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.

Figure 1

Beyond-mean-field contribution to the ground state energy in a harmonic trap as a function of particle number for (a) repulsive and (b) attractive interactions. Inset: comparison with exact diagonalization results for $|g|=0.3$ (red crosses, data from Ref. [62]).

Figure 2

Conformal tower of states created from a primary state $|P⟩$, ordered by energy and angular momentum. Nonprimary states are c.m. excitations, which are created by the operators $Q_{\pm }^{†}$ (blue arrows) that increase the energy by $\hslash \omega$ and the angular momentum by $\pm 1$, and internal breathing mode excitations, which are created by $R^{†}$ (orange arrows) which increases the internal energy by $2\hslash \omega$ without changing the angular momentum.

Figure 3

Excitations energies for $N=2$, 6, 9, and 12 particles in a harmonic trap ordered by angular momentum for an attractive interaction $g=-1$. Blue points represent primary states and red points are nonprimary states (cf. Fig. 2). Overlapping points are moved horizontally for clarity. Insets: Magnified spectrum near the second excitation level.

Figure 4

Distribution of the internal hyperradius $\tilde{R}$ for the 77 lowest eigenstates of $N=6$ particles. Gray points are results of the Monte Carlo sampling of the wave function, and continuous lines show the analytical result (14). The inset shows the same energy spectrum as Fig. 3 with a color coding that matches the distribution.