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Cluster Dynamical Mean Field Approach to Strongly Correlated Materials

K Haule Rutgers University. Cluster Dynamical Mean Field Approach to Strongly Correlated Materials. Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study, K. H. and G. Kotliar, Phys. Rev. B 76, 104509 (2007).

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Cluster Dynamical Mean Field Approach to Strongly Correlated Materials

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  1. K Haule Rutgers University Cluster Dynamical Mean Field Approach to Strongly Correlated Materials

  2. 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 CeIrIn5, 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 Bi2Sr2CaCu2O8+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). References and Collaborators Thanks to Ali Yazdani for unpublished data!

  3. M. Van Schilfgarde 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: • total energies, • stability of crystal phases • optical transitions

  4. Strong correlation – Standard theory fails • 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.

  5. 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

  6. Bad metal Bad insulator 1B HB model (plaquette): Mott phenomena at half filling Single site DMFT Georges, Kotliar, Krauth, Rozenber, Rev. Mod. Phys. 1996

  7. D 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)

  8. DMFT in single site approximation Successfully describes spectra and response functions of numerous correlated materials: Mott transition in V2O3 LaTiO3 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).

  9. In eV Ce In Band structure and optics of heavy fermion CeIrIn5 Non-f spectra at 10K 300K J.H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007).

  10. Anomalous Hall coefficient Later verified by Yang & Pines Remarkable agreement with Y. Yang & D. Pines cond-mat/0711.0789!

  11. High Tc: Need non-local self-energy d-wave pairing: 2x2 cluster-DMFT necessary to capture the order parameter Fermi surface evolution with doping can not be understood within single site DMFT.

  12. SR=(1,1) SR=(0,0) SR=(1,0) 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 ontoa self consistentimpurity model F[Gplaquette] • Impurity solvers: • Continuous time QMC • Hirsh-Fye QMC • NCA • ED periodization

  13. 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

  14. Approach • Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models. • 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…..] • Approach the problem from high temperatures where physics is more local. Address issues of finite frequency– and finite temperature crossovers. • Leave out disorder, electronic structure, phonons … [CDMFT+LDA second step, under way]

  15. S(iw) with CTQMC next nearest neighbor important in underdoped regime on-site largest nearest neighbor smaller Hubbard model, T=0.005t

  16. Normal state T>Tc: Very large scattering rate at optimal doping Normal state T>Tc SC state T<<Tc Momentum space differentiation …gets replaced by coherent SC state with large anomalous self-energy t-J model, T=0.005t

  17. SC Tunneling DOS K. H. and G. Kotliar, Phys. Rev. B 76, 104509 (2007). NM d=0.20 SC d=0.08 SC d=0.20 NM d=0.08 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

  18. Exp:Bi2212 with STM McElroy,.. JC Davis, PRL 94, 197005 (2005) Ratio AS/AN Ratio more universal, more symmetric With decreasing doping gap increases, coherence peaks less sharp->Non BCS

  19. Yazdani’s experiment on Bi2212, 30K, 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 Ratio AS/AN CDMFT calculation ratio almost symmetric Pronounced dip-hump feature can not be fitted with BCS

  20. Non-BCS DOS in normal state decreases Gap increases

  21. Yazdani’s experiment on Bi221,30K, 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 Normal state DOS and SC gap CDMFT Not using realistic band structure (t’)

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

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

  24. A(w) D(w) Eliashberg theory Real part constant phonon frequency W D0 Gap up toD0+W No scattering up toD0+W

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

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

  27. Fermi surface d=0.09 Cumulant is short in ranged: Arcs FS in underdoped regime pockets+lines ofzeros of G == arcs Single site DMFT PD

  28. Nodal quasiparticles

  29. Nodal quasiparticles Vnod almost constant up to 20% the slope=vnod almost constant vD dome like shape Superconducting gap tracks Tc! M. Civelli, cond-mat 0704.1486

  30. Two energy scales in Raman Spectrum in the SC State of Underdoped Cuprates doping 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)

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

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

  33. Basov et.al.,PRB 72,54529 (2005) Optical conductivity • Low doping: two components Drude peak + MIR peak at 2J • For x>0.12 the two components merge • In SC state, the partial gap opens – causes redistribution of spectral weight up to 1eV

  34. Kinetic energy in Hubbard model: • Moving of holes • Excitations between Hubbard bands ~1eV Hubbard model Experiments U Drude interband transitions intraband t2/U • Kinetic energy in t-J model • Only moving of holes Optical spectral weight - Hubbard versus t-J model f-sumrule Excitations into upper Hubbard band Drude t-J model J no-U

  35. Optical spectral weight & Optical mass mass does not diverge approaches ~1/J Bi2212 F. Carbone,et.al, PRB 74,64510 (2006) Weight increases because the arcs increase and Zn increases (more nodal quasiparticles) Basov et.al., PRB 72,60511R (2005)

  36. 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 Temperature/doping dependence of the optical spectral weight

  37. ~1eV Bi2212 Optical weight, plasma frequency Weight bigger in SC, K decreases (non-BCS) 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)

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

  39. Main origin of the condensation energy Scalapino&White, PRB 58, (1998) Origin of the condensation energy • Resonance at 0.16t~5Tc (most pronounced at optimal doping) • Second peak ~0.38t~120meV (at opt.d) substantially contributes to condensation energy

  40. 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

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