Cooper Instability of a Magnetic Wigner Crystal

Wangqian Miao; Chunli Huang

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arXiv · 2610.10710

Cooper Instability of a Magnetic Wigner Crystal

Wangqian Miao·Chunli Huang

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Abstract

Recent scanning tunneling microscopy has directly imaged electronic crystals in rhombohedral hexalayer graphene, and magnetotransport measurements suggest that superconductivity and charge-density-wave order emerge from a common spin-valley-polarized ferromagnetic metal at nearby carrier densities. Motivated by these observations, we ask whether collective modes in a magnetic Wigner crystal can mediate pairing in this setting. We address this question by studying the Cooper problem for two electrons in the first conduction band of the Wigner crystal. Unlike conventional phonon-mediated pairing, the collective mode (boson) energies and single-particle (fermion) dispersion are comparable scales controlled by the electron density, leaving no small parameter for an adiabatic expansion. Moreover, because the magnetic Wigner crystal is not time-reversal invariant, the usual Cooper logarithmic divergence in the pairing susceptibility is absent, so the attraction must exceed a finite threshold for pairing to occur. These unusual features motivate a numerical experiment to see whether the crystal's own vibrations can bind electrons. We obtain the crystalline ground state and its collective mode spectrum within Hartree--Fock and time-dependent Hartree--Fock approximation, and derive the electron--phonon Hamiltonian using the quasiboson approximation. Solving the resulting two-electron problem, we find that phonon-mediated attraction exceeds the pairing threshold, with chiral $p$-wave pairing as the leading instability. Within these approximations, we show that collective modes of a Wigner crystal can indeed favor chiral pairing despite broken time-reversal symmetry. Our results provide an encouraging starting point for developing a full many-body theory to study electronic crystallization and superconductivity.

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Wangqian Miao, Chunli Huang. 2026-10-07. Cooper Instability of a Magnetic Wigner Crystal. https://arxiv.org/abs/2610.10710

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