Lee-Yang paradigm of phase transition in eigenstate thermalized systems

As phase transitions in isolated quantum systems remain elusive, here we show how a thermodynamic-like phase transition, falling into the Lee-Yang paradigm, can arise in systems displaying eigenstate thermalization. Specifically, we show that in holographic conformal field theories, the eigenstate expectation of the auto-correlation function can be mapped to the partition function ${\cal Z}_{{gauge}}(z)$ of a virtual interacting instanton gas, with the conformal mapping of the imaginary time: $z=1-e^{-\tau}$ and the central charge $c$ mimicking the instanton fugacity and volume, respectively. We find that akin to the Lee-Yang paradigm, for $c\to\infty$ a pair of complex conjugate zeros of ${\cal Z}_{{gauge}}(z)$ move to the real axis located at the famous forbidden singularity. Passing through the singularity the system transits from the low- to high-fugacity phase, accompanied by dramatic changes in scaling behaviors of the free energy and dominant microscopic configurations. Our findings indicate that physics of phase transitions from eigenstate thermalization is very rich.