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(x) = O_L(x)O_L(0) with h_L ≪ c. As a light probe, O_obs(x) is constrained by ETH and satisfies 〈O_obs(x)〉_hH ≈ 〈O_obs(x)〉_micro for a high energy energy eigenstate |h_H 〉. In the CFTs of interests, 〈O_obs(x)〉_hH 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(x) 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(h_L/c) 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 → ∞. 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.