Apparent low-energy scale invariance in two-dimensional Fermi gases

Recent experiments on a $\2d$ Fermi gas find an undamped breathing mode at twice the trap frequency over a wide range of parameters. To understand this seemingly scale-invariant behavior in a system with a scale, we derive two exact results valid across the entire BCS-BEC crossover at all temperatures. First, we relate the shift of the mode frequency from its scale-invariant value to $\gamma_d \equiv (1+2/d)P-\rho(\partial P/\partial\rho)_s$ in $d$ dimensions. Next, we relate $\gamma_d$ to dissipation via a new low-energy bulk viscosity sum rule. We argue that $\2d$ is special, with its logarithmic dependence of the interaction on density, and thus $\gamma_2$ is small in both the BCS and BEC regimes, even though $P - 2\varepsilon/d$, sensitive to the dimer binding energy that breaks scale invariance, is not.