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We apply QCD-inspired techniques to study nonrelativistic N-componentdegenerate fermions with attractive interactions. By analyzing thesingular-value spectrum of the fermion matrix in the Lagrangian, we deriveseveral exact relations that characterize the spontaneous symmetry breakingU(1)xSU(N)$\to$Sp(N) through bifermion condensates. These are nonrelativisticanalogues of the Banks-Casher relation and the Smilga-Stern relation in QCD.Non-local order parameters are also introduced and their spectralrepresentations are derived, from which a nontrivial constraint on the phasediagram is obtained. The effective theory of soft collective excitations isderived and its equivalence to random matrix theory is demonstrated in theepsilon-regime. We numerically confirm the above analytical predictions inMonte Carlo simulations.

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