Virial expansion for the Tan contact and Beth-Uhlenbeck formula from two-dimensional SO(2,1) anomalies

Abstract

The relationship between 2D SO(2, 1) conformal anomalies in nonrelativistic systems and the virial expansion is explored using recently developed path-integral methods. In the process, the Beth-Uhlenbeck formula for the shift of the second virial coefficient δb2 is obtained, as well as a virial expansion for the Tan contact. A possible extension of these techniques for higher orders in the virial expansion is discussed.

Third- and fourth-order virial coefficients of harmonically trapped fermions in a semiclassical approximation

“…In the following we explore their behavior as a function of d and βω, focusing in particular on the unitary limit of the 3D Fermi gas. While numerical results exist for these quantities in some cases, in particular in 2D [9][10][11][12] (see also [13][14][15]) and in 3D at unitarity [16][17][18]20], most of those correspond to homogeneous systems and do not feature explicit, analytic dependence on the dimension nor on βω, as shown here. Our results are therefore useful in that they are able to provide analytic insight into the behavior of virial coefficients across dimensions, and as a function of the temperature (or trapping frequency) as well as the coupling strength.…”

Leading- and next-to-leading-order semiclassical approximation to the first seven virial coefficients of spin-1/2 fermions across spatial dimensions

“…Refs. [14][15][16][17][18]). That choice allowed us to express our results in powers of ∆b 2 and to perform cross-dimensional comparisons by varying d at fixed ∆b 2 .…”

Virial coefficients of one-dimensional and two-dimensional Fermi gases by stochastic methods and a semiclassical lattice approximation

“…The generalization of our approaches to higher dimensions is straightforward. In fact, the generic system studied here (a nonrelativistic gas with zero-range interactions) has been under intense investigation both theoretically and experimentally in the last decade in 1D, 2D, and 3D, and analytic results exist for b 2 in all dimensions based on the Beth-Uhlenbeck formula mentioned above [8,13,[31][32][33].…”

“…Thus, the proposed approach effectively consists in the Fourier projection of the virial coefficients from the density equation of state, which is reminiscent of other approaches such as those of Refs. [15,16,18,33].…”