1. Gordon Conference 2007 Superconductivity near the Mott transition: what can we learn from plaquette DMFT? K Haule Rutgers University

2. Les Diablerets 2007 References and Collaborators  Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study, K. H. and G. Kotliar, cond-mat/0709.0019 (37 pages and 42 figures)cond-mat/0709.0019  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, cond-mat/0704.1486. cond-mat/0704.1486.  Quantum Monte Carlo Impurity Solver for Cluster DMFT and Electronic Structure Calculations in Adjustable Base, K. H., Phys. Rev. B 75, 155113 (2007).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).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).Phys. Rev. B 74, 064510 (2006)  Avoided Quantum Criticality near Optimally Doped High Temperature Superconductors, K.H. and G. Kotliar, cond-mat/0605149cond-mat/0605149

3. Les Diablerets 2007 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…..]  Leave out disorder, electronic structure, phonons … [CDMFT+LDA second step, under way]  Approach the problem from high temperatures where physics is more local. Address issues of finite frequency– a nd finite temperature crossovers.

4. Les Diablerets 2007 Impurity solvers: ED NCA Continuous time QMC Cluster DMFT approach Exact Baym Kadanoff functional of two variables  [ ,G]. Restriction to the degrees of freedom that live on a plaquette and its supercell extension..  R=(0,0)  R=(1,0)  R=(1,1)  [G plaquette ] periodization Maps the many body problem onto a self consistent impurity model

5. Les Diablerets 2007 Momentum versus real space In plaquette CDMFT cluster quantities are diagonal matrices in cluster momentum base In analogy with multiorbital Hubbard model exist well defined orbitals But the inter-orbital Coulomb repulsion is nontrivial and tight-binding Hamiltonian in this base is off-diagonal

6. Les Diablerets 2007  (i  ) with CTQMC Hubbard model, T=0.005t on-site largest nearest neighbor smaller next nearest neighbor important in underdoped regime

7. Les Diablerets 2007 Normal state T>Tc  (0,0) orbital reasonable coherent Fermi liquid t-J model, T=0.01t Momentum space differentiation  ( ,0) very incoherent around optimal doping (   ~0.16 for t-J and   ~0.1 for Hubbard U=12t)  (  ) most incoherent and diverging at another doping (  1 ~0.1 for t-J and  1 ~0 for Hubbard U=12t)

8. Les Diablerets 2007 Momentum space differentiation t-J model, T=0.005t Normal state T>Tc SC state T<<Tc with large anomalous self- energy …gets replaced by coherent SC state Normal state T>Tc: Very large scattering rate at optimal doping  ( ,0) orbital T

9. Les Diablerets 2007 Fermi surface  =0.09 Arcs FS in underdoped regime pockets+lines of zeros of G == arcs Cumulant is short in ranged: Single site DMFT PD

10. Les Diablerets 2007 Nodal quasiparticles

11. Les Diablerets 2007 Nodal quasiparticles the slope=v nod almost constant V nod almost constant up to 20% v  dome like shape Superconducting gap tracks Tc! M. Civelli, cond-mat 0704.1486

12. Les Diablerets 2007 Antinodal gap – two gaps M. Civelli, using ED, cond-mat 0704.1486 Superconducting gap has a dome like shape (like v  ) Normal state “pseudogap” monotonically decreasing

13. Les Diablerets 2007 Superfluid density at low T Low T expansion using imaginary axis QMC data. Current vertex corrections are neglected In RVB the coefficient b~  2 at low  [Wen&Lee, Ioffe&Millis]

14. Les Diablerets 2007 Superfluid density close to T c Computed by NCA, current vertex corrections neglected underdoped

15. Les Diablerets 2007 Anomalous self-energy and order parameter Anomalous self-energy: Monotonically decreasing with i  Non-monotonic function of doping (largest at optimal doping) Of the order of t at optimal doping at T=0,  =0 Order parameter has a dome like shape and is small (of the order of 2Tc) Hubbard model, CTQMC

16. Les Diablerets 2007 Anomalous self-energy on real axis Computed by the NCA for the t-J model Many scales can be identified J,t,W It does not change sign at certain frequency  D ->attractive for any  Although it is peaked around J, it remains large even for  >W

17. Les Diablerets 2007 SC Tunneling DOS Large asymmetry at low doping Gap decreases with doping DOS becomes more symmetric Normal state has a pseudogap with the same asymmetry Computed by the NCA for the t-J model Approximate PH symmetry at optimal doping SC  =0.08 SC  =0.20 NM  =0.08 NM  =0.20 also B. Kyung et.al, PRB 73, 165114 2006

18. Les Diablerets 2007 Optical conductivity Basov et.al.,PRB 72,54529 (2005)  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

19. Les Diablerets 2007 Optical spectral weight - Hubbard versus t-J model t-J model J Drude no-U Experiments intraband interband transitions ~1eV Excitations into upper Hubbard band Kinetic energy in Hubbard model: Moving of holes Excitations between Hubbard bands Hubbard model Drude U t 2 /U Kinetic energy in t-J model Only moving of holes f-sumrule

20. Les Diablerets 2007 Optical spectral weight & Optical mass F. Carbone,et.al, PRB 74,64510 (2006) Bi2212 Weight increases because the arcs increase and Zn increases (more nodal quasiparticles) mass does not diverge approaches ~1/J Basov et.al., PRB 72,60511R (2005)

21. Les Diablerets 2007 Temperature/doping dependence o f the optical spectral weight Single site DMFT gives correct order of magni tude (Toshi&Capone) At low doping, single site DMFT has a small cohere nce scale -> big change Cluser DMF for t-J: Carriers become more coherent In the overdoped regime -> bigger change in kinetic energy for large 

22. Les Diablerets 2007 Bi2212 ~1eV Weight bigger in SC, K decreases (non-BCS) Weight smaller in SC, K increases (BCS-like) Optical weight, plasma frequency F. Carbone,et.al, PRB 74,64510 (2006) A.F. Santander-Syro et.al, Phys. Rev. B 70, 134504 (2004)

23. Les Diablerets 2007 Phys Rev. B 72, 092504 (2005) cluster-DMFT, Eu. Lett. 77, 27007 (2007). Kinetic energy change Kinetic energy decreases 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)

24. Les Diablerets 2007 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 Scalapino&White, PRB 58, (1998) Main origin of the condensation energy

25. Les Diablerets 2007 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 Superfluid density linear temperature coefficient approaches constant at low doping Superfuild density close to Tc is linear in temperature Tunneling DOS is very asymmetric in UR and becomes more symmetric at ODR Optical conductivity shows a two component behavior at low doping Optical mass ~1/J at low doping and optical weigh increases linearly with  In the underdoped system -> kinetic energy saving mechanism overdoped system -> kinetic energy loss mechanism exchange energy is always optimized in SC state

26. Les Diablerets 2007 Issues The mean field phase diagram and finite temperature crossover between underdoped and over doped regime Study only plaquette (2x2) cluster DMFT in the strong coupling limit (at large U=12t) Can not conclude if SC phase is stable in the exact solution of the model. If the mean field SC phase is not stable, other interacting term in H could stabilize the mean-field phase (long range U, J)

27. Les Diablerets 2007 Doping dependence of the spectral weight F. Carbone,et.al, PRB 74,64510 (2006) Comparison between CDMFT&Bi2212

28. Les Diablerets 2007 RVB phase diagram of the t-J m. Problems with the RVB slave bosons:  Mean field is too uniform on the Fermi surfa ce, in contradiction with ARPES.  Fails to describe the incoherent finite temperature regime and pseudogap regime.  Temperature dependence of the penetration depth. Theory:  [T]=x-Ta x 2, Exp:  [T]= x-T a.  Can not describe two distinctive gaps: normal state pseudogap and superconducting gap

29. Les Diablerets 2007 Similarity with experiments Louis Taillefer, Nature 447, 565 (2007). A. Kanigel et.al., Nature Physics 2, 447 (2006) Arcs FS in underdoped regime pockets+lines of zeros of G == arcs de Haas van Alphen small Fermi surface Shrinking arcs On qualitative level consistent with

30. Les Diablerets 2007 Fermi surface  =0.09 Arcs FS in underdoped regime pockets+lines of zeros of G == arcs Arcs shrink with T! Cumulant is short in ranged:

31. Les Diablerets 2007 Insights into superconducting state (BCS/non-BCS)? J. E. Hirsch, Science, 295, 5563 (2002) BCS: upon pairing potential energy of electrons decreases, kinetic energy increases (cooper pairs propagate slower) Condensation energy is the difference non-BCS: kinetic energy decreases upon pairing (holes propagate easier in superconductor)

32. Les Diablerets 2007 Pengcheng et.al., Science 284, (1999) YBa 2 Cu 3 O 6.6 (Tc=62.7K) Origin of the condensation energy local susceptibility Resonance at 0.16t~5Tc (most pronounced at optimal doping) Second peak ~0.38t~120meV (at opt.d) substantially contributes to condensation energy Scalapino&White, PRB 58, (1998) Main origin of the condensation energy

33. Les Diablerets 2007 Similarity with experiments Louis Taillefer, Nature 447, 565 (2007). Arcs FS in underdoped regime pockets+lines of zeros of G == arcs de Haas van Alphen small Fermi surface On qualitative level consistent with