Elliptic Integrable Models and Their Spectra from Superconformal Indices

In this contribution we summarize our recent progress in understanding the relation between $\mathcal{N}=1$ superconformal indices and relativistic elliptic integrable models. We start briefly reviewing the emergence of such models in computations of the index in presence of surface defect. Next we give an example of such relation considering $4d$ theories obtained in the compactificaiton of $6d$ $(D_{N+3},D_{N+3})$ minimal conformal matter theories. In this case we obtain van Diejen model as well as its higher rank generalizations on $A_N$ and $C_2$ root systems. Finally we review a novel algorithm for computation of the ground states of elliptic integrable systems from superconformal indices that was recently proposed by us.