Strongly Correlated Superconductivity

1. Strongly Correlated Superconductivity G. Kotliar Physics Department and Center for Materials Theory Rutgers

2. Cluster Dynamical Mean Field Theory . Plaquette as Reference Frame. • Mott transition and Superconductivity [Anderson RVB] • The Mott transition in the actinide series. Strongly Correlated Superconductivity in Am ?

3. References • Model for kappa organics. [O. Parcollet, G. Biroli and G. Kotliar PRL, 92, 226402. (2004)) ] • Model for cuprates [ M. Civelli [ Rutgers] Ph.D ThesisO. Parcollet (Saclay), M. Capone (U. Rome) V. Kancharla (Sherbrooke) GK(2005). PRL in press. • Cluster Dynamical Mean Field Theories a Strong Coupling Perspective. T. Stanescu and G. Kotliar (in preparation 2005) • Work on Am and Pu S. Savrasov K. Haule and GK.

4. RVB phase diagram of the 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 (1987)

5. RVB phase diagram of the Cuprate Superconductors. Superexchange. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) • 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. 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)

6. Problems (or non problems) with the approach. • Neel order • Stability of the pseudogap state at finite temperature. [Ubbens and Lee] • Missing incoherent spectra . [ fluctuations of slave bosons ] • Dynamical Mean Field Methods are ideal to remove address these difficulties.

7. COHERENCE INCOHERENCE CROSSOVER T/W Phase diagram of a Hubbard model with partial frustration at integer filling.  M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .

9. Impurity Model-----Lattice Model D , Weiss Field

10. Focus of this work • Generalize and extend these early mean field approaches to systems near the Mott transition. • Obtain the solution of the 2X 2 plaquette and gain physical understanding of the different CDMFT states. • Even if the results are changed by going to larger clusters, the short range physics is general and will teach us important lessons. Follow states as a function of parameters. Adiabatic continuity. Furthermore the results can be stabilized by adding further interactions.

12. Finite T Mott tranisiton in CDMFT Parcollet Biroli and GK PRL, 92, 226402. (2004))

13. 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.

14. Evolution of the k resolved Spectral Function at zero frequency. (QMC study 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

15. Momentum Space Differentiation the high temperature story T/W=1/88

16. Physical Interpretation • Momentum space differentiation. The Fermi liquid –Bad Metal, and the Bad Insulator - Mott Insulator regime are realized in two different regions of momentum space. • Cluster of impurities can have different characteristic temperatures. Coherence along the diagonal incoherence along x and y directions. • Connection with slave Boson theory divergence of Sigma13 . Connections with RVB (Schmalian and Trivedi)

17. Cuprate superconductors and the Hubbard Model . PW Anderson 1987

18. CDMFT study of cuprates . • 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 (Katsnelson and Lichtenstein, M. Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett 85, 1524 (2000) ) used QMC as an impurity solver and DCA as cluster scheme. • We use exact diag ( Krauth Caffarel 1995 with effective temperature 32/t=124/D ) as a solver and Cellular DMFT as the mean field scheme.

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

20. 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) .

21. Order Parameter and Superconducting Gap .

22. 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 ?

23. 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.

24. 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 .

25. D wave Superconductivity and Antiferromagnetism t’=0 M. Capone V. Kancharla (see also VCPT Senechal and Tremblay ). Antiferromagnetic (left) and d wave superconductor (right) Order Parameters

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

28. Follow the “normal state” with doping. 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 Ek = const and a height increasing as we approach the Fermi surface.

29. : Spectral Function A(k,ω→0)= -1/π G(k, ω→0) vs k U=16 t hole doped K.M. Shen et.al. 2004 2X2 CDMFT

30. 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. • Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, 226402. (2004) .

31. What about the electron doped semiconductors ?

32. Spectral Function A(k,ω→0)= -1/π G(k, ω→0) vs k electron doped P. Armitage et.al. 2001 Momentum space differentiation a we approach the Mott transition is a generic phenomena. Location of cold and hot regions depend on parameters. Civelli et.al. 2004

33. Qualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near (p/2,p/2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping. • Electron doped case, has in the underlying normal state quasiparticles leave in the (p, 0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (p/2, p/2).

34. Can we connect the superconducting state with the “underlying “normal” state “ ? • Yes, within our resolution in the hole doped case. • No in the electron doped case. • What does the underlying “normal state “ look like ? • Unusual distribution of spectra (Fermi arcs) in the normal state.

35. To test if the formation of the hot and cold regions is the result of the proximity to Antiferromagnetism, we studied various values of t’/t, U=16.

36. Introduce much larger frustration: t’=.9t U=16tn=.69 .92 .96

37. Approaching the Mott transition: • Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! • General phenomena, but the location of the cold regions depends on parameters. • With the present resolution, t’ =.9 and .3 are similar. However it is perfectly possible that at lower energies further refinements and differentiation will result from the proximity to different ordered states.

38. Fermi Surface Shape Renormalization ( teff)ij=tij+ Re(Sij(0))

39. Fermi Surface Shape Renormalization • Photoemission measured the low energy renormalized Fermi surface. • If the high energy (bare ) parameters are doping independent, then the low energy hopping parameters are doping dependent. Another failure of the rigid band picture. • Electron doped case, the Fermi surface renormalizes TOWARDS nesting, the hole doped case the Fermi surface renormalizes AWAY from nesting. Enhanced magnetism in the electron doped side.

40. Understanding the location of the hot and cold regions. Interplay of lifetime and fermi surface.

41. Large Doping

42. Small Doping

43. How is the Mott insulatorapproached from the superconducting state ? Work in collaboration with M. Capone, see also V. Kancharla’s talk.

44. Superconductivity is destroyed by transfer of spectral weight.

46. Evolution of the low energy tunneling density of state with doping. Decrease of spectral weight as the insulator is approached.

47. Superconductivity is destroyed at half filling due to a reduction of the one electron weight. Just like in the slave boson. • High energy ph asymmetry. • Low energy ph symmetry.

48. Dynamical RVB, real and imaginary parts of the anomalous self energies.

49. Conclusions • DMFT is a useful mean field tool to study correlated electrons. Provide a zeroth order picture of a physical phenomena. • Provide a link between a simple system (“mean field reference frame”) and the physical system of interest. [Sites, Links, and Plaquettes] • Formulate the problem in terms of local quantities (which we can usually compute better). • Allows to perform quantitative studies and predictions . Focus on the discrepancies between experiments and mean field predictions. • Generate useful language and concepts. Follow mean field states as a function of parameters. • K dependence gets strong as we approach the Mott transition. Fermi surfaces and lines of zeros of G.

50. Conclusions • Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! • General phenomena, but the location of the cold regions depends on parameters. Study the “normal state” of the Hubbard model is useful. • Character of the superconductivity is different for small and large U.