Superconductivity and the Pseudogap in the Two-Dimensional Hubbard Model

Abstract

Recently developed numerical methods have enabled the explicit construction of the superconducting state of the Hubbard model of strongly correlated electrons in parameter regimes where the model also exhibits a pseudogap and a Mott insulating phase. d x 2 −y 2 symmetry superconductivity is found to occur in proximity to the Mott insulator, but separated from it by a pseudogapped nonsuperconducting phase. The superconducting transition temperature and order parameter amplitude are found to be maximal at the onset of the normal-state pseudogap. The emergence of superconductivity from the normal state pseudogap leads to a decrease in the excitation gap. All of these features are consistent with the observed behavior of the copper-oxide superconductors. where ε p = −2t(cos p x + cos p y ) + 4t ′ cos p x cos p y an electron dispersion and U > 0 a local interaction which disfavors double occupancy of a site.In the years since Anderson's paper, the interplay of the pseudogap and superconductivity and the relation of both to the Hubbard model have been of central interest to condensed matter physicists. The existence of dwave superconductivity in the Hubbard model has been demonstrated by perturbative analytic calculations [4] (later improved by renormalization group methods [5, 6]) and by numerics [7, 8]. The issue of the pseudogap has been more controversial. It has been variously argued that the pseudogap is a signature of unusual superconducting fluctuations [9][10][11], of a competing nonsuperconducting phase or regime [3, 12], or of physics not contained in the Hubbard model [13]. Theoretical determination of the interplay of the pseudogap and superconductivity in the Hubbard model is important in helping resolve this controversy, and will provide insight into the pseudogap phenomenon and into strongly correlated superconductivity more generally, but this requires access to intermediate or strong couplings for which perturbation theory is inadequate. The development of cluster dynamical mean field theory [15] has provided important nonperturbative information about the Hubbard model. Dynamical mean field theory approximates the electron self-energy in terms of a finite number of auxiliary functions determined from the solution of an N -site quantum impurity model and becomes exact as N tends to infinity. In this Letter we use dynamical mean field methods to determine the interplay of superconductivity and the pseudogap in the Hubbard model. This is challenging because the theory

Resilience ofd-wave superconductivity to nearest-neighbor repulsion

“…40 The results obtained for d-wave superconductivity with the plaquette are essentially identical to those obtained with larger clusters. 41 As in previous studies of the 2D Hubbard model with this approach 39 , the plaquette is hybridized with a set of 8 bath orbitals, as illustrated in Fig. 1, with 6 parameters to be determined through the CDMFT self-consistency relation: the hybridizations θ 1,2 between the bath and the cluster, the bath energies ε 1,2 and the bath d-wave pairing parameters d 1,2 which, when nonzero, signal the presence of superconductivity and lead to a non-vanishing cluster d-wave order parameter…”

“…Generalizations of Dynamical Mean-Field Theory are particularly suited for the strong coupling limit, but they are also an excellent guide to the physics at weak to intermediate coupling. [33][34][35][36][37][38][39][40][41][42] , These calculations suggest that pairing is maximized at intermediate coupling, where the on-site interaction U is of order the bandwidth W = 8t. Some non-perturbative calculations based on weak coupling ideas even agree at intermediate coupling 43 with strong-coupling based approaches.…”

Dynamical mean-field theory study of stripe order and d -wave superconductivity in the two-dimensional Hubbard model

“…However, large-scale cluster DMFT calculations [59][60][61] have provided evidence that the superconductivity exhibited by the plaquette DMFT is not just an artefact of the approximation, but indeed exists also for larger clusters. The stripe and FFLO states have also been investigated within real-space DMFT calculations where correlations do not extend beyond a single site [62][63][64][65][66], but inhomogeneous solutions are allowed.…”

Origin of the pseudogap in cuprate superconductors from quantum cluster theories

“…There has been much work relating the pseudogap to preformed superconducting pairs [15], which have have not reached phase coherence and superconductivity at the temperature T studied. On the other hand it has been argued that the pseudogap and superconductivity phases compete [16].…”