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arXiv:2103.13943 · Journal of High Energy Physics

Generalized Gibbs Ensemble of 2D CFTs with U(1) charge from the AGT correspondence

Fábio Novaes

Published May 1, 2021• 4 citations
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Abstract

A bstract The Generalized Gibbs Ensemble (GGE) is relevant to understand the thermalization of quantum systems with an infinite set of conserved charges. In this work, we analyze the GGE partition function of 2D Conformal Field Theories (CFTs) with a U(1) charge and quantum Benjamin-Ono 2 (qBO 2 ) hierarchy charges. We use the Alday-Gaiotto-Tachikawa (AGT) correspondence to express the thermal trace in terms of the Alba-Fateev-Litvinov-Tarnopolskiy (AFLT) basis of descendants, which diagonalizes all charges. We analyze the GGE partition function in the thermodynamic semiclassical limit, including the first order quantum correction. We find that the equality between GGE averages and primary eigenvalues of the qBO 2 charges is attainable in the strict large c limit and potentially violated at the subleading 1 /c order. We also obtain the finite c partition function when only the first non-trivial charge is turned on, expressed in terms of partial theta functions. Our results should be relevant to the eigenstate thermalization hypothesis for charged CFTs, Warped CFTs and effective field theory descriptions of condensed matter systems.

DOI
10.1007/jhep05(2021)276

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A bstract The Generalized Gibbs Ensemble (GGE) is relevant to understand the thermalization of quantum systems with an infinite set of conserved charges. In this work, we analyze the GGE partition function of 2D Conformal Field Theories (CFTs) with a U(1) charge and quantum Benjamin-Ono 2 (qBO 2 ) hierarchy charges. We use the Alday-Gaiotto-Tachikawa (AGT) correspondence to express the thermal trace in terms of the Alba-Fateev-Litvinov-Tarnopolskiy (AFLT) basis of descendants, which diagonalizes all charges. We analyze the GGE partition function in the thermodynamic semiclassical limit, including the first order quantum correction. We find that the equality between GGE averages and primary eigenvalues of the qBO 2 charges is attainable in the strict large c limit and potentially violated at the subleading 1 /c order. We also obtain the finite c partition function when only the first non-trivial charge is turned on, expressed in terms of partial theta functions. Our results should be relevant to the eigenstate thermalization hypothesis for charged CFTs, Warped CFTs and effective field theory descriptions of condensed matter systems.
Infinite-dimensional conformal symmetry in two dimensions leads to integrability of 2D conformal field theories (CFTs) by giving rise to an infinite tower of local conserved quantum Korteweg-de Vries (qKdV) charges in involution. We discuss how the presence of conserved charges constrains equilibration in 2D CFTs. We propose that in the thermodynamic limit large central charge 2D CFTs satisfy generalized eigenstate thermalization, with the values of qKdV charges forming a complete set of thermodynamically relevant quantities, which unambiguously determine expectation values of all local observables from the vacuum family. Equivalence of ensembles further provides that local properties of an eigenstate can be described by the generalized Gibbs ensemble that includes only qKdV charges. In the case of a general initial state, upon equilibration, the emerging generalized Gibbs ensemble will necessarily include negative chemical potentials and holographically will be described by a quasiclassical black hole with quantum soft hair.
Abstract We discuss partition function of 2d CFTs decorated by higher qKdV charges in the thermodynamic limit when the size of the spatial circle goes to infinity. In this limit the saddle point approximation is exact and at infinite central charge generalized partition function can be calculated explicitly. We show that leading 1/c corrections to free energy can be reformulated as a sum over Young tableaux which we calculate for the first two qKdV charges. Next, we compare generalized ensemble with the “eigenstate ensemble” that consists of a single primary state. At infinite central charge the ensembles match at the level of expectation values of local operators for any values of qKdV fugacities. When the central charge is large but finite, for any values of the fugacities the aforementioned ensembles are distinguishable.
We investigate the modular properties of Generalised Gibbs Ensembles (GGEs) in two dimensional conformal field theories. These are obtained by inserting higher spin charges in the expressions for the partition function of the theory. We investigate the particular case where KdV charges are inserted in the GGE. We first determine an asymptotic expression for the transformed GGE. This expression is an expansion in terms of the zero modes of all the quasi-primary fields in the theory, not just the KdV charges. While these charges are non-commuting they can be re-exponentiated to give an asymptotic expression for the transformed GGE in terms of another GGE. As an explicit example we focus on the Lee-Yang model. We use the Thermodynamic Bethe Ansatz in the Lee-Yang model to first replicate the asymptotic results, and then find additional energies that need to be included in the transformed GGE in order to find the exact modular transformation.
We revisit certain aspects of thermalization in 2D conformal field theory (CFT). In particular, we consider similarities and differences between the time dependence of correlation functions in various states in rational and non-rational CFTs. We also consider the distinction between global and local thermalization and explain how states obtained by acting with a diffeomorphism on the ground state can appear locally thermal, and we review why the time-dependent expectation value of the energy-momentum tensor is generally a poor diagnostic of global thermalization. Since all 2D CFTs have an infinite set of commuting conserved charges, generic initial states might be expected to give rise to a generalized Gibbs ensemble rather than a pure thermal ensemble at late times. We construct the holographic dual of the generalized Gibbs ensemble and show that, to leading order, it is still described by a Banados-Teitelboim-Zanelli black hole. The extra conserved charges, while rendering c<1c<1c<1 theories essentially integrable, therefore seem to have little effect on large-ccc conformal field theories.
We develop a Fock-space formulation that reduces expectation values of vacuum-family quasi-primaries in qKdV eigenstates of arbitrary highest-weight conformal families to a free-bosonic problem, and employ it to test the generalized eigenstate thermalization hypothesis (GETH), in 2D conformal field theories with large central charge ccc. We demonstrate that the first two quantum Korteweg--de Vries (qKdV) charges Q1Q_1Q1​ and Q3Q_3Q3​ lift all degeneracies except along at most countably many curves in the (c,Δ)(c,Δ)(c,Δ) plane, where ΔΔΔ is the highest weight of the conformal family, so that the full-qKdV simultaneous eigenstates are unambiguously determined by Q1Q_1Q1​ and Q3Q_3Q3​. 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-ccc 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.
Abstract Two-dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges. We study the Generalized Gibbs Ensemble with chemical potentials for these charges at high temperature. In a large central charge limit, the partition function can be computed in a saddle-point approximation. We compare the ensemble values of the KdV charges to the values in a microstate, and find that they match irrespective of the values of the chemical potentials. We study the partition function at finite central charge perturbatively in the chemical potentials, and find that this degeneracy is broken. We also study the statistics of the KdV charges at high level within a Virasoro representation, and find that they are sharply peaked.