Thermodynamic properties of matter are conveniently expressed as functional relations between variables known as equations of state. Here we experimentally determine the compressibility, density, and pressure equations of state for an attractive 2D Fermi gas in the normal phase as a function of temperature and interaction strength. In 2D, interacting gases exhibit qualitatively different features to those found in 3D. This is evident in the normalized density equation of state, which peaks at intermediate densities corresponding to the crossover from classical to quantum behavior.

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...

Ultracold Fermi gases subject to tight transverse confinement offer a highly controllable setting to study the two-dimensional (2D) BCS to Berezinskii-Kosterlitz-Thouless superfluid crossover. Achieving the 2D regime requires confining particles to their transverse ground state which presents challenges in interacting systems. Here, we establish the conditions for an interacting Fermi gas to behave kinematically 2D. Transverse excitations are detected by measuring the transverse expansion rate which displays a sudden increase when the atom number exceeds a critical value N2D signifying a density driven departure from 2D kinematics. For weak interactions N2D is set by the aspect ratio of the trap. Close to a Feshbach resonance, however, the stronger interactions reduce N2D and excitations appear at lower density.PACS numbers: 03.75. Ss, 03.75.Hh, 05.30.Fk, 67.85.Lm Fermions confined to two-dimensional (2D) planes represent an important paradigm in many-body physics in settings ranging from thin films of superfluid helium-3 [1, 2] to the superconducting planes in high-T c cuprates [3]. Ultracold atomic gases confined in oblate potentials allow access to the 2D regime [4][5][6][7][8][9][10][11][12][13][14][15] where interactions between particles can be controlled using a Feshbach resonance [16]. In 2D Fermi gases, one can realize the BCS to Berezinskii-Kosterlitz-Thouless (BKT) superfluid crossover [17][18][19][20][21][22][23][24][25] by tuning the attractive interaction between particles in different spin states. Of particular interest is the enhanced pairing due to the transverse confinement [26][27][28][29][30] and the consequences this has for the phase diagram of the crossover [15,[31][32][33].Theoretical studies of the BCS-BKT crossover generally assume only two spatial dimensions, however, all atomic gases exist in 3D environments. Lower dimensional behaviour can be realized by freezing out dynamics along one or more directions. For atoms in a harmonic potential, with frequencies ω x , ω y and ω z , the 2D regime is achieved when the transverse (z) confinement is strong enough that occupation of transverse excited states is energetically forbidden. When a gas is frozen in the transverse ground state, dynamics in the x-y plane become decoupled from z and the gas is kinematically 2D. In an ideal gas this requires the thermal energy and chemical potential be much smaller than the transverse confinement energy k B T, µ ω z , where k B is Boltzmann's constant, T is the temperature and µ the chemical potential. When interactions are present, however, these can provide another means for generating transverse excitations which go beyond purely 2D models.In this Rapid Communication, we examine the criteria for an interacting Fermi gas to behave kinematically 2D.By measuring the transverse cloud width after time of flight we observe a rapid growth in the expansion rate when transverse excitations are present. Both the trap geometry and interaction strength are seen to limit the parameter space where interacting sys...