Probing beyond ETH at large $c$

We study probe corrections to the Eigenstate Thermalization Hypothesis (ETH) in the context of 2D CFTs with large central charge and a sparse spectrum of low dimension operators. In particular, we focus on observables in the form of non-local composite operators $O_{obs}\left(x\right)=O_L\left(x\right)O_L\left(0\right)$ with $h_L\ll c$. As a light probe, $O_{obs}\left(x\right)$ is constrained by ETH and satisfies $⟨O_{obs}\left(x\right){⟩}_{h_H}\approx ⟨O_{obs}\left(x\right){⟩}_{\text{micro}}$ for a high energy energy eigenstate $|h_H⟩$. In the CFTs of interests, $⟨O_{obs}\left(x\right){⟩}_{h_H}$ is related to a Heavy-Heavy-Light-Light (HL) correlator, and can be approximated by the vacuum Virasoro block, which we focus on computing. A sharp consequence of ETH for $O_{obs}\left(x\right)$ is the so called "forbidden singularities", arising from the emergent thermal periodicity in imaginary time. Using the monodromy method, we show that finite probe corrections of the form $O\left(h_L/c\right)$ drastically alter both sides of the ETH equality, replacing each thermal singularity with a pair of branch-cuts. Via the branch-cuts, the vacuum blocks are connected to infinitely many additional "saddles". We discuss and verify how such violent modification in analytic structure leads to a natural guess for the blocks at finite $c$: a series of zeros that condense into branch cuts as $c\to \infty$. We also discuss some interesting evidences connecting these to the Stoke's phenomena, which are non-perturbative $e^{-c}$ effects. As a related aspect of these probe modifications, we also compute the Renyi-entropy $S_n$ in high energy eigenstates on a circle. For subsystems much larger than the thermal length, we obtain a WKB solution to the monodromy problem, and deduce from this the entanglement spectrum.