Thermodynamic Bethe Ansatz for Fishnet CFT

We present the TBA equations and the Y-system for the exact spectrum of general multi-magnon local operators in the $D$-dimensional anisotropic version of the bi-scalar fishnet CFT. The mixing matrix of such operators is given in terms of fishnet planar graphs of multi-wheel and multi-spiral type. These graphs probe the two main building blocks of the TBA approach that are the magnon dispersion relation and the magnon scattering matrix and which we both obtain by diagonalising suitable graph-building operators. We also obtain the dual version of the TBA equations, which relates, in the continuum limit, $D$-dimensional graphs to two dimensional sigma models in $AdS_{D+1}$. It allows us to verify a general formula obtained by A.~Zamolodchikov for the critical coupling.