Superconducting correlations in any dimensionality

TL;DR

This paper derives an analytical expression for the anomalous Cooper loop across arbitrary dimensions, revealing unique features in the momentum space and implications for high-temperature superconductivity.

Contribution

It provides a closed-form expression for the Cooper loop in any dimension and explores its implications for superconductivity and high-T_c cuprates.

Findings

Analytical expression for Cooper loop in any dimension.

Logarithmic singularity at q=0 in q-space.

Possible bound state in 2D even as k_F approaches zero.

Abstract

We consider the static anomalous Cooper loop for the electron gas of arbitrary spatial dimensionality, D. This object enters the mean-field equation for the superconducting temperature, T_c. The closed expression in r- space is found at T=0 as an analytical function of D. Its counterpart in q-space has a logarithmic singularity at q=0 and no singularities at q=2k_F for any D. We extend our analysis for the case of finite T and note that in two dimensions the bound state of two electrons is possible even in the limit k_F -> 0. We comment on the possible implications of this fact for the theory of high-T_c cuprates.