Subsystem entropy in 2d CFT and KdV ETH

We study subsystem entropy in 2d CFTs, for subsystems constituting a finite fraction of the full system. We focus on the extensive contribution, which scales linearly with the subsystem size in the thermodynamic limit. We employ the so-called diagonal approximation to evaluate subsystem entropy for the chaotic CFTs in thermal state (canonical ensemble), microcanonical ensemble, and in a primary state, matching previously known results. We then proceed to find analytic expressions for the subsystem entropy at leading order in $c$, when the global CFT state is the KdV generalized Gibbs ensemble or the KdV microcanonical ensemble. Previous studies of primary eigenstates have shown that, akin to fixed-area states in AdS/CFT, corresponding subsystem entanglement spectrum is flat. This behavior is seemingly in sharp contradiction with the one for the thermal (microcanonical) state, and thus in apparent contradiction with the subsystem Eigenstate Thermalization Hypothesis (ETH). In this work, we resolve this issue by comparing the primary state with the KdV (micro)canonical ensemble. We show that the results are consistent with the KdV-generalized version of the subsystem ETH, in which local properties of quantum eigenstates are governed by their values of conserved KdV charges. Our work solidifies evidence for the KdV-generalized ETH in 2d CFTs and emphasizes Renyi entropy as a sensitive probe of the reduced-density matrix.