Exact few-body results for strongly correlated quantum gases in two dimensions

The study of strongly correlated quantum gases in two dimensions has important ramifications for understanding many intriguing phenomena in solid materials, such as high-$T_c$ superconductivity and the fractional quantum-Hall effect. However, theoretical methods are plagued by the existence of significant quantum fluctuations. Here, we present two- and three-body exact solutions for both fermions and bosons trapped in a two-dimensional harmonic potential with an arbitrary $s$-wave scattering length. These few-particle solutions link in a natural way to the high-temperature properties of many-particle systems via a quantum virial expansion. As a concrete example, using the energy spectrum of few fermions, we calculate the second and third virial coefficients of a strongly interacting Fermi gas in two dimensions, and consequently investigate its high-temperature thermodynamics. Our thermodynamic results may be useful for ongoing experiments on two-dimensional Fermi gases. These exact results also provide an unbiased benchmark for quantum Monte Carlo simulations of two-dimensional Fermi gases at high temperatures.

Figure 1

Relative energy spectrum of a two-particle system with $m_{rel}=0$ as a function of the dimensionless interaction parameter $d/a_{sc}$. The system goes to the strongly interacting limit when $d/a_{sc}$ increases to an infinitely large value.

Figure 2

Relative energy spectrum of three trapped interacting fermions in 2D, as a function of the dimensionless interaction parameter $\text{ln}\left(d/a_{sc}\right)$. We show the spectrum in different subspaces of relative angular momentum $m$. The ground-state energy level in the subspace $m=1$ has been highlighted by a thick line.

Figure 3

Relative energy spectrum of three trapped interacting bosons in 2D at difference subspace, as a function of the dimensionless interaction parameter $\text{ln}\left(d/a_{sc}\right)$. The ground-state energy level in the subspace $m=0$ has been highlighted by a thick line. The numerical accuracy with $m=0$ is greatly suppressed due to the existence of the self-bound dropletlike states. We thus plot the spectrum with $n_{\text{max}}=64$ (solid lines), 128 (dashed lines), and 256 (dotted-dashed lines), to show the slow convergence with respect to the number of expansion basis elements $n_{\text{max}}$.

Figure 4

(Color online) Second and third virial coefficients as a function of the interaction strength $E_B/E_F$ at different temperatures, $T/T_F=0.5$ (solid lines), 1.0 (dashed lines), and 2.0 (dotted-dashed lines).

Figure 5

(Color online) Temperature dependence of the second and third virial coefficients at two interaction strengths, $E_B=0.2E_F$ (solid lines) and $E_B=0.1E_F$ (dashed lines).

Figure 6

(Color online) Temperature dependence of the chemical potential, entropy, and energy of a strongly correlated Fermi gas in a 2D harmonic trap. The predictions of virial expansion up to the third and second order are shown, respectively, by the solid lines and dashed lines. For comparison, we also show the ideal, noninteracting results using dotted-dashed lines.