We present an experimental investigation of collective oscillations in harmonically trapped Fermi gases through the crossover from two to three dimensions. Specifically, we measure the frequency of the radial monopole or breathing mode as a function of dimensionality in Fermi gases with tunable interactions. The frequency of this mode is set by the adiabatic compressibility and probes the thermodynamic equation of state. In 2D, a dynamical scaling symmetry for atoms interacting via a δ-potential predicts the breathing mode to occur at exactly twice the harmonic confinement frequency. However, a renormalized quantum treatment introduces a new length scale which breaks this classical scale invariance resulting in a so-called quantum anomaly. Our measurements deep in the 2D regime lie above the scale-invariant prediction for a range of interaction strengths indicating the breakdown of a δ-potential model for atomic interactions. As the dimensionality is tuned from 2D to 3D we see the breathing oscillation frequency evolve smoothly towards the 3D limit.Two-dimensional (2D) materials exhibit many novel physical properties [1-4], but strong correlations and imperfections mean these are often difficult to understand theoretically. Quantum gases of neutral atoms may help address such fundamental challenges [5,6], as well as new phenomena, not readily accessible in other systems. One scenario, generally encountered in quantum field theories, is anomalous symmetry breaking. Specifically, a quantum anomaly occurs when a symmetry, present in a classical theory, is broken in the corresponding (renormalized) quantum theory. A paradigmatic example relevant to atomic collisions in ultracold alkali gases is the 2D δ-potential [7,8], where an additional length scale associated with the interactions is required to remove divergences in the elementary theory.Anomalous symmetry breaking has been considered in the context of 2D harmonically confined Bose [9] and Fermi gases [10][11][12] where interactions can be enhanced near a Feshbach resonance. In both cases, the anomaly leads to an increase in the frequency of the radial monopole mode, or breathing oscillation, above the value set by the scaling symmetry of the classical theory [13]. Previous experiments have studied the radial breathing mode in 2D Bose [14,15] and Fermi gases [16], although the anomalous upshift has not yet been observed. More broadly, the breathing mode is a sensitive probe of the adiabatic compressibility and hence the thermodynamic equation of state [17,18] of the gas being studied.In this Letter, we present measurements of the radial breathing mode frequency ω B for highly oblate Fermi gases as a function of the interaction strength and dimensionality. The dimensionality of a harmonically trapped gas can be tuned by varying the chemical potential µ relative to the confinement energies ω i , (i = x, y, z) in each dimension. When ω z µ, k B T ω x,y , where k B is Boltzmann's constant and T is the temperature, motion in the transverse (z) dimension can be frozen...

We study dipolar excitons confined in a flat trap of a GaAs bilayer heterostructure. By quantifying density and density fluctuations at sub-Kelvin temperatures, we unveil the excitons' universal equation of state. This allows us to infer thermodynamically the critical phase-space density for the transition expected between normal and superfluid regimes. The region of this crossover is directly confirmed by a net change in the excitons' temporal coherence, from an exponential to an algebraic decay with an exponent compatible with the Berezinskii-Kosterlitz-Thouless theory.