Exact solution for finite center-of-mass momentum Cooper pairing

TL;DR

This paper provides an exact two-electron solution demonstrating that finite center-of-mass momentum pairing is energetically favored under certain conditions, supporting the theoretical foundation for pair density waves in superconductors.

Contribution

It introduces an exact solution for two electrons with anisotropic attraction, establishing conditions for finite momentum pairing and constructing a BCS-like wave function to support PDWs.

Findings

Finite momentum pairs are favored above a critical interaction strength.

The two-body results are consistent with many-body energy calculations.

Supports the microscopic foundation for pair density waves.

Abstract

Pair density waves (PDWs) are superconducting states formed by ``Cooper pairs" of electrons containing a non-zero center-of-mass momentum. They are characterized by a spatially modulated order parameter and may occur in a variety of emerging quantum materials such as cuprates, transition metal dichalcogenides (TMDs) and Kagome metals. Despite extensive theoretical and numerical studies seeking PDWs in a variety of lattices and interacting settings, there is currently no generic and robust mechanism that favors a modulated solution of the superconducting order parameter in the presence of time reversal symmetry. Here, we study the problem of two electrons subject to an anisotropic ($d$-wave) attractive potential. We solve the two-body Schrodinger wave equation exactly to determine the pair binding energy as a function of the center-of-mass momentum. We find that a modulated (finite momentum) pair is favored over a homogeneous (zero momentum) solution above a critical interaction. Using this insight from the exact two-body solution, we construct a BCS-like variational many-body wave function and calculate the free energy and superconducting gap as a function of the center-of-mass momentum. A zero temperature analysis of the energy shows that the conclusions of the two-body problem are robust in the many-body limit. Our results lay the theoretical and microscopic foundation for the existence of PDWs.