Superconductivity near the Mott transition what can we

Superconductivity near the Mott transition: what can we learn from plaquette DMFT? K Haule Rutgers University n n Gordon Conference 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) ¡ 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 Bi 2 Sr 2 Ca. Cu 2 O 8+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 Les Diablerets 2007

Approach q. Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models. q. 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…. . ] q. Approach the problem from high temperatures where physics is more local. Address issues of finite frequency– and finite temperature crossovers. q. Leave out disorder, electronic structure, phonons … [CDMFT+LDA second step, under way] Les Diablerets 2007

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 onto a self consistent impurity model F[Gplaquette] SR=(1, 1) SR=(0, 0) S Impurity solvers: • ED • NCA • Continuous time QMC R=(1, 0) periodization 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 Les Diablerets 2007

S(iw) with CTQMC on-site largest nearest neighbor smaller next nearest neighbor important in underdoped regime Hubbard model, T=0. 005 t Les Diablerets 2007

Momentum space differentiation §(0, 0) orbital reasonable coherent Fermi liquid Normal state T>Tc §(p, 0) very incoherent around optimal doping (d 2~0. 16 for t-J and d 2~0. 1 for Hubbard U=12 t) §(p, p) most incoherent and diverging at another doping (d 1~0. 1 for t-J and d 1~0 for Hubbard U=12 t) t-J model, T=0. 01 t Les Diablerets 2007

SC state T<<Tc Normal state T>Tc Momentum space differentiation Normal state T>Tc: Very large scattering rate at optimal doping …gets replaced by coherent SC state with large anomalous selfenergy S(p, 0) orbital T t-J model, T=0. 005 t Les Diablerets 2007

Fermi surface d=0. 09 Cumulant is short in ranged: Arcs FS in underdoped regime pockets+lines of zeros of G == arcs Single site DMFT PD Les Diablerets 2007

Nodal quasiparticles Les Diablerets 2007

Nodal quasiparticles Vnod almost constant up to 20% the slope=vnod almost constant v. D dome like shape Superconducting gap tracks Tc! M. Civelli, cond-mat 0704. 1486 Les Diablerets 2007

Antinodal gap – two gaps Normal state “pseudogap” monotonically decreasing M. Civelli, using ED, cond-mat 0704. 1486 Superconducting gap has a dome like shape (like v. D) 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~d 2 at low d [Wen&Lee, Ioffe&Millis] Les Diablerets 2007

Superfluid density close to Tc underdoped Computed by NCA, current vertex corrections neglected Les Diablerets 2007

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 t at optimal doping at T=0, w=0 Order parameter has a dome like shape and is small (of the order of 2 Tc) Hubbard model, CTQMC Les Diablerets 2007

Anomalous self-energy on real axis • Many scales can be identified J, t, W • It does not change sign at certain frequency w. D->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 Les Diablerets 2007

SC Tunneling DOS Large asymmetry at low doping Gap decreases with doping DOS becomes more symmetric NM d=0. 08 SC d=0. 20 Normal state has a pseudogap with the same asymmetry Approximate PH symmetry at optimal doping also B. Kyung et. al, PRB 73, 165114 2006 Computed by the NCA for the t-J model Les Diablerets 2007

Optical conductivity q. Low doping: two components Drude peak + MIR peak at 2 J q. For x>0. 12 the two components merge q. In SC state, the partial gap opens – causes redistribution of spectral weight up to 1 e. V Basov et. al. , PRB 72, 54529 (2005) Les Diablerets 2007

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

Optical spectral weight & Optical mass does not diverge approaches ~1/J Bi 2212 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, 60511 R (2005) Les Diablerets 2007

Temperature/doping dependence of the optical spectral weight 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 Les Diablerets 2007

Optical weight, plasma frequency Weight bigger in SC, K decreases (non-BCS) ~1 e. V Bi 2212 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) Les Diablerets 2007

Kinetic energy change Kinetic energy increases cluster-DMFT, Eu. Lett. 77, 27007 (2007). Kinetic energy decreases Kinetic energy increases Phys Rev. B 72, 092504 (2005) same as RVB (see P. W. Anderson Physica C, 341, 9 (2000) Exchange energy decreases and gives largest contribution to condensation energy Les Diablerets 2007

Origin of the condensation energy Main origin of the condensation energy Scalapino&White, PRB 58, (1998) • Resonance at 0. 16 t~5 Tc (most pronounced at optimal doping) • Second peak ~0. 38 t~120 me. V (at opt. d) substantially contributes to condensation energy 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 d • In the underdoped system -> kinetic energy saving mechanism overdoped system -> kinetic energy loss mechanism exchange energy is always optimized in SC state Les Diablerets 2007

Issues n n The mean field phase diagram and finite temperature crossover between underdoped and overdoped regime Study only plaquette (2 x 2) cluster DMFT in the strong coupling limit (at large U=12 t) 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) Les Diablerets 2007

Doping dependence of the spectral weight Comparison between CDMFT&Bi 2212 F. Carbone, et. al, PRB 74, 64510 (2006) Les Diablerets 2007

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

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

Fermi surface d=0. 09 Cumulant is short in ranged: Arcs FS in underdoped regime pockets+lines of zeros of G == arcs Arcs shrink with T! Les Diablerets 2007

Insights into superconducting state (BCS/non-BCS)? 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) J. E. Hirsch, Science, 295, 5563 (2002) Les Diablerets 2007

Origin of the condensation energy Main origin of the condensation energy Scalapino&White, PRB 58, (1998) • Resonance at 0. 16 t~5 Tc (most pronounced at optimal doping) • Second peak ~0. 38 t~120 me. V (at opt. d) substantially contributes to condensation energy local susceptibility YBa 2 Cu 3 O 6. 6 (Tc=62. 7 K) Pengcheng et. al. , Science 284, (1999) Les Diablerets 2007

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