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 h L ≪ c. As a light probe, \( {\mathcal{O}}_{\mathrm{obs}}(x) \) is constrained by ETH and satisfies \( {\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_{h_H}\approx {\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_{\mathrm{micro}} \) for a high energy energy eigenstate |h H 〉. In the CFTs of interests, \( {{\left\langle {\mathcal{O}}_{\mathrm{obs}}(x)\right\rangle}_h}_{{}_H} \) 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 \( \mathcal{O}\left({h}_L/c\right) \) 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 → ∞. 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.
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Faulkner, T., Wang, H. Probing beyond ETH at large c. J. High Energ. Phys. 2018, 123 (2018). https://doi.org/10.1007/JHEP06(2018)123
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DOI: https://doi.org/10.1007/JHEP06(2018)123