Anomalous Breaking of Scale Invariance in a Two-Dimensional Fermi Gas

The frequency of the breathing mode of a two-dimensional Fermi gas with zero-range interactions in a harmonic confinement is fixed by the scale invariance of the Hamiltonian. Scale invariance is broken in the quantized theory by introducing the two-dimensional scattering length as a regulator. This is an example of a quantum anomaly in the field of ultracold atoms and leads to a shift of the frequency of the collective breathing mode of the cloud. In this work, we study this anomalous frequency shift for a two-component Fermi gas in the strongly interacting regime. We measure significant upwards shifts away from the scale-invariant result that show a strong interaction dependence. This observation implies that scale invariance is broken anomalously in the strongly interacting two-dimensional Fermi gas.

Figure 1

Breathing mode in a harmonic confinement. (a)–(c) In situ images of the cloud along the strongly confined axial direction at different hold times $t=68–80\mathrm{ms}$ after quenching the trap depth at $t=0\mathrm{ms}$. The inner (outer) dashed lines indicate the $1\sigma$ ($2\sigma$) width of a Gaussian fit to the cloud profile. The images are averaged over several experimental realizations and were taken at larger amplitudes to make the oscillation more apparent. (d) The cloud width ${\sigma }_x$ as a function of the hold time $t$ after the quench (blue circles). The inset shows the complete data set from $t=0$ to 400 ms. The center-of-mass oscillations of the same cloud are shown in red as a comparison (right, $y$ axis). The solid lines are fits of a damped sine function to the measurements.

Figure 2

Measured average breathing and dipole frequencies versus magnetic offset field $\mathit{B}$. Statistical errors are on the order of the symbol size. The dipole frequencies were scaled by a factor of 2 to facilitate the comparison to the breathing mode. We fit a model ${\omega }_R(B,\sigma )$ for our trap frequencies to the dipole frequency measurements (solid orange line). The solid black line shows the same model for a fixed cloud size.

Figure 3

Anomalous shift of the breathing mode frequency. At low temperatures the breathing mode shows a significant shift away from the scale-invariant result of ${\omega }_B=2{\omega }_R$ towards higher frequencies, even after accounting for both the effects of anharmonicity and anisotropy of our trap. Our data agree with a beyond mean-field approximation from Ref. [24] at $T/T_F=0.2$ (solid black line). The inset compares our data to a zero temperature calculation from Ref. [21]. Circles and squares represent measurements that were taken with different spin mixtures.

Figure 4

Influence of the third dimension on the oscillation frequency. As we increase the particle number we observe a strong shift towards lower frequencies. On the $x$ axis we plot the measured particle number of one spin component $N$ divided by $N_{2\mathrm{D}}=48000$. $N_{2\mathrm{D}}$ is the maximal number of noninteracting atoms per spin state in the axial ground state of our trap at this magnetic field. The dashed line is a straight connection between the data points.