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$_{B}$ . 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 . (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. 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 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 5 S = ∫ d $^{d}$x dt ( iφ $^{†}$∂$_{t}$ φ − | ⃗ ∇ φ | 2 2 m − g 4 ( φ $^{†}$φ ) 2 ) (2) S = ∫ d $^{d}$x dt ( ∑ α = ↑ , ↓ iψ † $_{α}$∂$_{t}$ ψ$_{α}$ − | ⃗ ∇ ψ$_{α}$ | 2 2 m − g 2 ψ † $_{↑}$ψ$_{↑}$ ψ † $_{↓}$ψ$_{↓}$ ) (3) t → Λ − $^{2}$t, x → Λ − $^{1}$x II. RENORMALIZATION GROUP AND THE UNITARY LIMIT