Dynamical Mean Field Approach to strongly Correlated Electrons Theoretical and Experimental Magnetism Meeting Gabriel Kotliar Rutgers University 3-4 August Cosener house , Abingdon, Oxfordhisre UK Support :National Science Foundation. Department of Energy (BES).
Outline • Motivation. Introduction to DMFT ideas. • Application to the late actinides. • Application to Cuprate Supeconductors. Collaborators M. Civelli K. Haule (Rutgers ) Ji-Hoon Shim (Rutgers) S. Savrasov (UCDavis ) A.M. Tremblay B. Kyung V. Kancharla (Sherbrook) M. Capone (Rome) O Parcollet(Saclay).
DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Happy marriage of atomic and band physics. Extremize a functional of the local spectra. Local self energy. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti (to appear in RMP).
Mott transition in one band model. Review Georges et.al. RMP 96 T/W Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U,and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993)..
DMFT + electronic structure method Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites problem, in a medium of non interacting electrons obeying a self-consistency condition. (A. Georges et al., RMP 68, 13 (1996)). DMFT in the language of functionals: DMFT sums up all local diagrams in BK functional Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated bands (s,p): use LDA or GW For correlated bands (f or d): with DMFT add all local diagrams. Gives total energy and spectra Technical Implementation is Involved. Different Impurity Solvers. [ED-NCA- Expansions in t and U , etc ] Different forms of Self consistency conditions for the bath in the clusters case. Different levels of complexity in the description of the electronic structure, simple models to all electron calculations. Review: G. Kotliar, S. Savrasov K. Haule, V. Oudovenko O Parcollet, C. Marianetti . Review of Modern Physics 2006.
Mean Field Approach T Follow different “states” as a function of parameters. • Second step compare free energies. • Work in progress. Solving the DMFT equations are non trivial. Configurational cordinate, doping, T, U, structure
Photoemission and Localization Trends in Actinides Curie-Weiss Tc alpa->delta volume collapse transition F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11 Curium has large magnetic moment and orders antif Pu does is non magnetic.
Minimum in melting curve and divergence of the compressibility at the Mott endpoint
C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 DMFT Phonons in fcc d-Pu ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)
Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, 097002 (2005).
Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule Kotliar (2005)
Theoretical Approach [P.WAnderson,1987] • Connection of the cuprate anomalies to the proximity to a doped Mott insulator without magnetic long range order.[Spin Liquid] • Study low energy one band models, Hubbard and t-J. Needed. a good mean field theory of the problem. RVB physics requires a plaquette as a reference frame.
. CDMFT study of cuprates • A functional 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. al T. Maier et. al. (2000) . ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8t ) • Use exact diag ( Krauth Caffarel 1995 ) and vertex corrected NCA as a solvers to study larger U’s and CDMFT as the mean field scheme.
RVB phase diagram of the Cuprate Superconductors. Superexchange. Flux-S+iD spin liquid. [Affleck and Marston , G Kotliar] 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)
Superconductivity in the Hubbard model role of the Mott transition and influence of the super-exchange. ( work with M. Capone et.al V. Kancharla.et.al CDMFT+ED, 4+ 8 sites t’=0) .
cond-mat/0508205Anomalous superconductivity in doped Mott insulator:Order Parameter and Superconducting Gap . They scale together for small U, but not for large U. S. Kancharla M. Civelli M. Capone B. Kyung D. Senechal G. Kotliar andA.Tremblay. Cond mat 0508205M. Capone (2006).
Superconducting DOS • = .06 d=.08 • = .1 d = .16 Superconductivity is destroyed by transfer of spectral weight.. Similar to slave bosons d wave RVB. M. Capone et. al
Doping Driven Mott transiton at low temperature, in 2d (U=16 t=1, t’=-.3 ) Hubbard model Spectral Function A(k,ω→0)= -1/π G(k, ω→0) vs k K.M. Shen et.al. 2004 Antinodal Region 2X2 CDMFT Senechal et.al PRL94 (2005) Nodal Region Civelli et.al. PRL 95 (2005)
Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK cond-matt 0508302
Optics and RESTRICTED SUM RULES Below energy Low energy sum rule can have T and doping dependence . For nearest neighbor it gives the kinetic energy. Use it to extract changes in KE in superconducing state
Optics and RESTRICTED SUM RULES <T>n is defined for T> Tc, while <T>s exists only for T<Tc . <T>n is a strong function of temperature in the normal state. Carbone et. al (2006) .
~1eV Experiments interband transitions intraband Hubbard versus t-J model • Kinetic energy in Hubbard model: • Moving of holes • Excitations between Hubbard bands Hubbard model U Drude t2/U t Excitations into upper Hubbard band • Kinetic energy in t-J model • Only moving of holes Drude t-J model J-t no-U
Kinetic energy change in t-J K Haule and GK Kinetic energy increases cluster-DMFT, cond-mat/0601478 Kinetic energy decreases Kinetic energy increases cond-mat/0503073 Exchange energy decreases and gives largest contribution to condensation energy Phys Rev. B 72, 092504 (2005)
Haule and Kotliar (2006) Coarsed grained or “local “ susceptibility around (pp) Scalapino White PRB 58, 8222 (1988)
Conclusion • DMFT versatile tool for advancing our understanding, and predicting properties of strongly correlated materials. • Theoretical spectroscopy in the making. Substantial work is needed to refine the tool. • Great opportunity for experimental-theoretical interactions. • Refine the questions and our understanding by focusing on differences between the DMFT results and the experiments.
Mean-Field : Classical vs Quantum Classical case Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497
Anomalous Self Energy. (from Capone et.al.) Notice the remarkable increase with decreasing doping! True superconducting pairing!! U=8t Significant Difference with Migdal-Eliashberg.
J. Tobin et.al. PRB 72,085109 (2005) XAS and EELS
Double well structure and d Pu Qualitative explanation of negative thermal expansion[Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)] F(T,V)=Fphonons+Finvar 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.
“Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag. (2006) Data fits if the excited state has zero stiffness.
References and Collaborators • References: • M. Capone et. al. in preparation • M. Capone and G. Kotliar cond-mat cond-mat/0603227 • Kristjan Haule, Gabriel Kotliar cond-mat/0605149 • M. Capone and G.K cond-mat/0603227 • Kristjan Haule, Gabriel Kotliar cond-mat/0601478 • Tudor D. Stanescu and Gabriel Kotliar cond-mat/0508302 • S. S. Kancharla, M. Civelli, M. Capone, B. Kyung, D. Senechal, G. Kotliar, A.-M.S. Tremblay cond-mat/0508205 • M. Civelli M. Capone S. S. Kancharla O. Parcollet and G. Kotliar Phys. Rev. Lett. 95, 106402 (2005)
Mott Phenomeman and High Temperature Superconductivity Began Study of minimal model of a doped Mott insulator within plaquette Cellular DMFT • Rich Structure of the normal state and the interplay of the ordered phases. • Work needed to reach the same level of understanding of the single site DMFT solution. • A) Either that we will understand some qualitative aspects found in the experiment. In which case the next step LDA+CDMFT or GW+CDMFT could be then be used make realistic modelling of the various spectroscopies. • B) Or we do not, in which case other degrees of freedom, or inhomogeneities or long wavelength non Gaussian modes are essential as many authors have surmised. • Too early to tell, talk presented some evidence for A. .
Correlations Magnetism and Structure across the actinide series : a Dynamical Mean Field Theory Perspective G.Kotliar Physics Department and Center for Materials Theory Rutgers University. . Collaborators K. Haule (Rutgers ) Ji-Hoon Shim (Rutgers) S. Savrasov (UCDavis ) A.M. Tremblay B. Kyung (Sherbrook) M. Capone (Rome) O Parcollet(Saclay). Plutonium Futures Asilomar July 9-13 (2006). Support: DOE- BES DOE-NNSA . Expts. : M. Fluss J. C Griveaux G Lander A. Lawson A. Migliori J.Singleton J.Smith J Thompson J. Tobin
M. Capone and GK cond-mat 0511334 . Competition fo superconductivity and antiferromagnetism.
Temperature dependence of the spectral weight of CDMFT in normal state. Carbone, see also ortholani for CDMFT.
Finite temperature view of the phase diagram t-J model.K. Haule and GK (2006)
Outline • Introduction. Mott physics and high temperature superconductivity. Early Ideas: slave boson mean field theory. Successes and Difficulties. • Dynamical Mean Field Theory approach and its cluster extensions. • Results for optical conductivity. • Anomalous superconductivity and normal state. • Future directions.
UPS of alpha-U GGA - He I (hv=21.21eV), He II (hv=40.81eV) - f-electron features is enhanced in He II spectra. Opeil et al. PRB(2006)