Determination of scale-invariant equations of state without fitting parameters: application to the two-dimensional Bose gas across the Berezinskii-Kosterlitz-Thouless transition

Tan's Two-Body Contact in a Planar Bose Gas: Experiment versus Theory

We determine the two-body contact in a planar Bose gas confined by a transverse harmonic potential, using the nonperturbative functional renormalization group. We use the three-dimensional thermodynamic definition of the contact where the latter is related to the derivation of the pressure of the quasi-two-dimensional system with respect to the three-dimensional scattering length of the bosons. Without any free parameter, we find a remarkable agreement with the experimental data of Zou et al. [Tan's two-body contact across the superfluid transition of a planar Bose gas, Nat. Commun. 12, 760 (2021).NCAOBW2041-172310.1038/s41467-020-20647-6] from low to high temperatures, including the vicinity of the Berezinskii-Kosterlitz-Thouless transition. We also show that the short-distance behavior of the pair distribution function and the high-momentum behavior of the momentum distribution are determined by two contacts: the three-dimensional contact for length scales smaller than the characteristic length ℓ_{z}=sqrt[ℏ/mω_{z}] of the harmonic potential and, for length scales larger than ℓ_{z}, an effective two-dimensional contact, related to the three-dimensional one by a geometric factor depending on ℓ_{z}.

Nonrelativistic Conformal Invariance in Mesoscopic Two-Dimensional Fermi Gases

Two-dimensional Fermi gases with universal short-range interactions are known to exhibit a quantum anomaly, where a classical scale and conformal invariance is broken by quantum effects at strong coupling. We argue that in a quasi two-dimensional geometry, a conformal window remains at weak interactions. Using degenerate perturbation theory, we verify the conformal symmetry by computing the energy spectrum of mesoscopic particle ensembles in a harmonic trap, which separates into conformal towers formed by so-called primary states and their center-of-mass and breathing-mode excitations, the latter having excitation energies at precisely twice the harmonic oscillator energy. In addition, using Metropolis importance sampling, we compute the hyperradial distribution function of the many-body wave functions, which are predicted by the conformal symmetry in closed analytical form. The weakly interacting Fermi gas constitutes a system where the nonrelativistic conformal symmetry can be revealed using elementary methods, and our results are testable in current experiments on mesoscopic Fermi gases.

Observation of Scale Invariance in Two-Dimensional Matter-Wave Townes Solitons

We report near-deterministic generation of two-dimensional (2D) matter-wave Townes solitons and a precision test on scale invariance in attractive 2D Bose gases. We induce a shape-controlled modulational instability in an elongated 2D matter wave to create an array of isolated solitary waves of various sizes and peak densities. We confirm scale invariance by observing the collapse of solitary-wave density profiles onto a single curve in a dimensionless coordinate rescaled according to their peak densities and observe that the scale-invariant profiles measured at different coupling constants g can further collapse onto the universal profile of Townes solitons. The reported scaling behavior is tested with a nearly 60-fold difference in soliton interaction energies and allows us to discuss the impact of a non-negligible magnetic dipole-dipole interaction (MDDI) on 2D scale invariance. We confirm that the effect of MDDI in our alkali cesium quasi-2D samples effectively conforms to the same scaling law governed by a contact interaction to well within our experiment uncertainty.

Measurement of the Canonical Equation of State of a Weakly Interacting 3D Bose Gas

Using a multiple-image reconstruction method applied to a harmonically trapped Bose gas, we determine the equation of state of uniform matter across the critical transition point, within the local density approximation. Our experimental results provide the canonical description of pressure as a function of the specific volume, emphasizing the dramatic deviations from the ideal Bose gas behavior caused by interactions. They also provide clear evidence for the nonmonotonic behavior with temperature of the chemical potential, which is a consequence of superfluidity and Bose-Einstein condensation. The measured thermodynamic quantities are compared to mean-field predictions available for the interacting Bose gas. The limits of applicability of the local density approximation near the critical point are also discussed, focusing on the behavior of the isothermal compressibility.

Quantum scale anomaly and spatial coherence in a 2D Fermi superfluid

Quantum anomalies are violations of classical scaling symmetries caused by divergences that appear in the quantization of certain classical theories. Although they play a prominent role in the quantum field theoretical description of many-body systems, their influence on experimental observables is difficult to discern. In this study, we discovered a distinctive manifestation of a quantum anomaly in the momentum-space dynamics of a two-dimensional (2D) Fermi superfluid of ultracold atoms. The measured pair momentum distributions of the superfluid during a breathing mode cycle exhibit a scaling violation in the strongly interacting regime. We found that the power-law exponents that characterize long-range phase correlations in the system are modified by the quantum anomaly, emphasizing the influence of this effect on the critical properties of 2D superfluids.

Quantum Anomaly and 2D-3D Crossover in Strongly Interacting Fermi Gases

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 oscillation or breathing mode in highly oblate gases with tunable interactions. The breathing mode frequency 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 providing evidence for the quantum anomaly and signifying the breakdown of an elementary δ-potential model of atomic interactions. By varying the atom number we can tune the chemical potential and see the breathing mode frequency evolve smoothly between the 2D to 3D thermodynamic limits.

Equation of State of Ultracold Fermions in the 2D BEC-BCS Crossover Region

We report the experimental measurement of the equation of state of a two-dimensional Fermi gas with attractive s-wave interactions throughout the crossover from a weakly coupled Fermi gas to a Bose gas of tightly bound dimers as the interaction strength is varied. We demonstrate that interactions lead to a renormalization of the density of the Fermi gas by several orders of magnitude. We compare our data near the ground state and at finite temperature with predictions for both fermions and bosons from quantum Monte Carlo simulations and Luttinger-Ward theory. Our results serve as input for investigations of close-to-equilibrium dynamics and transport in the two-dimensional system.

Thermodynamics of an Attractive 2D Fermi Gas

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.