Phase transition from eigenstate thermalization: forbidden singularity and instanton proliferation via AGT correspondence

In theoretical physics, finding connections between problems that appear in distinct contexts is an important way to leapfrog progresses, often by illuminating deep aspects that may otherwise seem obscure. In this paper, we consider in 2d CFTs the phenomenon of forbidden singularities in auto-correlation functions -- a key signature of eigenstate thermalization. We show that they correspond to phase transitions in the context of eigenstates. The connection is made explicit by utilizing the AGT correspondence, which relates eigenstate auto-correlations to the Nekrasov partition functions describing an instanton gas of the $\mathcal{N}=2$ SUSY gauge theories. We show that by taking the counter-part of the heavy-light limit, two phases emerge for the instanton gas. They are dominated by configurations represented by string-like Young tableaux with distinct structures and thermodynamic properties, which bare resemblance to the confined and the deconfined phases. A phase transition occurs as instantons proliferate from one side, in a manner that mimics the Lee-Yang theory. We work out the critical fugacity and find it corresponding exactly to the forbidden singularity.