### 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 O_{obs}(x) = O_{L}(x)O_{L}(0) with h_{L} ≪ c. As a light probe, O_{obs}(x) is constrained by ETH and satisfies 〈O_{obs}(x)〉_{hH} ≈ 〈O_{obs}(x)〉_{micro} for a high energy energy eigenstate |h_{H} 〉. In the CFTs of interests, 〈O_{obs}(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 O_{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 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.

Original language | English (US) |
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Article number | 123 |

Journal | Journal of High Energy Physics |

Volume | 2018 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

*Journal of High Energy Physics*,

*2018*(6), [123]. https://doi.org/10.1007/JHEP06(2018)123