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## Gabriel Kotliar Physics Department and Center for Materials Theory

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**Electronic Structure of Strongly Correlated**Materials:Insights from Dynamical Mean Field Theory (DMFT). Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers UniversityCenter 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.**VanSchilfgaarde 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 RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)**Mean-Field : Classical vs Quantum**Classical case Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497**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 V2O3 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**V2O3: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, NiSeS.**[Miyasaka and Takagi 2000]**Anomalous Resistivity and Mott transition Ni Se2-x Sx**Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator. M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)**Ising critical endpoint! In V2O3 P. Limelette et.al.**Science 302, 89 (2003)**ARPES measurements on NiS2-xSexMatsuura et. Al Phys. Rev B**58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)Mo et al., Phys. Rev.Lett. 90, 186403 (2003). .**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 x2 , 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.!!**EDC along different parts of the zone, from Zhou et.al.**Photoemission spectra near the antinodal direction in a Bi2212 underdoped sample. Campuzano 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.**Superconductivity in the Hubbard model role of the Mott**transition and influence of the super-exchange. ( 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=8t. Capone et.al. :**Superconductivity is driven by transfer of spectral weight , slave boson b2 !**Anomalous Self Energy. (from Capone et.al 2006.) Notice the**remarkable increase with decreasing doping! True superconducting pairing!! U=8t 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. Different for electron doped! 2X2 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.**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)**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 3d 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.**Functional formulation to achieve more realistic**calculations For a review see Kotliar et.al. to appear in RMP.. • Spectral Density Functional • LDA+DMFT Final Goal Savrasov Kotliar and Abrahams Nature 410,793 (2001). 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 La2−xSrxCuO4 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 G=Gloc+Gnonloc . Ex. Ir>=|R, r> Gloc=G(R r, R r’) dR,R’’ Sum of 2PI graphs One can also view as an approximation to an exact Spectral Density Functional of Gloc and Wloc.**Order in Perturbation Theory**n=1 Order in PT n=2 Basis set size. l=1 r=1 DMFT l=2 r site CDMFT r=2 GW+ first vertex correction l=lmax GW Range of the clusters