Cluster Dynamical Mean Field Approach to Strongly Correlated

Cluster Dynamical Mean Field Approach to Strongly Correlated Materials K Haule Rutgers University n n 48 Sanibel Symposium 2008

References and Collaborators ¡ Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study, K. H. and G. Kotliar, Phys. Rev. B 76, 104509 (2007). ¡ Nodal/Antinodal Dichotomy and the Energy-Gaps of a doped Mott Insulator, M. Civelli, M. Capone, A. Georges, K. H. , O. Parcollet, T. D. Stanescu, G. Kotliar, Phys. Rev. Lett. 100, 046402 (2008). ¡ Modelling the Localized to Itinerant Electronic Transition in the Heavy Fermion System Ce. Ir. In 5, J. H. Shim, K. Haule and G. Kotliar, Science 318, 1615 (2007), ¡ Quantum Monte Carlo Impurity Solver for Cluster DMFT and Electronic Structure Calculations in Adjustable Base, K. H. , Phys. Rev. B 75, 155113 (2007). ¡ Optical conductivity and kinetic energy of the superconducting state: a cluster dynamical mean field study, K. H. , and G. Kotliar, Europhys Lett. 77, 27007 (2007). ¡ Doping dependence of the redistribution of optical spectral weight in Bi 2 Sr 2 Ca. Cu 2 O 8+delta, F. Carbone, A. B. Kuzmenko, H. J. A. Molegraaf, E. van Heumen, V. Lukovac, F. Marsiglio, D. van der Marel, K. H. , G. Kotliar, H. Berger, S. Courjault, P. H. Kes, and M. Li, Phys. Rev. B 74, 064510 (2006). ¡ Avoided Quantum Criticality near Optimally Doped High Temperature Superconductors, K. H. and G. Kotliar, Phys. Rev. B 76, 092503 (2007). Thanks to Ali Yazdani for unpublished data! Sanibel 2008

Standard theory of solids Band Theory: electrons as waves: Rigid band picture: En(k) versus k Landau Fermi Liquid Theory applicable Very powerful quantitative tools: LDA, LSDA, GW Predictions: M. Van Schilfgarde • total energies, • stability of crystal phases • optical transitions Sanibel 2008

Strong correlation – Standard theory fails n n n Fermi Liquid Theory does NOT work. Need new concepts to replace rigid bands picture! Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). Non perturbative problem. Sanibel 2008

Non perturbative methods On site correlations usually the strongest -> Mott phenomena at integer fillings Successful theory which can describe Mott transition: Dynamical Mean Field Theory Sanibel 2008

Mott phenomena at half filling eta l 1 B HB model (plaquette): insu dm Bad Ba lator Single site DMFT Georges, Kotliar, Krauth, Rozenber, Rev. Mod. Phys. 1996 Sanibel 2008

Dynamical Mean Field Theory For a given lattice site, DMFT envisions the neighboring sites on the lattice as a Weiss field of conduction electrons, exchanging electrons with that site. Maps lattice model to an effective quantum impurity model More rigorously: DMFT sumps up all local diagrams (to all orders in perturbation theory) D Sanibel 2008

DMFT in single site approximation Successfully describes spectra and response functions of numerous correlated materials: Mott transition in V 2 O 3 La. Ti. O 3 actinides (Pu, …) Lanthanides (Ce, …) and … far to many to mention all , KH et al. 2007 , KH et al. 2003 Recent review: (G. Kotliar S. Savrasov K. H. , V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006). Sanibel 2008

Band structure and optics of heavy fermion Ce. Ir. In 5 300 K e. V Non-f spectra at 10 K In Ce In J. H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007). Sanibel 2008

Remarkable agreement with Y. Yang & D. Pines cond-mat/0711. 0789! Anomalous Hall coefficient Later verified by Yang & Pines Sanibel 2008

High Tc: Need non-local self-energy d-wave pairing: 2 x 2 cluster-DMFT necessary to capture the order parameter Fermi surface evolution with doping can not be understood within single site DMFT. Sanibel 2008

Cluster DMFT approaches • Momentum space approach-Dynamical cluster approximation (Hettler, Maier, Jarrel) • Real space approach – Cellular DMFT (Kotliar, Savrasov, Palson) In the Baym Kadanoff functional, the interacting part F is restrictied to the degrees of freedom (G) that live on the cluster. Maps the many body problem onto a self consistent impurity model F[Gplaquette] SR=(1, 1) SR=(0, 0) S periodization R=(1, 0) Impurity solvers: • Continuous time QMC • Hirsh-Fye QMC • NCA • ED Sanibel 2008

An exact impurity solver, continuous time QMC - expansion in terms of hybridization K. H. Phys. Rev. B 75, 155113 (2007) ; P Werner, PRL (2007); N. Rubtsov PRB 72, 35122 (2005). General impurity problem Diagrammatic expansion in terms of hybridization D +Metropolis sampling over the diagrams • Exact method: samples all diagrams! • No severe sign problem Sanibel 2008

Approach q. Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models. q. Construct mean-field type of theory and follow different “states” as a function of parameters – superconducting & normal state. [Second step compare free energies which will depend more on the detailed modeling and long range terms in Hamiltonian…. . ] q. Approach the problem from high temperatures where physics is more local. Address issues of finite frequency– and finite temperature crossovers. q. Leave out disorder, electronic structure, phonons … [CDMFT+LDA second step, under way] Sanibel 2008

S(iw) with CTQMC on-site largest nearest neighbor smaller next nearest neighbor important in underdoped regime Hubbard model, T=0. 005 t Sanibel 2008

SC state T<<Tc Normal state T>Tc Momentum space differentiation Normal state T>Tc: Very large scattering rate at optimal doping …gets replaced by coherent SC state with large anomalous selfenergy t-J model, T=0. 005 t Sanibel 2008

SC Tunneling DOS NM d=0. 20 d=0. 08 SC d=0. 20 d=0. 08 K. H. and G. Kotliar, Phys. Rev. B 76, 104509 (2007). Large asymmetry at low doping Gap decreases with doping DOS becomes more symmetric Asymmetry is due to normal state DOS -> Mottness Computed by the NCA for the t-J model Sanibel 2008

Ratio AS/AN Ratio more universal, more symmetric With decreasing doping gap increases, coherence peaks less sharp->Non BCS Exp: Bi 2212 with STM Mc. Elroy, . . JC Davis, PRL 94, 197005 (2005) Sanibel 2008

Ratio AS/AN CDMFT calculation ratio almost symmetric Pronounced dip-hump feature can not be fitted with BCS Yazdani’s experiment on Bi 2212, 30 K, slightly overdoped A. N. Pasupathy (1), A. Pushp (1, 2), K. K. Gomes (1, 2), C. V. Parker (1), J. Wen (3), Z. Xu (3), G. Gu (3), S. Ono (4), Y. Ando (5), and Ali Yazdani (1), (1)Princeton University, (2)Urbana-Champaign (3)Brookhaven N. L. , (4)CRIEPI, Tokyo, Japan, (5)ISIR, Osaka, Japan. Unpublished, shown with permission Sanibel 2008

Non-BCS DOS in normal state decreases Gap increases Sanibel 2008

CDMFT Normal state DOS and SC gap Not using realistic band structure (t’) Yazdani’s experiment on Bi 221, 30 K, slightly overdoped A. N. Pasupathy (1), A. Pushp (1, 2), K. K. Gomes (1, 2), C. V. Parker (1), J. Wen (3), Z. Xu (3), G. Gu (3), S. Ono (4), Y. Ando (5), and Ali Yazdani (1), (1)Princeton University, (2)Urbana-Champaign (3)Brookhaven N. L. , (4)CRIEPI, Tokyo, Japan, (5)ISIR, Osaka, Japan. Unpublished, shown with permission Sanibel 2008

Gap changes, mode does not J. C. Davis, Nature 442, 546 (2006) Sanibel 2008

Where does the dip-hump structure come from? J. C. Davis, Nature 442, 546 (2006) Sanibel 2008

Eliashberg theory Real part constant phonon frequency W D(w) A(w) D 0 Gap up to D 0+W No scattering up to D 0+W Sanibel 2008

AS/AN Dip-hump structure Kink in normal self-energy Sharp rise of scattering rate Normal self-energy in SC state Normal self-energy in NM state Most important: Dip and peak in anomalous self-energy Sanibel 2008

Phenomenology Similar frequency dependence of gap recently introduced by W. Sacks and B. Doucot PRB 74, 174517 (2006) to fit experiments. Sanibel 2008

Fermi surface d=0. 09 Cumulant is short in ranged: Arcs FS in underdoped regime pockets+lines of zeros of G == arcs Single site DMFT PD Sanibel 2008

Nodal quasiparticles Sanibel 2008

Nodal quasiparticles Vnod almost constant up to 20% the slope=vnod almost constant v. D dome like shape Superconducting gap tracks Tc! M. Civelli, cond-mat 0704. 1486 Sanibel 2008

doping Two energy scales in Raman Spectrum in the SC State of Underdoped Cuprates Energy scale of peak in antinodal (nodal) region increases (decreases) with decreasing doping in underdoped cuprates. Le Tacon et al, Nat. Phys. 2, 537 (2006) Sanibel 2008

Evolution of Nodal and Antinodal energy scales with doping Le Tacon et al, Nat. Phys. 2, 537 (2006) Sanibel 2008

Antinodal gap – two gaps Normal state “pseudogap” monotonically increasing with underdoping M. Civelli, using ED, PRL. 100, 046402 (2008). “true” superconducting gap has a dome like shape (like v. D) Sanibel 2008

Optical conductivity q. Low doping: two components Drude peak + MIR peak at 2 J q. For x>0. 12 the two components merge q. In SC state, the partial gap opens – causes redistribution of spectral weight up to 1 e. V Basov et. al. , PRB 72, 54529 (2005) Sanibel 2008

Optical spectral weight - Hubbard versus t-J model Kinetic energy in Hubbard model: • Moving of holes • Excitations between Hubbard bands Hubbard model U Drude t 2/U Excitations into upper Hubbard band Kinetic energy in t-J model • Only moving of holes Drude J f-sumrule Experiments intraband t-J model no-U interband transitions ~1 e. V Sanibel 2008

Optical spectral weight & Optical mass does not diverge approaches ~1/J Bi 2212 F. Carbone, et. al, PRB 74, 64510 (2006) Weight increases because the arcs increase and Zn increases (more nodal quasiparticles) Basov et. al. , PRB 72, 60511 R (2005) Sanibel 2008

Temperature/doping dependence of the optical spectral weight Single site DMFT gives correct order of magnitude (Toshi&Capone) At low doping, single site DMFT has a small coherence scale -> big change Cluser DMF for t-J: Carriers become more coherent In the overdoped regime -> bigger change in kinetic energy for large d Sanibel 2008

Optical weight, plasma frequency Weight bigger in SC, K decreases (non-BCS) ~1 e. V Bi 2212 Weight smaller in SC, K increases (BCS-like) A. F. Santander-Syro et. al, Phys. Rev. B 70, 134504 (2004) F. Carbone, et. al, PRB 74, 64510 (2006) Sanibel 2008

Kinetic energy change Kinetic energy increases cluster-DMFT, Eu. Lett. 77, 27007 (2007). Kinetic energy decreases Kinetic energy increases Phys Rev. B 72, 092504 (2005) same as RVB (see P. W. Anderson Physica C, 341, 9 (2000) Exchange energy decreases and gives largest contribution to condensation energy Sanibel 2008

Origin of the condensation energy Main origin of the condensation energy Scalapino&White, PRB 58, (1998) • Resonance at 0. 16 t~5 Tc (most pronounced at optimal doping) • Second peak ~0. 38 t~120 me. V (at opt. d) substantially contributes to condensation energy Sanibel 2008

Conclusions • Plaquette DMFT provides a simple mean field picture of the underdoped, optimally doped and overdoped regime • One can consider mean field phases and track them even in the region where they are not stable (normal state below Tc) • Many similarities with high-Tc’s can be found in the plaquette DMFT: • Strong momentum space differentiation with appearance of arcs in UR • Superconducting gap tracks Tc while the PG increases with underdoping • Nodal fermi velocity is almost constant • Tunneling DOS As/An has a dip hump dip structure -> comes from the structure in the anomalous self-energy • Optical conductivity shows a two component behavior at low doping • Optical mass ~1/J at low doping and optical weigh increases linearly with d • In the underdoped system -> kinetic energy saving mechanism overdoped system -> kinetic energy loss mechanism exchange energy is always optimized in SC state Sanibel 2008