Probing beyond ETH at large c

Abstract

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 {mathcal{O}}_{mathrm{obs}}(x)={mathcal{O}}_L(x){mathcal{O}}_L(0) with hL ≪ c. As a light probe, {mathcal{O}}_{mathrm{obs}}(x) is constrained by ETH and satisfies {leftlangle {mathcal{O}}_{mathrm{obs}}(x)rightrangle}_{h_H}approx {leftlangle {mathcal{O}}_{mathrm{obs}}(x)rightrangle}_{mathrm{micro}} for a high energy energy eigenstate |hH〉. In the CFTs of interests, {{leftlangle {mathcal{O}}_{mathrm{obs}}(x)rightrangle}_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 {mathcal{O}}_{mathrm{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 mathcal{O}left({h}_L/cright) 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 Sn 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.