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Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators.

This paper explores the application of Dynamical Mean Field Theory (DMFT) to study doped Mott Insulators, specifically in the context of high-temperature superconductivity in copper oxides. The paper discusses the successes and difficulties of the DMFT approach, as well as the competition between superconductivity and antiferromagnetism. The authors also address open problems and future outlook in this field.

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Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators.

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  1. Dynamical RVB: Cluster Dynamical Mean Field Studies of Doped Mott Insulators. Gabriel Kotliar and Center for Materials Theory & CPHT Ecole Polytechnique Palaiseau & SPHT CEA Saclay, France Itzykson Meeting 2006, Strongly Correlated Electrons Cea Saclay June 21-23 2006 Support : Chaire Blaise Pascal Fondation de l’Ecole Normale.

  2. Outline • Introduction. Mott physics and high temperature superconductivity. Early Ideas: slave boson mean field theory. Successes and Difficulties. • The Dynamical Mean Field Theory approach and cluster extensions. • CDMFT results. Normal state. • Competition of superconductivity and antiferromagnetism. • Comparing superconductivity and the normal state. • Outlook.

  3. References-Collaborators • A. Georges (Ecole Polytechnique) • G. Biroli (CEA-Saclay)-S. Savrasov (UCDavis) O. Parcollet (CEA-Saclay). • M Civelli (ILL-Grenoble)T. Stanescu and M.Capone (U. Rome). • B. Kyung, D. Senechal A. M. Tremblay (Sherbrooke) • K. Haule (Rutgers).

  4. Cuprate Experimental Phase diagram Damascelli, Shen, Hussain, RMP 75, 473 (2003)

  5. Kappa Organics F. Kagawa, K. Miyagawa, + K. Kanoda PRB 69 (2004) +Nature 436 (2005) Phase diagram of (X=Cu[N(CN)2]Cl) S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)

  6. Cuprates Open Problems • What is the mechanism for high temperature superconductivity. Why is it realized in the copper oxides? • What are the relevant energy degrees of freedom to describe the physics of these materials at a given energy scale? • Proper reference frame for understanding the correlated solid, e.g. are there other competing phases besides SC, and quantum critical points controlling the physical properties of this material.

  7. Approach • Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models. [ Leave out disorder, electronic structure …] • Follow different “states” as a function of parameters. [Second step compare free energies which will depend more on the detailed modelling…..] • Mean Field Approach. No R.G. analysis of QCP. • Approach the problem directly from finite temperatures,not from zero temperature. Address issues of finite frequency –temperature crossovers, coherent QP incoherent Hubbard bands. • Work in progress. The framework and the resulting equations are very non trivial to solve and to interpret.

  8. Hubbard Hamiltonians Hubbard Hamiltonian t-J Hamiltonian Slave Boson Formulation: Baskaran Zhou Anderson (1987) Ruckenstein Hirschfeld and Appell (1987) b+i bi +f+si fsi = 1

  9. Perspective U/t Doping Driven Mott Transition . Cuprates Pressure Driven Mott transtion k-organics d A Tale of Two Phase Diagrams: G. Kotliar, J. of Low Temp. Phys. 126, pp.1009-27. t’/t

  10. 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. t-J limit. • Slave boson approach. <b> coherence order parameter. k, D singlet formation order parameters.Baskaran Zhou Anderson , (1987)Ruckenstein Hirshfeld and Appell (1987) .Uniform Solutions. S-wave superconductors. Uniform RVB states. Other RVB states with d wave symmetry. Flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988) . Spectrum of excitation have point zerosUpon doping they become a d –wave superconductor. (Kotliar and Liu 1988). .

  11. 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)

  12. Problems with the approach. • Stability of the MFT. Ex. Neel order. Slave boson MFT with Neel order predicts AF AND SC. [Inui et.al. 1988] Giamarchi and L’huillier. • Mean field is too uniform on the Fermi surface, in contradiction with ARPES.[Penetration depth, Wen and Lee , Ioffe and Millis, Photoemission ] • Description of the incoherent regime. Fluctuations. Development of cluster DMFT may solve some of these problems.!!

  13. Dynamical Mean Field Theory. Cavity Construction.A. Georges and G. Kotliar PRB 45, 6479 (1992). Reviews: A. Georges W. Krauth G.Kotliar and M. Rozenberg RMP (1996)G. Kotliar and D. Vollhardt Physics Today (2004).

  14. Mean-Field : Classical vs Quantum IPT: Georges Kotliar (1992). . QMC: M. Jarrell, (1992), NCA T.Pruschke D. Cox and M. Jarrell (1993), ED:Caffarel Krauth and Rozenberg (1994) Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999) ,……………………………………... • Pruschke et. al Adv. Phys. (1995) • Georges et. al RMP (1996) Classical case Quantum case Hard!!! Easy!!! QMC: J. Hirsch R. Fye (1986) NCA : T. Pruschke and N. Grewe (1989) PT : Yoshida and Yamada (1970) NRG: Wilson (1980) A. Georges, G. Kotliar (1992)

  15. CDMFT: removes limitations of single site DMFT • No k dependence of the self energy. • No d-wave superconductivity. • No Peierls dimerization. • No (R)valence bonds. Various cluster approaches, DCA momentum spcace. Cellular DMFT G. Kotliar et.al. PRL (2004). O Parcollet G. Biroli and G. Kotliar B 69, 205108 (2004) T. D. Stanescu and G. Kotliar cond-mat/0508302 Reviews: Georges et.al. RMP(1996). Th. Maier, M. Jarrell, Th.Pruschke, M.H. Hettler RMP (2005); G. Kotliar S. Savrasov K. Haule O. Parcollet V. Udovenko and C. Marianetti RMP in Press. Tremblay Kyung Senechal cond-matt 0511334

  16. . CDMFT : methodological comments • 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. • Can study different states on the same footing allowing for the full frequency dependence of all the degrees of freedom contained in the plaquette. • DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS • w-S(k,w)+m= w/b2 -(D+b2 t) (cos kx + cos ky)/b2 +l • b--------> b(k), D ----- D(w), l ----- l (k ) • Better description of the incoherent state, more general functional form of the self energy to finite T and higher frequency. Further extensions by periodizing cumulants rather than self energies. Stanescu and GK (2005)

  17. DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W Georges et.al. RMP (1996) Kotliar Vollhardt Physics Today (2004)

  18. Single site DMFT and kappa organics. Qualitative phase diagram Coherence incoherence crosover.

  19. Finite T Mott tranisiton in CDMFT O. Parcollet G. Biroli and GK PRL, 92, 226402. (2004)) CDMFT results Kyung et.al. (2006)

  20. 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.

  21. Evolution of the k resolved Spectral Function at zero frequency. (Parcollet Biroli and GK PRL, 92, 226402. (2004)) ) U/D=2.25 U/D=2 Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W

  22. 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)

  23. Larger frustration: t’=.9t U=16tn=.69 .92 .96 M. Civelli M. CaponeO. Parcollet and GK PRL (20050

  24. Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK cond-matt 0508302

  25. Finite temperature view of the phase diagram t-J model.K. Haule (2006)

  26. Lower Temperature, AF and SCM. Capone and GK, Kancharla et. al. SC AF SC AF AF+SC d d

  27. M. Capone and GK cond-mat 0511334 . Competition fo superconductivity and antiferromagnetism.

  28. Can we continue the superconducting state towards the Mott insulating state ? For U > ~ 8t YES. For U ~ < 8t NO, magnetism really gets in the way.

  29. 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.M. Capone (2006).

  30. Energetics and phase separation. Right U=16t Left U=8t

  31. 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

  32. E Energy difference between the normal and superconducing state of the t-J model. K. Haule (2006)

  33. . Spectral weight integrated up to 1 eV of the three BSCCO films. a) under-doped, Tc=70 K; b) ∼ optimally doped, Tc=80 K; c) overdoped, Tc=63 K; the fullsymbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the superfuid). H.J.A. Molegraaf et al., Science 295, 2239 (2002). A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, 092504(2005) . Recent review:

  34. 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 LDA+CDMFT or GW+CDMFT could be then be used to account semiquantitatively for the large body of experimental data by studying more realistic models of the material. • B) Or we do not, in which case other degrees of freedom, or inhomgeneities or long wavelength non Gaussian modes are essential as many authors have surmised. • Too early to tell, talk presented some evidence for A. .

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