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

Pair density waves (PDWs) are superconducting states formed by Cooper pairs of electrons containing a nonzero 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 exact mechanism that spontaneously favors a modulated solution of the superconducting order parameter. Here, we study the problem of two electrons subject to an anisotropic attractive potential. We solve the two-body Schrödinger 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, intermediate interaction strength. Hence our exact result justifies previous mean-field approximations that obtain modulated ground states at finite but large interactions. Using this insight from the exact two-body solution, we construct a variational many-body wave function and show that the conclusions of the two-body problem are robust in the many-body limit. Our results thus lay the theoretical and microscopic foundation for the existence of PDWs.