Cluster dynamical mean-field methods ford-wave superconductors: Role of geometry

Abstract

We compare the accuracy of two cluster extensions of dynamical mean-field theory in describing $d$-wave superconductors, using as a reference model a saddle-point $t\text{\ensuremath{-}}J$ model which can be solved exactly in the thermodynamic limit and at the same time reasonably describes the properties of high-temperature superconductors. The two methods are cellular dynamical mean-field theory, which is based on a real-space perspective, and dynamical cluster approximation, which enforces a momentum-space picture by imposing periodic boundary conditions on the cluster, as opposed to the open boundary conditions of the first method. We consider the scaling of the methods for large cluster size, but we also focus on the behavior for small clusters, such as those accessible by means of present techniques, with particular emphasis on the geometrical structure, which is definitely a relevant issue in small clusters.