S-matrix approach to quantum gases in the unitary limit: I. The two-dimensional case

Abstract

In three spatial dimensions, in the unitary limit of a non-relativistic quantum Bose or Fermi gas, the scattering length diverges. This occurs at a renormalization group fixed point, thus these systems present interesting examples of interacting scale-invariant models with dynamical exponent z = 2. We study this problem in two and three spatial dimensions using the S-matrix based approach to the thermodynamics we recently developed. It is well suited to the unitary limit where the S-matrix S = −1, since it allows an expansion in the inverse coupling. We define a meaningful scale-invariant, unitary limit in two spatial dimensions, where again the scattering length diverges. In the two-dimensional case, the integral equation for the pseudo-energy becomes transcendentally algebraic, and we can easily compute the various universal scaling functions as a function of µ/T , such as the energy per particle. The ratio of the shear viscosity to the entropy density η/s is above the conjectured lower bound of /4πk B for all cases except attractive bosons. For attractive 2-component fermions, η/s ≥ 6.07 /4πk B , whereas for attractive bosons η/s ≥ 0.4 /4πk B .

On the viscosity-to-entropy density ratio for unitary Bose and Fermi gases

“…Consider first the bosonic model. (There were two errors by factors of 2 in the original discussion presented in [29] which are here corrected. However they turn out to cancel and the main result below eq.…”

“…There have been some proposals to use the AdS/CFT correspondence to learn about non-relativistic systems [25][26][27][28]. One difficulty is that the conformal symmetry of relativistic systems is larger than the Schrödinger symme- For the remainder of this paper we will describe a novel, but more conventional approach to the problem [29]. The main approximation made is that we only consider 2-body interactions, and consistently resum their contributions to the free energy via an integral equation.…”

Quantum Bose and Fermi gases with large negative scattering length in the two-bodyS-matrix approximation

“…(15), and the virial coefficients are pure numbers by the scale invariance. The above integrals for b 2,3 are easily performed analytically by making the change of variables k 1 = k − k , k 2 = k + ak , where a is chosen to cancel the cross term in the exponential; the result then factorizes into two Gaussian integrals.…”

“…As expected, the virial coefficients are different than those above, and this is traced to the discontinuity of the kernel G as the scattering length goes from minus to plus infinity; it in fact changes sign, Eq. (15), which allows us to perform the calculation in a very simple way. In other words, apart from this change of sign of the kernel, the calculation is identical to that of the last section.…”

Virial coefficients for trapped Bose and Fermi gases beyond the unitary limit: AnS-matrix approach

“…The method is a "foam diagram" approximation which is valid for high temperatures and low densities and considers only contributions from two-body processes to the free energy. It is explained in [16] and has been already used to study the thermodynamical and critical properties of quantum gases in two and three dimensions in the unitary limit [17,18] and beyond the unitary limit in three dimensions [19]. In [20] the method was used to calculate the ratio of the viscosity to entropy density and the results were in well agreement with experimental data [21].…”