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 Bardeen-Cooper-Schrieffer–Bose-Einstein condensation (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–BEC regimes, even though $P-2\epsilon /d$, sensitive to the dimer binding energy that breaks scale invariance, is not.

Figure 1

Schematic plot of the bulk viscosity spectral function $\zeta (\omega )$ (solid line) and the scaled “vacuum” contribution $C{\zeta }_0(\omega )/C_0$ (dashed line) given by (6). The smallness of the subtracted sum rule (3) for $[\zeta (\omega )-C{\zeta }_0(\omega )/C_0]$ implies very little spectral weight below the dimer binding energy $|{\epsilon }_b|$ for excitations that break scale invariance.

Figure 2

Bulk viscosity sum rule:—The 2D sum rule $S_{2\mathrm{D}}$ at $T=0$ in units of $n{ϵ}_F$ plotted as a function of the coupling $g_2=\ln (k_F{a}_2)$. Inset: The corresponding 3D result [4] as a function of $g_3=-1/(k_F{a}_3)$. In both 2D and 3D, $g_d\to +\infty$, the BCS limit, while $g_d\to -\infty$ is the BEC limit.

Figure 3

${\gamma }_2$ (solid blue line) and ${\gamma }_3$ (dashed red line) shown in units of $n{ϵ}_F$, where $n$ and ${ϵ}_F$ are the two and three dimensional density and Fermi energy, respectively. The error bars on ${\gamma }_2$ are associated with numerical derivatives of QMC data (with errors) [20]. The coupling $g_d$ is $\ln (k_F{a}_2)$ for $d=2$ and $-1/(k_F{a}_3)$ for $d=3$.