Continuum limit of fishnet graphs and AdS sigma model

We consider the continuum limit of 4d planar fishnet diagrams using integrable spin chain methods borrowed from the $\mathcal{N}=4$ Super-Yang-Mills theory. These techniques give us control on the scaling dimensions of single-trace operators for all values of the coupling constant in the fishnet theory. We use them to study the thermodynamical limit of the BMN operator corresponding to the spin chain ferromagnetic vacuum. We find that its scaling dimension exhibits a critical behaviour when the coupling constant approaches Zamolodchikov's critical coupling. Analysis close to that point suggests that the continuum limit of the fishnet graphs is controlled by the two-dimensional $AdS_{5}$ non-linear sigma model. More generally, we present evidence that the fishnet diagrams define an integrable lattice regularization of the $AdS_{5}$ model. A system of massless TBA equations is derived for the tachyon energy by dualizing the TBA equations of the weakly coupled planar $\mathcal{N}=4$ SYM theory.