Probing beyond ETH at large c

Abstract

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 $$ {\mathcal{O}}_{\mathrm{obs}}(x)={\mathcal{O}}_L(x){\mathcal{O}}_L(0) $$ O o b s x = O L x O L 0 with h L ≪ c. As a light probe, $$ {\mathcal{O}}_{\mathrm{obs}}(x) $$ O o b s x is constrained by ETH and satisfies $$ {\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_{h_H}\approx {\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_{\mathrm{micro}} $$ O o b s x h H ≈ O o b s x micro for a high energy energy eigenstate |h H 〉. In the CFTs of interests, $$ {{\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_h}_{{}_H} $$ O o b s x 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 $$ {\mathcal{O}}_{\mathrm{obs}}(x) $$ O o b s 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 $$ \mathcal{O}\left({h}_L/c\right) $$ 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.