content_image/369427.jpg content_image/369428.jpg content_image/369429.jpg
in section VI. If one disregards unphysical, potentially metastable regions, one finds η/s ≥ 6 . 07 ℏ / 4 πk$_{B}$ . The other cases of repulsive bosons or fermions also satisfy the bound. However the attractive boson is below it: η/s ≥ 0 . 4 ℏ / 4 πk$_{B}$ . 

## II. RENORMALIZATION GROUP AND THE UNITARY LIMIT 

The models considered in this paper are the simplest models of non-relativistic bosons and fermions with quartic interactions. The bosonic model is defined by the action for a complex scalar field φ . 

<img src='content_image/369427.jpg'>

(Throughout this paper, d refers to the number of spatial dimensions). For fermions, due to the fermionic statistics, one needs at least a 2-component field ψ$_{↑}$$_{,}$$_{↓}$ : 

In both cases, positive g corresponds to repulsive interactions. 

<img src='content_image/369428.jpg'>

The bosonic theory only has a U (1) symmetry. The fermionic theory on the other hand has the much larger SO(5) symmetry. This is evident from the work[26] which considered a relativistic version, since the same arguments apply to a non-relativistic kinetic term. This is also clear from the work[15] which considered an N -component version with Sp(2N) symmetry, and noting that Sp(4) = SO(5). 

The free versions of the above theories have a scale invariance with dynamical exponent z = 2, i.e. are invariant under 

<img src='content_image/369429.jpg'>

At a renormalization group fixed point, i.e. quantum critical point, the model is expected to have the same scale invariance. It is natural to define scaling dimensions 

