The Mott Transition and the Challenge of Strongly Correlated Electron Systems. G. Kotliar Physics Department and Center for Materials Theory Rutgers PIPT Showcase Conference UBC Vancouver May 12th 2005

Outline Correlated Electron Materials. Dynamical Mean Field Theory. The Mott transition problem: qualitative insights from DMFT. Towards first principles calculations of the electronic structure of correlated materials. Pu Am and the Mott transition across the actinide series.

The Standard Model of Solids Itinerant limit. Band Theory. Wave picture of the electron in momentum space.. Pauli susceptibility. Localized model. Real space picture of electrons bound to atoms. Curie susceptibility at high temperatures, spin- orbital ordering at low temperatures.

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, colossal magneto-resistance, huge volume collapses……………..

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

Breakdown of the standard model of solids. Large metallic resistivities exceeding the Mott limit. Maximum metallic resistivity 200  ohm cm Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics. The quantitative tools of the standard model fail.

MODEL HAMILTONIAN AND OBSERVABLES Limiting case itinerant electrons Limiting case localized electrons Hubbard bands Local Spectral Function Parameters: U/t, T, carrier concentration, frustration :

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)

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). 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 correalted 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 Under development. Functional of G and W, self consistent determination of the Coulomb interaction and the Greens functions.

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

The Mott transition and Universality Same behavior at high tempeartures, completely different at low T

T/W Phase diagram of a Hubbard model with partial frustration at integer filling. M. Rozenberg et.al., Phys. Rev. Lett. 75, (1995)..Phys. Rev. Lett. 75, (1995). COHERENCE INCOHERENCE CROSSOVER

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)

Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi 2000]

Anomalous Resistivity and Mott transition Ni Se 2-x S x Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

Single-site DMFT and expts

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.

Mott Transition in the Actinide Series

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  Pu to be magnetic with a 5  b moment. Experimentally it is not. Treating f electrons as core overestimates the volume by 30 %

Total Energy as a function of volume for PU (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu.

Double well structure and  Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Phys vol.126, (2002)]See also A. Lawson et.al.Phil. Mag. B 82, 1837 ] Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal.

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

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory Experiment ( 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, ,2003

First Principles DMFT Studies of Pu 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. Qualitative Insights and quantitative studies. Double well. Alpha and Delta Pu.

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

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

Cluster Extensions of Single Site DMFT

Conclusions Future Directions DMFT: Method under development, but it already gives new insights into materials……. Exciting development: cluster extensions. Allows us to see to check the accuracy of the single site DMFT corrections, and obtain new physics at lower temperatures and closer to the Mott transition where the single site DMFT breaks down. Captures new physics beyond single site DMFT, i.e. d wave superconductivity, and other novel aspects of the Mott transition in two dimensional systems. Allow us to focus on deviations of experiments from DMFT. DMFT and RG developments

Some 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)

Am Equation of State: LDA+DMFT Predictions (Savrasov Kotliar Haule Murthy 2005) LDA+DMFT predictions:  Non magnetic f 6 ground state with J=0 ( 7 F 0 )  Equilibrium Volume: V theory /V exp =0.93  Bulk Modulus: B theory =47 GPa Experimentally B=40-45 GPa Theoretical P(V) using LDA+DMFT Self-consistent evaluations of total energies with LDA+DMFT. Accounting for full atomic multiplet structure using Slater integrals: F (0) =4.5 eV, F (2) =8 eV, F (4) =5.4 eV, F (6) =4 eV New algorithms allow studies of complex structures. Predictions for Am II Predictions for Am IV Predictions for Am III Predictions for Am I

Photoemission Spectrum from 7 F 0 Americium LDA+DMFT Density of States Experimental Photoemission Spectrum (after J. Naegele et.al, PRL 1984) Matrix Hubbard I Method F (0) =4.5 eV F (2) =8.0 eV F (4) =5.4 eV F (6) =4.0 eV

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

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

Cluster Extensions of DMFT

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

More important, one would like to be able to evaluate from the theory itself when the approximation is reliable!! And captures new fascinating aspects of the immediate vecinity of the Mott transition in two dimensional systems…..

Cluster Extensions of Single Site DMFT

Some 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)

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)

Total Energy as a function of volume for Pu Total Energy as a function of volume for Pu W (ev) vs (a.u ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein Savrasov and Kotliar (2004)

DMFT : What is the dominant atomic configuration,what is the fate of the atomic 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 realistic calculations, (GGA+U),itineracy, crystal fields     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:(5f) 6

Anomalous Resistivity PRL 91, (2003)

Approach the Mott transition, if the localized configuration has an OPEN shell the mass increases as the transition is approached. Consistent theory, entropy increases monotonically as U  Uc. Approach the Mott transition, if the localized configuration has a CLOSED shell. We have an apparent paradox. To approach the Mott transitions the bands have to narrow, but the insulator has not entropy.. SOLUTION: superconductivity intervenes.

Mott transition into an open (right) and closed (left) shell systems. AmAt room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0 apply pressure ? S S U U  T Log[2J+1] Uc  ~1/(Uc-U) S=0 ???

BACKUPS

C. Urano et. al. PRL 85, 1052 (2000) Strong Correlation Anomalies cannot be understood within the standard model of solids, based on a RIGID BAND PICTURE,e.g.“Metallic “resistivities that rise without sign of saturation beyond the Mott limit, temperature dependence of the integrated optical weight up to high frequency

RESTRICTED SUM RULES M. Rozenberg G. Kotliar and H. Kajueter PRB 54, 8452, (1996). ApreciableT dependence found. Below energy

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

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

Am under pressure. Lindbaum et.al. PRB 63, (2001)

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

Model Hamiltonians and Observables  U/t  Doping  or chemical potential  Frustration (t’/t)  T temperature

Outlook  The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR  Strategy: advancing our understanding scale by scale  Generalized cluster methods to capture longer range magnetic correlations  New structures in k space?

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, (2002); (and neglecting electronic entropy). TC ~ 600 K.

Further Approximations. o The light, SP (or SPD) electrons are extended, well described by LDA.The heavy, d(or f) electrons are localized treat by DMFT.LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term). o Truncate the W operator act on the H sector only. i.e. Replace W(  ) by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation with light electrons only. e.g. M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998) T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein cond-matt (2004)

or the U matrix can be adjusted empirically. At this point, the approximation can be derived from a functional (Savrasov and Kotliar 2001) FURTHER APPROXIMATION, ignore charge self consistency, namely set LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) See also. A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988). Reviews: Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Blumer, A. K. McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, 2003, Psi-k Newsletter #56, 65. Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron Correlations and Materials Properties 2, edited by A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New York), p Georges, A., 2004, Electronic Archive,.lanl.gov, condmat/

LDA+DMFT Self-Consistency loop DMFT U Edc

Realistic DMFT loop

LDA+DMFT functional  Sum of local 2PI graphs with local U matrix and local G

Anomalous Resistivity PRL 91, (2003)

The Mott Transiton across the Actinides Series.

Medium of free electrons : impurity model. Solve for the medium using Self Consistency G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, (2001)

Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme. Causality issues O. Parcollet, G. Biroli and GK Phys. Rev. B 69, (2004)Phys. Rev. B 69, (2004)

Mott transition in layered organic conductors S Lefebvre et al. cond-mat/ , Phys. Rev. Lett. 85, 5420 (2000)

Insulating anion layer  -(ET) 2 X are across Mott transition ET = X -1 [(ET) 2 ] +1 conducting ET layer t’ t modeled to triangular lattice t’ t modeled to triangular lattice

Single-site DMFT as a zeroth order picture ?

Finite T Mott tranisiton in CDMFT Parcollet Biroli and GK PRL, 92, (2004))

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 t(k) = const and a height increasing as we approach the Fermi surface.

Evolution of the k resolved Spectral Function at zero frequency. (QMC study Parcollet Biroli and GK PRL, 92, (2004)) ) Uc= , Tc/D=1/44. Tmott~.01 W U/D=2 U/D=2.25

Momentum Space Differentiation the high temperature story T/W=1/88

Actinies, role of Pu in the periodic table

CMDFT Studies of the Mott Transition cond-mat/ [ PRL, 92, (2004) ] Cluster Dynamical Mean Field analysis of the Mott transition : O. Parcollet, G. Biroli, G. Kotliar cond-mat/ [abs, ps, pdf, other] : Dynamical Breakup of the Fermi Surface in a doped Mott Insulator M. Civelli (1), M. Capone (2), S. S. Kancharla (3), O. Parcollet (4), G. Kotliar cond-mat/ Title: Short-Range Correlation Induced Pseudogap in Doped Mott Insulators B. Kyung, S. S. Kancharla, D. Sénéchal, A. -M. S. Tremblay, M. Civelli, G. Kotliar

Two paths for calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

Band Theory: electrons as waves. Landau Fermi Liquid Theory. Electrons in a Solid:the Standard Model Quantitative Tools. Density Functional Theory+Perturbation Theory. Rigid bands, optical transitions, thermodynamics, transport………

Mean-Field Classical vs Quantum Quantum case A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992) Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

Phase Diag: Ni Se 2-x S x

Mott transition in systems with close shell. Resolution: as the Mott transition is approached from the metallic side, eventually superconductivity intervenes to for a continuous transition to the localized side. DMFT study of a 2 band model for Buckminster fullerines Capone et. al. Science Mechanism is relevant to Americium.

Mott transition in systems with close shell. Resolution: as the Mott transition is approached from the metallic side, eventually superconductivity intervenes to for a continuous transition to the localized side. DMFT study of a 2 band model for Buckminster fullerines Capone et. al. Science Mechanism is relevant to Americium.

Mott transition in layered organic conductors S Lefebvre et al. cond-mat/