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

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.

24 pages, 6 figures

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• "Quantum Geometric Nesting" and Solvable Model Flat-Band Systems
• Self-consistent theory of $2\times 2$ pair density waves in kagome superconductors
• Pair-density-wave superconductivity: a microscopic model on 2D honeycomb lattice
• Density matrix renormalization group study of quantum-geometry-facilitated pair density wave order
• Enhanced pair-density-wave vertices in a bilayer Hubbard model at half-filling
• Efficiency of the superconducting diode effect of pair-density-wave states in two-dimensional $d$-wave altermagnets
• Strong pair-density-wave fluctuations in an exactly solvable doped Mott insulator
• Variational approach for the two-body problem in a multiband extended-Hubbard model
• Spin-triplet pair density wave superconductors