Relativistic Cooper pairing in the microscopic limit of chiral random matrix theory

Abstract

Random matrix theory (RMT) provides a powerful framework for analyzing universal features of strongly coupled physical systems. In quantum chromodynamics (QCD), cold quark matter at asymptotically high density is expected to exhibit color superconductivity (CSC), the analogue of superconductivity in condensed-matter systems. Although CSC phases have been studied within RMT primarily in the macroscopic large-$N$ limit, where $N$ denotes the matrix size, it has remained unclear whether an RMT exists that realizes CSC in the microscopic large-$N$ limit. Here we answer this question in the affirmative by introducing a novel non-Hermitian chiral random matrix model. For three quark flavors, we show that the model exhibits spontaneous breaking of color $\mathrm{SU}(3)$ and flavor $\mathrm{SU}(3)$ symmetries down to the diagonal $\mathrm{SU}(3)$ subgroup, thereby reproducing color-flavor locking in dense QCD. For two flavors, we find that color $\mathrm{SU}(3)$ is spontaneously broken to $\mathrm{SU}(2)$ while the chiral symmetry $\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{SU}(2)_{\mathrm{R}}$ remains unbroken, consistent with the two-flavor color-superconducting phase.