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

• Published in: arXiv.org
• Main Authors: Setty, Chandan, Zhao, Jinchao, Fanfarillo, Laura, Huang, Edwin W, Hirschfeld, Peter J, Phillips, Philip W, Yang, Kun
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 21.09.2022
• Subjects: Center of mass; Cooper pairs; Cuprates; Electrons; Exact solutions; Free energy; Momentum; Order parameters; Physics - Strongly Correlated Electrons; Physics - Superconductivity; Robustness (mathematics); Superconductivity; Transition metal compounds; Two body problem; Wave equations; Wave functions
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2209.10568

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.