Non-Analytic Crossover Behavior of SU($\mathcal{N}_c$) Fermi Liquid

TL;DR

This paper analytically explores the non-analytic crossover behavior in the thermodynamic potential of an SU(N) Fermi liquid with contact interactions, highlighting the effects of temperature and magnetic fields.

Contribution

It derives the equation of state for a dilute Fermi gas considering non-analytic dependencies and extends the analysis to SU(N) symmetric systems, revealing potential enhancement with larger N.

Findings

Identifies non-analytic crossover behavior in thermodynamic potential.

Shows the crossover is analogous to Ginzburg-Landau critical scaling.

Suggests larger N enhances the crossover effects.

Abstract

We consider the thermodynamic potential of a dilute Fermi gas with a contact interaction, at both finite temperature $T$ and non-zero effective magnetic fields $\mathbf{H}$, and derive the equation of state analytically using second order perturbation theory. Special attention is paid to the non-analytic dependence of $\Omega$ on temperature $T$ and (effective) magnetic field $\mathbf{H}$, which exhibits a crossover behavior as the ratio of the two is continuously varied. This non-analyticity is due to the particle-hole pair excitation being always gapless and long-ranged. The non-analytic crossover found in this paper can therefore be understood as an analog of the Ginzberg-Landau critical scaling, albeit only at the sub-leading order. We extend our results to an $\mathcal{N}_c$- component Fermi gas with an $\mathrm{SU}(\mathcal{N}_c)$-symmetric interaction, and point out possible enhancement of the crossover behavior by a large $\mathcal{N}_c$.