Nonrelativistic Banks-Casher relation and random matrix theory for multi-component fermionic superfluids

TL;DR

This paper extends QCD-inspired techniques to nonrelativistic multi-component fermions, deriving exact relations for symmetry breaking, introducing non-local order parameters, and demonstrating the equivalence to random matrix theory through analytical and numerical methods.

Contribution

It provides the first nonrelativistic analogues of Banks-Casher and Smilga-Stern relations, linking spectral properties to phase transitions in multi-component fermionic superfluids.

Findings

Derived exact spectral relations for symmetry breaking

Introduced non-local order parameters with spectral representations

Confirmed predictions through Monte Carlo simulations

Abstract

We apply QCD-inspired techniques to study nonrelativistic N-component degenerate fermions with attractive interactions. By analyzing the singular-value spectrum of the fermion matrix in the Lagrangian, we derive several exact relations that characterize the spontaneous symmetry breaking U(1)xSU(N)$\to$Sp(N) through bifermion condensates. These are nonrelativistic analogues of the Banks-Casher relation and the Smilga-Stern relation in QCD. Non-local order parameters are also introduced and their spectral representations are derived, from which a nontrivial constraint on the phase diagram is obtained. The effective theory of soft collective excitations is derived and its equivalence to random matrix theory is demonstrated in the epsilon-regime. We numerically confirm the above analytical predictions in Monte Carlo simulations.