Electronic Structure of Strongly Correlated Materials Insights from

Electronic Structure of Strongly Correlated Materials: Insights from Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University CPTH Ecole Polytechnique Palaiseau, and CPTH CEA Saclay , France REUNIÓN NACIONAL DE FÍSICA DEL ESTADO SÓLIDO. GEFES IV Alicante Spain. February 1 -3 (2006) $upport : NSF DMR. Blaise Pascal Chair Fondation de l’Ecole Normale.

Electrons in a Solid: the Standard Model Band Theory: electrons as waves. Landau Fermi Liquid Theory. At low energies the electrons behave as non interacting quasiparticles. Rigid bands , optical transitions , thermodynamics, transport……… • Quantitative Tools. Density Functional Theory+ GW Perturbation Theory.

LDA+GW: semiconducting gaps. Reviews J. Wilkins, M. Van. Schilfgaarde Success story : Density Functional linear response Review: Baroni et. al, Rev. Mod. Phys, 73, 515, 2001

Correlated Electron Materials • Are not well described by either the itinerant or the localized framework. • Compounds with partially filled f and d shells. Need new starting point for their description. Non perturbative problem. New reference frame for computing their physical properties. • Have consistently produce spectacular “big” effects thru the years. High temperature superconductivity, huge resistivity changes across the MIT, colossal magneto-resistance, huge volume collapses, large masses in heavy Fermions, ……………. .

Breakdown of the Standard Model : Large Metallic Resistivities

Transfer of optical spectral weight non local in frequency Schlesinger et. al. (1994), Vander Marel (2005) Takagi (2003 ) Neff depends on T

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.

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 RMP 68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today

Mean-Field : Classical vs Quantum Classical case Phys. Rev. B 45, 6497 Quantum case A. Georges, G. Kotliar (1992)

Cluster Extensions of Single Site DMFT Many Techniques for solving the impurity model: QMC, (Fye. Hirsch), NCA, ED(Krauth –Caffarel), IPT, …………For a review see Kotliar et. Al to appear in RMP (2006)

For reviews of cluster methods see: Georges et. al. RMP (1996) Maier et. al RMP (2005), Kotliar et. al cond-mat 0511085. to appear in RMP (2006) Kyung et. al cond-mat 0511085 Parametrizes the physics in terms of a few functions. D , Weiss Field Alternative (T. Stanescu and G. K. ) periodize the cumulants rather than the self energies.

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.

Mott transition in V 2 O 3 under pressure or chemical substitution on V-site. How does the electron go from localized to itinerant.

Pressure Driven Mott transition 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

V 2 O 3: Anomalous transfer of 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, Ni. Se. S. [Miyasaka and Takagi 2000]

Anomalous Resistivity and Mott transition Ni Se 2 -x Sx Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic M. insulator. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

Single site DMFT and kappa organics

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

. 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) Mo al. , Phys. Rev. Lett. 90, 186403 (2003). et

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.

Cuprate superconductors and the Hubbard Model. PW Anderson 1987

RVB physics and Cuprate Superconductors • P. W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987) • Connection between the anomalous normal state of a doped Mott insulator and high Tc. • Slave boson approach. <b> coherence order parameter. k, D singlet formation order parameters. Baskaran Zhou Anderson , Ruckenstein et. al (1987). Other states flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988) have point zeros.

RVB phase diagram of the Cuprate Superconductors. Superexchange. • The approach to the Mott insulator renormalizes the kinetic energy Trvb increases. • The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero. • Superconducting dome. Pseudogap evolves continously into the superconducting state. G. Kotliar and J. Liu Phys. Rev. B 38, 5412 (1988) Related approach using wave functions: T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria N. Trivedi , A. Paramenkanti PRL 87, 217002 (2001)

Problems with the approach. • Neel order. How to continue a Neel insulating state ? Need to treat properly finite T. • Temperature dependence of the penetration depth [Wen and Lee , Ioffe and Millis ]. Theory: rs[T]=x-Ta x 2 , Exp: r[T]= x-T a. • Mean field is too uniform on the Fermi surface, in contradiction with ARPES. • No quantitative computations in the regime where there is a coherent-incoherent crossover which compare well with experiments. [e. g. Ioffe Kotliar 1990] The development of CDMFT solves may solve many of these problems. !!

Photoemission spectra near the antinodal direction in a Bi 2212 underdoped sample. Campuzano et. al EDC along different parts of the zone, from Zhou et. al.

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

The development of CDMFT solves may solve many of the problems of the early slave bosons RVB theory. !! Theoretical approach: study the plaquette CDMFT equations. • Ignore inhomogeneities and phase separation. • Follow separately each mean field state. • Focus on the physics results from the proximity to a Mott insulating state and to which extent it accounts for the experimental observations.

Competition of AF and SC M. Capone M. Civelli and GK (2006)

Superconductivity in the Hubbard model role of the Mott transition and influence of the superexchange. ( M. Capone et. al. V. Kancharla et. al. CDMFT+ED, 4+ 8 sites t’=0). Pd

Order Parameter and Superconducting Gap do not always scale! ED study in the SC state Capone Civelli Parcollet and GK (2006)

Evolution of DOS with doping U=8 t. Capone et. al. : Superconductivity is driven by transfer of spectral weight , slave boson b 2 !

Anomalous Self Energy. (from Capone et. al 2006. ) Notice the remarkable increase with decreasing doping! True superconducting pairing!! U=8 t Significant Difference with Migdal-Eliashberg.

Follow the “normal state” with doping. Civelli et. al. PRL 95, 106402 (2005) Spectral Function A(k, ω→ 0)= -1/π G(k, ω → 0) vs k U=16 t, t’=-. 3 K. M. Shen et. al. 2004 If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface. 2 X 2 CDMFT

Interpretation in terms of lines of zeros and lines of poles of G T. D. Stanescu and G. K cond-matt 0508302

Conclusion • CDMFT delivers the spectra. • Path between d-wave and insulator. Dynamical RVB! • Lines of zeros. Connection with other work. of A. Tsvelik and collaborators. (Perturbation theory in chains , see however Biermann et. al). T. Stanescu, fully self consistent scheme. • Weak coupling RG (T. M. Rice and collaborators). Truncation of the Fermi surface. CDMFT presents it as a strong coupling instability that begins FAR FROM FERMI SURFACE.

Realistic Descriptions of Materials and a First Principles Approach to Strongly Correlated Electron Systems. • Incorporate realistic band structure and orbital degeneracy. • Incorporate the coupling of the lattice degrees of freedom to the electronic degrees of freedom. • Predict properties of matter without empirical information.

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). • Realistic band structure and orbital degeneracy. Describes the excitation spectra of many strongly correlated solids. . Spectral Density Functionals. Chitra and Kotliar PRB 2001 Savrasov et. al. Nature (2001) Savrasov and Kotliar PRB (2005) • Determine the self energy , the density and the structure of the solid by extremizing a functional of these quantities. Coupling of electronic degrees of freedom to structural degrees of freedom.

Mott Transition in Actinides The f electrons in Plutonium are close to a localization-delocalization transition (Johansson, 1974). Modern understanding of the phenomena with DMFT (Savrasov and Kotliar 2002 -2003) Mott Transition after G. Lander, Science (2003). after J. Lashley et. al, cond-mat (2005). This regime is not well described by traditional techniques of electronic structure techniques and require new methods which take into account the itinerant and the localized character of the electron on the same footing.

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34. 56 33. 03 26. 81 3. 88 Experiment 36. 28 33. 59 26. 73 4. 78 ( Dai, Savrasov, Kotliar, Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et. al, Science, 22 August 2003)

Summary • Review the standard model of solids. • Introduced some of the problems posed by strongly correlated electron materials. • Dynamical Mean Field Theory (DMFT). New reference frame to think about the physics of these materials and compute its properties. • The Mott Transition in 3 d frustrated transition metal oxides and in high temperature superconductors. • Future Directions. The field of correlated electrons is at the brink of a revolution. (C) DMFT : Rapid development of conceptual tools and computational abilities. Theoretical Spectroscopy in the making. Prelude to theoretical material design using strongly correlated elemenets. • Focus on the deviations between CDMFT and experiments to elucidate the role of long wavelength non Gaussian fluctuations.

Gracias por invitarme y por vuestra atencion!

Functional formulation to achieve more realistic calculations For a review see Kotliar et. al. to appear in RMP. . Final Goal q Spectral Density Functional Savrasov Kotliar and Abrahams Nature 410, 793 (2001). § LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) generalizing LDA+ U

Approaching the Mott transition: CDMFT Picture • Fermi Surface Breakup. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! • D wave gapping of the single particle spectra as the Mott transition is approached. Real and Imaginary part of the self energies grow approaching half filling. Unlike weak coupling! • Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, 226402. (2004). Both real and imaginary parts of the self energy get larger. Strong Coupling instability.

Two paths for the calculation of electronic structure of materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. Hubbard Model

ARPES spectra for La 2−x. Srx. Cu. O 4 at doping x = 0. 063, 0. 09, 0. 22. From Zhou et al

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev. B (2001) . Introduce Notion of Local Greens functions, Wloc, Gloc Ex. Ir>=|R, r> G=Gloc+Gnonloc . Gloc=G(R r, R r’) d. R, R’’ Sum of 2 PI graphs One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc.

Order in Perturbation Theory n=1 n=2 Basis set size. l=1 l=2 r=1 r=2 l=lmax Range of the clusters Order in PT DMFT r site CDMFT GW GW+ first vertex correction

q Spectral Density Functional Savrasov Kotliar and Abrahams Nature 410, 793 (2001). § LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997)

Conclusions sp systems. • Not well described by single site DMFT. But very well described by first principles cdmft with relatively small clusters. [2 or 3 coordination spheres] • Weakly correlated materials. Use cheap impurity solvers. • Fast, self consistent way of getting first principles electronic structure without LDA. Good trends for semiconducting gaps and band withds.

Earlier approximations as limiting cases; sinlge site DMFT for models A. Georges G. Kotliar PRB (1992) q Spectral Density Functional Savrasov Kotliar and Abrahams Nature 410, 793 (2001). § LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997)

Spectral shapes. Large Doping Stanescu and GK cond-mat 0508302

Small Doping. T. Stanescu and GK cond-matt 0508302

Interpretation in terms of lines of zeros and lines of poles of G T. D. Stanescu and G. K cond-matt 0508302

Lines of Zeros and Spectral Shapes. Stanescu and GK cond-matt 0508302

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.

• The combination of realistic band theory and many body physics, is a very broad subject. • Having a practical and tractable non perturbative method for solving many body Hamiltonians, the next step is to bring more realistic descriptions of the materials Orbital degeneracy and realistic band structure. • 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. • Functional formulation similar to DFT: Total Energy as functional of local Green function: Spectral Density Functional Theory(Chitra, Kotliar, PRB 2001, Savrasov, Kotliar, Abrahams, Nature 2001).

• Focus on the “local “ spectral function A(w) (and of the local screened Coulomb interaction W(w) ) of the solid. • Write a functional of the local spectral function such that its stationary point, give the energy of the solid. • No explicit expression for the exact functional exists, but good approximations are available. LDA+DMFT. • The spectral function is computed by solving a local impurity model in a medium. Which is a new reference system to think about correlated electrons. • Add non local perturbative corrections “GW+DMFT”. • Explosion of papers, refining the techniques and applying it to many different materials.

Actinies , role of Pu in the periodic table

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

Pu is not MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.

DMFT : What is the dominant atomic configuration , what is the fate of the atomic moment ? • Snapshots of the f electron : Dominant configuration: (5 f)5 • Naïve view Lz=-3, -2, -1, 0, 1, ML=-5 m. B, , S=5/2 Ms=5 m. B. Mtot=0 • More realistic calculations, (GGA+U), itineracy, crystal fields G 7 +G 8, ML=-3. 9 Mtot=1. 1. S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett. , 84, 3670 (2000) • This moment is quenched or screened by spd electrons, and other f electrons. (e. g. alpha Ce). § Contrast Am: (5 f)6

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 ]

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.

• CDMFT study of cuprates. AFunctional of the cluster Greens function. Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition. • Earlier studies use QMC (Katsnelson and Lichtenstein, (1998) M Hettler et. T. Maier et. al. (2000). ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8 t ) • Use exact diag ( Krauth Caffarel 1995 ) as a solver to reach larger U’s and smaller Temperature and CDMFT as the mean field scheme. • Recently (K. Haule and GK ) the region near the superconducting –normal state transition temperature near optimal doping was studied using NCA + DCA. • DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS • w-S(k, w)+m= w/b 2 -(D+b 2 t) (cos kx + cos ky)/b 2 +l • b----> b(k), D ----- D(w), l ----- l (k ) • Extends the functional form of self energy to finite T and higher frequency.

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = Ei - Ef Q =ki - kf

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34. 56 33. 03 26. 81 3. 88 Experiment 36. 28 33. 59 26. 73 4. 78 ( Dai, Savrasov, Kotliar, Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et. al, Science, 22 August 2003)

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 , 2003

Dynamical Mean Field View of Pu (Savrasov Kotliar and Abrahams, Nature 2001) • Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). • Is the natural consequence of earlier studies of the Mott transition phase diagram once electronic structure is about to vary.

• Pu strongly correlated element, at the brink of a Mott instability. • Realistic implementations of DMFT : total energy, photoemission spectra and phonon dispersions of delta Pu. • Clues to understanding other Pu anomalies.

Outline • Introduction to strongly correlated electrons. • Introduction to Dynamical Mean Field Theory (DMFT) • The Mott transition problem. Theory and experiments. • More realistic calculations. Pu the Mott transition across the actinide series. • Conclusions. Current developments and future directions.

Mott transition into an open (right) and closed (left) shell systems. Am. At room pressure a localised 5 f 6 system; j=5/2. S = -L = 3: J = 0 apply pressure ? S . g T Log[2 J+1] S ? ? ? Uc U S=0 g ~1/(Uc-U) U

Americium under pressure Experimental Equation of State (after Heathman et. al, PRL 2000) “Soft” Mott Transition? “Hard” Density functional based electronic structure calculations: q Non magnetic LDA/GGA predicts volume 50% off. q Magnetic GGA corrects most of error in volume but gives m~6 m. B (Soderlind et. al. , PRB 2000). q Experimentally, Am has non magnetic f 6 ground state with J=0 (7 F 0)

Mott transition in open (right) and closed (left) shell systems. S g. T Log[2 J+1] S Tc ? ? ? Uc U J=0 g ~1/(Uc-U) U

Am under pressure: J. C. Griveau. J. Rebizant G. Lander G. Kotliar PRL (2005)

Collaborators References • 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)

Conclusion • DMFT. Electronic Structure Method under development. Local Approach. Cluster extensions. • Quantitative results , connection between electronic structure, scales and bonding. • Qualitative understanding by linking real materials to impurity models. Concepts to think about correlated materials. • Closely tied to experiments. System specific. Many materials to be studied, realistic matrix elements for each spectroscopy. Optics. …… • Role of loclity. • Material design using strongly correlated systems.

Anomalous Resistivity PRL 91, 061401 (2003)

The delta –epsilon transition • The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. • What drives this phase transition? • LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.

Epsilon Plutonium.

Phonon entropy drives the epsilon delta phase transition • Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. 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 following Drumont and G. Ackland et. al. PRB. 65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Pu in the periodic table actinides

Small amounts of Ga stabilize the d phase (A. Lawson LANL)

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)

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34. 56 33. 03 26. 81 3. 88 Experiment 36. 28 33. 59 26. 73 4. 78 ( Dai, Savrasov, Kotliar, Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et. al, Science, 22 August 2003)

Mott transition into an open (right) and closed (left) shell systems. In single site DMFT, superconductivity must intervene before reaching the Mott insulating state. [Capone et. al. ] Am At room pressure a localised 5 f 6 system; j=5/2. S = -L = 3: J = 0 apply pressure ? S . g T Log[2 J+1] S ? ? ? Uc U S=0 g ~1/(Uc-U) U

J. C. Griveau et. al. (2004)

H. Q. Yuan et. al. Ce. Cu 2(Si 2 -x Gex). Am under pressure Griveau et. al. Superconductivity due to valence fluctuations ?

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)

Epilogue, the search for a quasiparticle peak and its demise, photoemission, transport. Confirmation of the DMFT predictions q q q 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) S. -K. Mo et al. , Phys Rev. Lett. 90, 186403 (2003). Limelette et. al. [Science] G. Kotliar [Science].

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)

One Particle Local Spectral Function and Angle Integrated Photoemission e • Probability of removing an electron and transfering energy w=Ei-Ef, f(w) A(w) M 2 • Probability of absorbing an electron and transfering energy w=Ei-Ef, (1 -f(w)) A(w) M 2 • Theory. Compute one particle greens function and use spectral function. n n e

Dynamical Mean Field Theory • Focus on the local spectral function A(w) of the solid. • Write a functional of the local spectral function such that its stationary point, give the energy of the solid. • No explicit expression for the exact functional exists, but good approximations are available. • The spectral function is computed by solving a local impurity model. Which is a new reference system to think about correlated electrons. • Ref: A. Georges G. Kotliar W. Krauth M. Rozenberg.

Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the

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 Kancharla C. Castellani and GK P. R B 69, 195105 (2004) ] U/t=4. V

Site Cellular DMFT. C-DMFT. Kotliar, S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) G. . tˆ(K) hopping expressed in the superlattice notations. • Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes , causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587.

Searching for a quasiparticle peak

Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V 2 O 3 from optics. M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

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)

QP in V 2 O 3 was recently found Mo et. al

k organics • ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene • K (ET)2 X Increasing pressure ----- increasing t’ -----X 0 X 1 X 2 X 3 • (Cu)2 CN)3 Cu(NCN)2 Cl Cu(NCN 2)2 Br Cu(NCS)2 • Spin liquid Mott transition

Large and ultrafast optical nonlinearities Sr 2 Cu. O 3 (T Ogasawara et. a Phys. Rev. Lett. 85, 2204 (2000) )

More examples • Li. Co. O 2 • Used in batteries, laptops, cell phones

Large thermoelectric power in a metal with a large number of carriers Na. Co 2 O 4

Vanadium Oxide Transport under pressure. Limelette etal

Mean-Field : Classical vs Quantum Classical case Phys. Rev. B 45, 6497 Quantum case A. Georges, G. Kotliar (1992)

Expt. Wong et. al.

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 Kancharla C. Castellani and GK P. R B 69, 195105 (2004) ] U/t=4. V

Two paths for ab-initio 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.

Failure of the standard model : Anomalous Resistivity: Li. V 2 O 4 Takagi et. al. PRL 2000

1 2 4 3 A. Georges and G. Kotliar PRB 45, 6479 (1992). G. Kotliar, S. Savrasov, G. Palsson and G. Biroli, PRL 87, 186401 (2001).

Mott Transition in Actinides The f electrons in Plutonium are close to a localization-delocalization transition (Johansson, 1974). Mott Transition after G. Lander, Science (2003). after J. Lashley et. al, cond-mat (2005). This regime is not well described by traditional techniques of electronic structure techniques and require new methods which take into account the itinerant and the localized character of the electron on the same footing.

Resistivity in Americium Resistivity behavior (after Griveau et. al, PRL 2005) Superconductivity • Under pressure, resistivity of Am raises almost an order of magnitude and reaches its value of 500 m. W*cm • Superconductivity in Am is observed with Tc ~ 0. 5 K

Photoemission in Am, Pu, Sm Atomic multiplet structure emerges from measured photoemission spectra in Am (5 f 6), Sm(4 f 6) Signature for f electrons localization. after J. R. Naegele, Phys. Rev. Lett. (1984).

Am Equation of State: LDA+DMFT Predictions Self-consistent evaluations of total energies with LDA+DMFT using matrix Hubbard I method. Accounting for full atomic multiplet structure using Slater integrals: F(0)=4. 5 e. V, F(2)=8 e. V, F(4)=5. 4 e. V, F(6)=4 e. V New algorithms allow studies of complex structures. Theoretical P(V) using LDA+DMFT Predictions for Am I LDA+DMFT predictions: q Non magnetic f 6 ground state with J=0 (7 F 0) q Equilibrium Volume: Vtheory/Vexp=0. 93 q Bulk Modulus: Btheory=47 GPa Experimentally B=40 -45 GPa Predictions for Am III Predictions for Am IV

Photoemission Spectrum from 7 F 0 Americium LDA+DMFT Density of States Matrix Hubbard I Method F(0)=4. 5 e. V F(2)=8. 0 e. V F(4)=5. 4 e. V F(6)=4. 0 e. V Experimental Photoemission Spectrum (after J. Naegele et. al, PRL 1984)

Atomic Multiplets in Americium LDA+DMFT Density of States Matrix Hubbard I Method F(0)=4. 5 e. V F(2)=8. 0 e. V F(4)=5. 4 e. V F(6)=4. 0 e. V Exact Diag. for atomic shell F(0)=4. 5 e. V F(2)=8. 0 e. V F(4)=5. 4 e. V F(6)=4. 0 e. V

Alpha and delta Pu

Failure of the Standard Model: Anomalous Spectral Weight Transfer Optical Conductivity o of Fe. Si for T=20, 40, 200 and 250 K from Schlesinger et. al (1993) Neff depends on T

DMFT Impurity cavity construction

A. C. Lawson et. al. LA UR 046008 F(T, V)=Fphonons+Finvar

Invar model A. C. Lawson et. al. LA UR 04 -6008

Small amounts of Ga stabilize the d phase (A. Lawson LANL)

Breakdown of standard model • Large metallic resistivities exceeding the Mott limit. • Breakdown of the rigid band picture. • Anomalous transfer of spectral weight in photoemission and optics. • LDA+GW loses its predictive power. • Need new reference frame, to think about and compute the properties of correlated materials. • Need new starting point to do perturbation theory.

Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

The electron in a solid: particle picture. • Array of hydrogen atoms is insulating if a>>a. B. Mott: correlations localize the electron e_ e_ • Superexchange Think in real space , solid collection of atoms High T : local moments, Low T spin-orbital order

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

Optical transfer of spectral weight , kappa organics. Eldridge, J. , Kornelsen, K. , Wang, H. , Williams, J. , Crouch, A. , and Watkins, D. , Sol. State. Comm. , 79, 583 (1991).

RESTRICTED SUM RULES Below energy Low energy sum rule can have T and doping dependence. For nearest neighbor it gives the kinetic energy.

Treatement needs refinement • The kinetic energy of the Hubbard model contains both the kinetic energy of the holes, and the superexchange energy of the spins. • Physically they are very different. • Experimentally only measures the kinetic energy of the holes.

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 temperature etc.

Single site DMFT cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field Semicircular density of states. Behte lattice.

Photoemission and the Theory of Electronic Structure Local Spectral Function Limiting case itinerant electrons Limiting case localized electrons Hubbard bands

Cellular DMFT studies of the doped Mott insulator : the plaquette as a reference frame. Dynamical RVB Collaborators: M. Civelli, K. Haule, M. Capone, O. Parcollet, T. D. Stanescu, (Rutgers) V. Kancharla (Rutgers+Sherbrook) A. M Tremblay, D. Senechal B. Kyung (Sherbrooke).

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)

One Particle Spectral Function and Angle Integrated Photoemission e • Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k f(w) A(w, K) M 2 • Probability of absorbing an electron and transfering energy w=Ei-Ef, and momentum k (1 -f(w)) A(w K ) M 2 • Theory. Compute one particle greens function and use spectral function. n n e