TY - JOUR

T1 - Probing beyond ETH at large c

AU - Faulkner, Thomas

AU - Wang, Huajia

N1 - Funding Information:
TF would like to thank Tom Hartman and Tarun Grover for early collaboration on the topic of Renyi entropies of highly excited states. We thank Alex Belin, Tolya Dymarsky, Liam Fitzpatrick, Jared Kaplan, Daliang Li, and Junpu Wang for discussions and comments on the draft. This research was supported by the DARPA YFA program, contract D15AP00108.
Publisher Copyright:
© The Authors.

PY - 2018/6

Y1 - 2018/6

N2 - 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 Oobs(x) = OL(x)OL(0) with hL ≪ c. As a light probe, Oobs(x) is constrained by ETH and satisfies 〈Oobs(x)〉hH ≈ 〈Oobs(x)〉micro for a high energy energy eigenstate |hH 〉. In the CFTs of interests, 〈Oobs(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 Oobs(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(hL/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 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.

AB - 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 Oobs(x) = OL(x)OL(0) with hL ≪ c. As a light probe, Oobs(x) is constrained by ETH and satisfies 〈Oobs(x)〉hH ≈ 〈Oobs(x)〉micro for a high energy energy eigenstate |hH 〉. In the CFTs of interests, 〈Oobs(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 Oobs(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(hL/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 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.

KW - 1/N expansion

KW - Black holes

KW - Conformal field theory

KW - Nonperturbative effects

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U2 - 10.1007/JHEP06(2018)123

DO - 10.1007/JHEP06(2018)123

M3 - Article

AN - SCOPUS:85051067508

SN - 1126-6708

VL - 2018

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

IS - 6

M1 - 123

ER -