition is approached.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS U/D=2, U/D=2.25 (Parcollet et.al.) Uc= , Tc/D=1/44

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conjecture Formation of hot regions is a more general phenomena due to the proximity to the Mott point. Plaquette reference system is good enough to contain the essential features of momentum space differentiation.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS System specific application : Pu

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu in the periodic table actinides

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Electronic Physics of Pu

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT studies. Underestimates the volume by 35 % Predicts Pu to be magnetic. Largest quantitative failure of DFT-LDA- GA Fails to predict a stable delta phase. (negative shear) DFT predicts correct volume of alpha phase.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What is the dominant atomic configuration? Local moment? Snapshots of the f electron Dominant configuration:(5f) 5 Naïve view Lz=-3,-2,-1,0,1 ML=-5  B S=5/2 Ms=5  B Mtot=0 More refined estimates ML=-3.9 Mtot=1.1 This bit is quenches by the f and spd electrons

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 Alpha and delta Pu

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Both alpha and delta plutonium are correlated phases. Susceptibility Specific heat. Resistivity

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity Maximum metallic resistivity

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Specific heat and susceptibility.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 was not measured.

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 Expt. Wong et. al.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expts’ Wong et. al. Science (2003) Theory Dai et. al. Science 300, 953, (2003)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more metallic than delta, but it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. Estimates of the phase transition neglecting the Electronic entropy: TC 600 K.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonons epsilon

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What do we want from materials theory? New concepts, qualitative ideas Understanding, explanation of existent experiments, and predictions of new ones. Quantitative capabilities with predictive power. Notoriously difficult to achieve in strongly correlated materials. DMFT is delivering on both counts.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Local approach to strongly correlated electrons offers a new starting point or reference frame to describe new physics. Breakdown of rigid band picture. Many extensions, make the approach suitable for getting insights and quantitative results in many correlated materials.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Plutonium is just one correlated element, there are many many more strongly correlated compounds which can be studies with this method.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS BACKUPS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice and cluster self energies

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mechanism for hot spot formation

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Deviations from single site DMFT

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V2O3 resistivity

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Specific heat and susceptibility.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Spectral Weight Transfer: Optics Schlesinger et.al (FeSi) 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). ApreciableT dependence found. Below energy

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Start with the TOE

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Rewrite the TOE as an electron boson problem.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Build effective action for the local greens functions of the fermion and Bose field r=R+  R unit cell vector  position within the unit cell. Ir>=|R,  Couple sources to

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Legendre transfor the sources, eliminating the field  Build exact functional of the correlation functionsW(r R,r’ R) and G (r R,r’ R)

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS “Kohn Sham “ decomposition.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS (E)DMFT pproximation to Sum over all LOCAL 2PI graphs (integrations are restricted over the unit cell ) built with W and G Map into impurity model to generate G and W Go beyond this approximation by returning to many body theory and adding the first non local correction.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Electronic structure of strongly correlated electrons. Insights from dynamical mean field theory.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight heavy fermions

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weight

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous transfer of spectral weigth heavy fermions

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 G ii (  ). Exact functional  [G ii (  )  DMFT Approximation to the functional.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Spectral Density Functional : effective action construction ( Chitra and GK ). Introduce local orbitals,   R (r-R)orbitals, and local GF G(R,R)(i  ) = The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for  (r) and G and performing a Legendre transformation,  (r),G(R,R)(i  )] Approximate functional using DMFT insights.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated systems are usually treated with model Hamiltonians

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

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] 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. DMFT, build functionals of the LOCAL spectral function. Exact functionals exist. We also have good approximations! Extension to an ab initio method.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Realistic applications of DMFT References V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, (1997). A Lichtenstein and M. Katsenelson Phys. Rev. B 57, 6884 (1988). S. Savrasov and G.Kotliar and Abrahams funcional formulation for full self consistent Nature {\bf 410}, 793(2001). Reviews: Held et.al., Psi-k Newsletter \#{\bf 56} (April 2003), p. 65 Lichtenstein Katsnelson and and Kotliar cond-mat/ :

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double Occupancy vs U CDMFT Parcollet, Biroli GK

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Compare with single site results Rozenberg Chitra Kotliar PRL 2002

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in cluster

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral FunctionU/D=2, U/D=2.25 (Parcollet et.al.) Uc= , Tc/D=1/44

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Model Hamiltonian. Exact functional of the local Greens function A +

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT for model Hamiltonians. Kohn Sham formulation. Introduce auxiliary field Exact “local self energy”

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS About XC functional. One can derive a coupling constant integration formulae (Harris Jones formula) for Generate approximations. The exact formalism generates the local Greens function and  ii is NOT the self energy. However one can use the approach as starting point for computing other quantities.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on functional construction Atoms as a reference point. Expansion in t. Locality does not necessarily mean a single point. Extension to clusters. Jii ---  Jii Ji i+  Aii ---  Ai i+   ii ---   i i+  Exact functional  Aii,Ai i+   he lattice self energy and other non local quantities extending beyond the cluster are OUTSIDE the formalism and need to be inferred.

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on funct. construction. Construction of approximations in the cluster case requires care to maintain causality. One good approximate construction is the cellular DMFT: a) take a supercell of the desired range,b) c) obtain estimate of the lattice self energy by restoring translational symmetry. Many other cluster approximations (eg. DCA, the use of lattice self energy in self consitency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK]

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice and cluster self energies

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mapping onto impurity models. The local Greens function A, and the auxilliary quantity  can be computed from the solution of an impurity model that describes the effects of the rest of the lattice on the on a selected central site. One can arrive at the same concept via the cavity construction.