arXiv:1912.13444 · Journal of High Energy Physics

Zero modes of local operators in 2d CFT on a cylinder

Anatoly Y. Dymarsky, Kirill Pavlenko, Dmitry Solovyev

Published Jul 1, 2020• 10 citations

Abstract

A bstract Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs call for calculation of the expectation values of local operators in highly excited energy eigenstates. This can be done efficiently by representing zero modes of these operators in terms of the Virasoro algebra generators. In this paper we present a pedagogical introduction explaining how this calculation can be performed analytically or using computer algebra. We illustrate the computation of zero modes by a number of examples and list explicit expressions for all local operators from the vacuum family with the dimension of less or equal than eight. Finally, we derive an explicit expression for the quantum KdV generator Q 7 in terms of the Virasoro algebra generators. The obtained results can be used for quantitative studies of ETH at finite value of central charge.

DOI: 10.1007/jhep07(2020)172

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arXiv:1912.13444 · Journal of High Energy Physics

Zero modes of local operators in 2d CFT on a cylinder

Anatoly Y. Dymarsky, Kirill Pavlenko, Dmitry Solovyev

Published Jul 1, 2020• 10 citations

arXiv HTML PDF

Abstract

DOI: 10.1007/jhep07(2020)172

Related papers

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{\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.

c

. We demonstrate that the first two quantum Korteweg--de Vries (qKdV) charges

Q_1

and

Q_3

lift all degeneracies except along at most countably many curves in the

(c,Δ)

plane, where

Δ

is the highest weight of the conformal family, so that the full-qKdV simultaneous eigenstates are unambiguously determined by

Q_1

and

Q_3

. Based on this, a near-complete toolkit for testing in the vacuum conformal family or falsifying the general form of GETH, i.e. equating qKdV eigenstates and thermal states of the generalized Gibbs ensemble in the sense of expectations of local observables, is provided in this paper, through the infinite tower of qKdV charges. A stronger form of the GETH, namely that all expectations of local observables of a definite level should be expressible of qKdV charges but not contain microscopic eigenstate information, is directly falsified at subleading order in the large-

c

expansion. At levels 6 and 8 in the vacuum family, expectation values of quasi-primaries in simultaneous eigenstates of the qKdV charges are not fixed by first finitely many qKdV charges, and microscopic information of the eigenstate is required.