Superconductivity near the Mott transition: what can we learn from plaquette DMFT?

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

2. Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study, K. H. and G. Kotliar, cond-mat/0709.0019 (37 pages and 42 figures) 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. 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, cond-mat/0605149 References and Collaborators

3. 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]

4. SR=(1,1) SR=(0,0) SR=(1,0) Cluster DMFT approach Exact Baym Kadanoff functional of two variables G[S,G]. Restriction to the degrees of freedom that live on a plaquette and its supercell extension.. Maps the many body problem ontoa self consistentimpurity model • Impurity solvers: • ED • NCA • Continuous time QMC F[Gplaquette] periodization

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

7. Normal state T>Tc Momentum space differentiation • (0,0) orbital reasonable coherent Fermi liquid • (p,0) very incoherent around optimal doping (d2~0.16 for t-J and d2~0.1 for Hubbard U=12t) • (p,p) most incoherent and diverging at another doping (d1~0.1 for t-J and d1~0 for Hubbard U=12t) t-J model, T=0.01t

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

9. 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

10. Nodal quasiparticles

11. 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

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

13. Superfluid density at low T Low T expansion using imaginary axis QMC data. Current vertex corrections are neglected In RVB the coefficient b~d2 at low d [Wen&Lee, Ioffe&Millis]

14. underdoped Computed by NCA, current vertex corrections neglected Superfluid density close to Tc

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

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

17. 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 NM d=0.08 SC d=0.08 SC d=0.20 NM d=0.20 Approximate PH symmetry at optimal doping also B. Kyung et.al, PRB 73, 165114 2006 Computed by the NCA for the t-J model

18. 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

19. 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

20. 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)

21. 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

22. ~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)

23. 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)

24. 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

25. 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 d • In the underdoped system -> kinetic energy saving mechanism • overdoped system -> kinetic energy loss mechanism • exchange energy is always optimized in SC state

26. The mean field phase diagram and finite temperature crossover between underdoped and overdoped 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) Issues

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

28. Problems with the RVB slave bosons: Mean field is too uniform on the Fermi surface, in contradiction with ARPES. Fails to describe the incoherent finite temperature regime and pseudogap regime. Temperature dependence of the penetration depth. Theory: r[T]=x-Ta x2 , Exp: r[T]= x-T a. Can not describe two distinctive gaps: normal state pseudogap and superconducting gap RVB phase diagram of the t-J m.

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

30. Fermi surface d=0.09 Cumulant is short in ranged: Arcs FS in underdoped regime pockets+lines ofzeros of G == arcs Arcs shrink with T!

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

32. 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 local susceptibility YBa2Cu3O6.6 (Tc=62.7K) Pengcheng et.al., Science 284, (1999)

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