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

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 spontaneous symmetry breaking $\mathrm{U}(1)\times \mathrm{SU}(N)\to \mathrm{Sp}(N)$ through bifermion condensates. These are nonrelativistic analogues of the Banks-Casher relation and the Smilga-Stern relation in QCD. Nonlocal 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 $ϵ$ regime. We numerically confirm the above analytical predictions in Monte Carlo simulations.

Figure 1

Classification of possible finite-temperature phase diagrams for even $N\ge 4$.

Figure 2

The four fermion bilinears that transform to each other under U(1) and $\mathrm{SU}(N)/\mathrm{Sp}(N)$ rotations.

Figure 3

Spectral density $R_1(\mathrm{\Lambda })$ in the lattice simulation for $N=2$ and $g/a^2=1$. The averages over 500 configurations are shown.

Figure 4

Microscopic spectral density $\rho (\lambda )$ and the smallest singular-value distribution $P({\lambda }_{\min })$ in the quenched lattice simulation with $g/a^2=3$. The green, blue, and red broken curves are the predictions (45) and (46) by RMT and the Poisson statistics (47), respectively. The averages over 5000 configurations are shown.

Figure 5

Spectral density in the noninteracting limit at $T=0$ in 2 (top) and 3 (bottom) spatial dimensions.