Few-Body Perspective of a Quantum Anomaly in Two-Dimensional Fermi Gases

A quantum anomaly manifests itself in the deviation of the breathing mode frequency from the scale invariant value of $2\omega$ in two-dimensional harmonically trapped Fermi gases, where $\omega$ is the trapping frequency. Its recent experimental observation with cold atoms reveals an unexpected role played by the effective range of interactions, which requires a quantitative theoretical understanding. Here we provide accurate, benchmark results on a quantum anomaly from a few-body perspective. We consider the breathing mode of a few trapped interacting fermions in two dimensions up to six particles and present the mode frequency as a function of scattering length for a wide range of effective range. We show that the maximum quantum anomaly gradually reduces as the effective range increases while the maximum position shifts towards the weak-coupling limit. We extrapolate our few-body results to the many-body limit and find a good agreement with the experimental measurements. Our results may also be directly applicable to a few-fermion system prepared in microtraps and optical tweezers.

Figure 1

(a) Pseudopotential $V(r)$ at $V_0/E_{\mathrm{ho}}=-79.04$ and $r_0/a_{\mathrm{ho}}=0.104$. Here, $E_{\mathrm{ho}}=\hslash \omega$ and $a_{\mathrm{ho}}=\sqrt{\hslash /(m\omega )}$ are the energy and length of the harmonic oscillator, respectively. This potential coincides with the potential used in obtaining the third diamond symbol of Fig. 2. Dashed line marks the line $V(r)=0$. (b) The 2D scattering length $a_{2\mathrm{D}}$ (solid line, scale on the left) and the effective range $R_s$ (dashed line, scale on the right) as a function of potential depth $V_0$ at $r_0/a_{\mathrm{ho}}=0.1$.

Figure 2

Symbols in (a), (b), and (c) show the breathing mode frequency ${\omega }_B$ as a function of logarithm of scattering length for (1,1), (2,2), and (3,3) systems, respectively. Blue circles, red squares, green diamonds, and orange triangles are for $k_F^2{R}_s=-0.0245$, $-0.245$, $-0.620$, and $-0.980$, respectively. The solid line shows the zero-range result in the many-body limit based on the AFQMC equation of state, while the dashed line shows the exact two-body zero-range solution [33].

Figure 3

(a) Circles, squares, diamonds, upper triangles, and lower triangles show the breathing mode frequency ${\omega }_B$ as a function of $1/N_{\mathrm{tot}}$ at $\ln (k_F{a}_{2\mathrm{D}})=0.284$, 0.794, 1.20, 1.49, and 2.40, respectively. The effective range is $k_F^2{R}_s=-0.0245$. The hollow symbols at $1/N_{\mathrm{tot}}=0$ show the many-body result at zero effective range. (b) Circles, squares, diamonds, upper triangles, and lower triangles show the breathing mode frequency ${\omega }_B$ as a function of $1/N_{\mathrm{tot}}$ at $\ln (k_F{a}_{2\mathrm{D}})=0.689$, 0.794, 1.20, 1.49, and 2.40, respectively. The effective range is $k_F^2{R}_s=-0.620$. In both plots, dashed lines show the linear extrapolation of (2,2) and (3,3) results towards the many-particle limit $N_{\mathrm{tot}}\to \infty$.

Figure 4

Circles and squares show the breathing mode frequency ${\omega }_B$ as a function of $\ln (k_F{a}_{2\mathrm{D}})$ for (3,3) system and for the extrapolated many-body results, respectively, with the rightmost two sets of circles and squares overlap. The effective range is fixed to $k_F^2{R}_s=-0.620$. Upper and lower triangles show the experimental data by Peppler et al. (Swinburne) [11] and Holten et al. (Heidelberg) [10]. The solid line shows the zero-range AFQMC result in the many-body limit while the dotted line shows the latest many-body result [14].