Gabriel Kotliar Center for Materials Theory Rutgers University

1. High Temperature Superconductors. What can we learn from the study of the doped Mott insulator within plaquette Cellular DMFT. • Gabriel Kotliar • Center for Materials Theory Rutgers University • CPhT Ecole Polytechnique Palaiseau, and SPhT CEA Saclay , France Geneve February 10th 2006 Collaborators: M. Civelli, K. Haule (Haule), M. Capone (U. Rome), O. Parcollet(SPhT Saclay), T. D. Stanescu, (Rutgers) V. Kancharla (Rutgers+Sherbrooke) A. M Tremblay, D. Senechal B. Kyung (Sherbrooke) $$Support : NSF DMR . Blaise Pascal Chair Fondation de l’Ecole Normale.

2. Outline • Strongly Correlated Electrons. Basic Dynamical Mean Field Ideas and Cluster Extensions. • High Temperature Superconductivity and Proximity to the Mott Transition. Early Ideas. Slave Boson Implementation. • CDMFT results for the 2x2 plaquette. • a) Normal State Photoemission. [Civelli et. al. PRL (2005) Stanescu and Kotliar cond-mat] b) Superconducting State Tunnelling Density of States. [ Kancharla et.al. Capone et.al] c) Optical Conductivity near optimal doping and near Tc [K. Haule and G. Kotliar]

3. Correlated Electron Materials • Are not well described by either the itinerant or the localized framework . Do not fit in the “Standard Model” Solid State Physics. Reference System: QP. [Fermi Liquid Theory and Kohn Sham DFT+GW ] • Compounds with partially filled f and d shells. • Have consistently produce spectacular “big” effects thru the years. High temperature superconductivity, colossal magneto-resistance, huge volume collapses…………….. • Need new starting point for their description. Non perturbative problem. DMFT New reference frame for thinking about correlated materials and computing their physical properties.

4. Breakdown of the Standard Model Large Metallic Resistivities (Takagi)

5. Transfer of optical spectral weight non local in frequency Schlesinger et. al. (1994), Van der Marel (2005) Takagi (2003 ) Neff depends on T

6. 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)

7. Mean-Field : Classical vs Quantum Classical case Quantum case A. Georges, G. Kotliar (1992) Phys. Rev. B 45, 6497

8. 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)

9. 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.

10. 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.

11. Effective Action point of view. • Identify observable, A. Construct a free energy functional of <A>=a, G [a] which is stationary at the physical value of a. • Example, density in DFT theory. (Fukuda et. al.). • DMFT Local Spectral Function. (R. Chitra and G.K (2000) (2001). • H=H0+l H1. G [a,J0]=F0[J0 ]–a J0 _ + Ghxc [a] • Functional of two variables, a ,J0. • H0 +A J0 Reference system to think about H. • J0 [a] Is the functional of a with the property <A>0 =a < >0 computed with H0+A J0 • Many choices for H0 and for A • Extremize a to get G [J0]=exta G [a,J0]

12. Finite T, DMFT and the Energy Landscape of Correlated Materials T

13. Pressure Driven Mott transition How does the electron go from the localized to the itinerant limit ?

14. 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

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

16. Ising critical endpoint: prediction Kotliar Lange Rozenberg Phys. Rev. Lett. 84, 5180 (2000)Observed! In V2O3 P. Limelette et.al. Science 302, 89 (2003)

17. Three peak structure, predicted Georges and Kotliar (1992) Transfer of spectral weight near the Mott transtion. Predicted Zhang Rozenberg and GK (1993) . 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). .

18. 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 the broad picture.

19. Some References • Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996). • Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2006). • Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)

20. Cuprate superconductors and the Hubbard Model . PW Anderson 1987 . Schematic Phase Diagram (Hole Doped Case)

21. Methodological Remarks • Leave out inhomogeneous states and ignore disorder. • What can we understand about the evolution of the electronic structure from a minimal model of a doped Mott insulator, using Dynamical Mean Field Theory ? • Approach the problem directly from finite temperatures,not from zero temperature. Address issues of finite frequency –temperature crossovers. As we increase the temperature DMFT becomes more and more accurate. • DMFT provides a reference frame capable of describing coherent and incoherent regimes within the same scheme.

22. 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. t-J limit. • 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 zeors.

23. 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)

24. 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:r[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,compare well with experiments. [e.g. Ioffe Kotliar 1989] CDMFT may solve some of these problems.!!

25. 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

26. 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

27. CDMFT study of cuprates . • 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. 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 ) as a solver to reach larger U’s and smaller Temperature and CDMFT as the mean field scheme. • Recently (K. Haule and GK ) the region near the superconducting –normal state transition temperature near optimal doping was studied using NCA + DCA . • 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 ) • Extends the functional form of the self energy to finite T and higher frequency.

28. Can we continue the superconducting state towards the Mott insulating state ?

29. Competition of AF and SC or SC AF SC AF AF+SC d d

30. Competition of AF and SC M. Capone M. Civelli and GK (2006)

31. 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.

32. Superconducting State t’=0 • Does the Hubbard model superconduct ? • Is there a superconducting dome ? • Does the superconductivity scale with J ? • Is it BCS like ?

33. Superconductivity in the Hubbard model role of the Mott transition and influence of the super-exchange. ( work with M. Capone V. Kancharla. CDMFT+ED, 4+ 8 sites t’=0) .

34. Order Parameter and Superconducting Gap do not always scale! ED study in the SC state Capone Civelli Parcollet and GK (2006)

35. How is the Mott insulatorapproached from the superconducting state ? Work in collaboration with M. Capone.

36. Evolution of DOS with doping U=12t. Capone et.al. : Superconductivity is driven by transfer of spectral weight , slave boson b2 !

37. Superconductivity is destroyed by transfer of spectral weight. M. Capone et. al. Similar to slave bosons d wave RVB.

38. In BCS theory the order parameter is tied to the superconducting gap. This is seen at U=4t, but not at large U. • How is superconductivity destroyed as one approaches half filling ?

39. Superconducting State t’=0 • Does it superconduct ? • Yes. Unless there is a competing phase. • Is there a superconducting dome ? • Yes. Provided U /W is above the Mott transition . • Does the superconductivity scale with J ? • Yes. Provided U /W is above the Mott transition . • Is superconductivity BCS like? • Yes for small U/W. No for large U, it is RVB like!

40. The superconductivity scales with J, as in the RVB approach. Qualitative difference between large and small U. The superconductivity goes to zero at half filling ONLY above the Mott transition.

41. Anomalous Self Energy. (from Capone et.al.) Notice the remarkable increase with decreasing doping! True superconducting pairing!! U=8t Significant Difference with Migdal-Eliashberg.

42. Can we connect the superconducting state with the “underlying “normal” state “ ? What does the underlying “normal” state look like ?

43. 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. 2X2 CDMFT

44. Dependence on periodization scheme.

45. Comparison of 2 and 4 sites

46. Spectral shapes. Large Doping Stanescu and GK cond-matt 0508302

47. Small Doping. T. Stanescu and GK cond-matt 0508302

48. Interpretation in terms of lines of zeros and lines of poles of G T.D. Stanescu and G.K cond-matt 0508302

49. Lines of Zeros and Spectral Shapes. Stanescu and GK cond-matt 0508302

50. Connection between superconducting and normal state. • Transfer of spectral weight in optics. Elucidate how the spin superexchange energy and the kinetic energy of holes changes upon entering the superconducting state! • Origin of the powerlaws discovered in the groups of N. Bontemps and D. VarDerMarel. • K. Haule and GK development of an ED+DCA+NCA approach to the problem. New tool for addressing the neighborhood of the dome.