In this Letter, we show that the classical SO(2,1) symmetry of a harmonically trapped Fermi gas in two dimensions is broken by quantum effects. The anomalous correction to the symmetry algebra is given by a two-body operator that is well known as the contact. Taking into account this modification, we are able to derive the virial theorem for the system and a universal relation for the pressure of a homogeneous gas. The existence of an undamped breathing mode is associated with the classical symmetry. We provide an estimate for the anomalous frequency shift of this oscillation at zero temperature and compare the result with a recent experiment by [E. Vogt et al., Phys. Rev. Lett. 108, 070404 (2012)]. Discrepancies are attributed to finite temperature effects.
A harmonically trapped strongly interacting Fermi gas in two dimensions is anticipated to exhibit a quantum anomaly and possesses a breathing mode at frequencies different from a classical scale invariant value
ωB=2ω⊥, where
ω⊥ is the frequency of the trapping potential. This quantum anomaly, however, has never been conclusively confirmed in current experiments {[}E. Vogt \textit{et al.}, Phys. Rev. Lett. \textbf{108}, 070404 (2012); Holten \textit{et al.}, arXiv:1803.08879; Peppler \textit{et al.}, arXiv:1804.05102{]}. Here, we theoretically investigate the universal density equation of state and the breathing mode frequency of a unitary Fermi gas at the dimensional crossover from three to two dimensions. We find that the simple model of a two-dimensional strongly interacting Fermi gas with a single
s-wave scattering length is not adequate to describe the experiments, as commonly believed. A more complete description of quasi-two-dimensions leads to a much weaker quantum anomaly in the breathing mode frequency, consistent with the experimental observations.
SO(2,1) dynamical symmetry makes a remarkable prediction that the breathing oscillation of a scale-invariant quantum gas in an isotropic harmonic trap is isentropic and can persist indefinitely. In two dimensions, this symmetry is broken due to quantum anomaly in the strongly interacting range, and consequently the lifetime of the breathing mode becomes finite. The persistent breather in a strongly interacting system has so far not been realized. Here, we experimentally achieve the long-lived breathing mode in a three-dimensional unitary Fermi gas, which is protected by the SO(2,1) symmetry. The nearly perfect SO(2,1) symmetry is realized by loading the ultracold Fermi gas in an isotropic trap and tuning the interatomic interaction to resonance. The breathing mode oscillates at twice the trapping frequency even for large excitation amplitudes. The ratio of damping rate to oscillation frequency is as small as 0.002, providing an interacting persistent breather. The oscillation frequency and damping rate are nearly constant for different atomic densities and temperatures, demonstrating the robustness of the SO(2,1) symmetry in three dimensions. The factors that lead to the residual damping have also been clarified. This work opens the way to study many-body nonequilibrium dynamics related to the dynamical symmetry.
The contact is an important concept that characterizes the universal properties of a strongly interacting quantum gas. It appears in both thermodynamic (energy, pressure, etc.) and dynamic quantities (radio-frequency and Bragg spectroscopies, etc.) of the system. Very recently, the concept of contact was extended to higher partial waves; in particular, the
p-wave contacts have been experimentally probed in recent experiments. So far, discussions on
p-wave contacts have been limited to three dimensions. In this paper, we generalize the
p-wave contacts to two dimensions and derive a series of universal relations, including the adiabatic relations, high-momentum distribution, virial theorem, and pressure relation. At the high-temperature and low-density limit, we calculate the
p-wave contacts explicitly using virial expansion. A formula which directly connects the shift of the breathing-mode frequency and the
p-wave contacts is given in a harmonically trapped system. Finally, we also derive the relationships between interaction parameters in three- and two-dimensional Fermi gases and discuss possible experimental realization of a two-dimensional Fermi gas with
p-wave interactions.
We examine the breathing mode of a strongly interacting two-dimensional Fermi gas and the role of temperature on the anomalous breaking of scale invariance. By calculating the equation of state with different many-body
T-matrix theories and the virial expansion, we obtain a hydrodynamic equation of the harmonically trapped Fermi gas (with trapping frequency ${\ensuremath{\omega}}_{0}$) through the local density approximation. By solving the hydrodynamic equations, we determine the breathing mode frequencies as a function of interaction strength and temperature. We find that the breathing mode anomaly depends sensitively on both interaction strength and temperature. In particular, in the strongly interacting regime, we predict a significant downshift of the breathing mode frequency, below the scale invariant value of $2{\ensuremath{\omega}}_{0}$, for temperatures of the order of the Fermi temperature.
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.
For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently, the frequency of the breathing mode ${\ensuremath{\omega}}_{B}$ of the 2D Fermi gas can be different from $2{\ensuremath{\omega}}_{0}$, with ${\ensuremath{\omega}}_{0}$ the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for ${\ensuremath{\omega}}_{B}$. We further calculate ${\ensuremath{\omega}}_{B}$ through the Euler equations in the hydrodynamic regime. The obtained value of ${\ensuremath{\omega}}_{B}$ satisfies the upper bounds and shows deviation from $2{\ensuremath{\omega}}_{0}$, which can be as large as about
8.