One-dimensional Cooper pairing

TL;DR

This paper investigates electron pairing in a one-dimensional fermion gas, revealing unique dispersion relations and the effects of interaction strength and range on Cooper pair properties at zero temperature.

Contribution

It provides a detailed analysis of 1D Cooper pairs, including novel dispersion relations with two branches and the influence of interaction parameters, extending previous 2D and 3D studies.

Findings

Cooper pair binding energy increases with attraction strength

Dispersion relation exhibits phonon-like and roton-like branches

Quadratic dispersion is recovered at zero Fermi wavenumber

Abstract

We study electron pairing in a one-dimensional (1D) fermion gas at zero temperature under zero- and finite-range, attractive, two-body interactions. The binding energy of Cooper pairs (CPs) with zero total or center-of-mass momentum (CMM) increases with attraction strength and decreases with interaction range for fixed strength. The excitation energy of 1D CPs with nonzero CMM display novel, unique properties. It satisfies a dispersion relation with \textit{two} branches: a\ phonon-like \textit{linear }excitation for small CP CMM; this is followed by roton-like \textit{quadratic} excitation minimum for CMM greater than twice the Fermi wavenumber, but only above a minimum threshold attraction strength. The expected quadratic-in-CMM dispersion \textit{in vacuo }when the Fermi wavenumber is set to zero is recovered for \textit{any% } coupling. This paper completes a three-part exploration initiated in 2D and continued in 3D.