Probing beyond ETH at large c

Thomas Faulkner, Huajia Wang

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number123
JournalJournal of High Energy Physics
Issue number6
StatePublished - Jun 2018


  • 1/N expansion
  • Black holes
  • Conformal field theory
  • Nonperturbative effects

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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