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

Newman Laboratory, Cornell University, Ithaca, NY 

Pye-Ton How and Andr´e LeClair 

## 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 ℏ / 4 πk$_{B}$ for all cases except attractive bosons. For attractive 2- component fermions, η/s ≥ 6 . 07 ℏ / 4 πk$_{B}$ , whereas for attractive bosons η/s ≥ 0 . 4 ℏ / 4 πk$_{B}$ . 

