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## B. 2d case 

The d = 2 case is somewhat more subtle due to the marginality of the coupling g . The fixed point ̂ g$_{∗}$ = 0 and the RG flows are depicted in Figure 1. The fixed point at ̂ g$_{∗}$ = 0 is just the free field theory, thus there is no analog of the BEC/BCS cross-over. One can nevertheless formally define the unitary limit as S = − 1, as in 1 d and 3 d . In this subsection we explore this possibility and interpret it using the renormalization group. As we’ll see, this limit occurs at g = ±∞ , and the scattering length diverges. 

First begin with the beta-function in 2d, eq. (110): 

<img src='content_image/333773.jpg'>

Let g = g$_{0}$ at some arbitrary scale Λ$_{0}$. Integrating the beta function one finds: 

Note that g diverges at the scale: 

<img src='content_image/333774.jpg'>

<img src='content_image/333775.jpg'>

This is the familiar Landau pole. Whereas in for example the relativistic φ 4 theory in d = 3 where the Landau pole is unphysical due to higher order corrections, here the beta function (11) is exact[15], thus this divergence is physical. There are two cases to consider: 

Attractive case: If g$_{0}$ is negative, Λ$_{∗}$ < Λ$_{0}$, and thus g = −∞ occurs in the infra-red. 

Repulsive case: If g$_{0}$ is positive, Λ$_{∗}$ > Λ$_{0}$, and g = + ∞ occurs in the ultra-violet. 

Both cases are consistent with the flows depicted in Figure 1. 

