The Optimal Inhomogeneity for Superconductivity: Finite Size Studies

TL;DR

This study uses exact diagonalization on a 4x4 Hubbard model to identify optimal inhomogeneity patterns and interaction strengths that maximize superconductivity, showing finite size effects are minimal at certain parameters.

Contribution

It provides the first detailed finite size analysis of inhomogeneous Hubbard models, identifying conditions that optimize superconductivity.

Findings

Superconductivity peaks at intermediate inhomogeneity levels.

Maximum pair-binding energy found is 0.32t for specific parameters.

Results are robust against boundary condition changes.

Abstract

We report the results of exact diagonalization studies of Hubbard models on a $4\times 4$ square lattice with periodic boundary conditions and various degrees and patterns of inhomogeneity, which are represented by inequivalent hopping integrals $t$ and $t^{\prime}$. We focus primarily on two patterns, the checkerboard and the striped cases, for a large range of values of the on-site repulsion $U$ and doped hole concentration, $x$. We present evidence that superconductivity is strongest for $U$ of order the bandwidth, and intermediate inhomogeneity, $0 <t^\prime< t$. The maximum value of the ``pair-binding energy'' we have found with purely repulsive interactions is $\Delta_{pb} = 0.32t$ for the checkerboard Hubbard model with $U=8t$ and $t^\prime = 0.5t$. Moreover, for near optimal values, our results are insensitive to changes in boundary conditions, suggesting that the correlation length is sufficiently short that finite size effects are already unimportant.