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

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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 thesesystems present interesting examples of interacting scale-invariant models with dynamical exponentz = 2. We study this problem in two and three spatial dimensions using theS-matrix-based approach to the thermodynamics we recently developed. It is well suited to the unitary limitwhere 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 thetwo-dimensional case, the integral equation for the pseudo-energy becomes transcendentallyalgebraic and we can easily compute the various universal scaling functions as a function ofμ/T, such as the energy per particle. The ratio of the shear viscosity to the entropy densityη/s is above the conjectured lower bound of for all cases except attractive bosons. For attractive two-component fermions, , whereas for attractive bosons .