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arXiv:2311.04564 · Journal of High Energy Physics

Free fermions, KdV charges, generalised Gibbs ensembles, modular transforms and line defects

Max Downing, G.M.T. Watts

Published Jan 9, 2024• 5 citations
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Abstract

A bstract In this paper we return to the question of the modular properties of a generalised Gibbs ensemble of a single free fermion. We extend our previous proposals to a GGE containing an arbitrary number of conserved charges and provide a physical interpretation of the result in terms of a line defect. The defect description perfectly explains the product formula for the modular transformation we found previously. We also give a proposal for a Hamiltonian approach to the line defect.

DOI
10.1007/jhep01(2024)041

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A bstract In this paper we return to the question of the modular properties of a generalised Gibbs ensemble of a single free fermion. We extend our previous proposals to a GGE containing an arbitrary number of conserved charges and provide a physical interpretation of the result in terms of a line defect. The defect description perfectly explains the product formula for the modular transformation we found previously. We also give a proposal for a Hamiltonian approach to the line defect.
The symplectic fermion is a much-studied non-unitary conformal field theory with c=−2c=-2c=−2 and is known to contain an infinite family of mutually commuting conserved charges. We derive expressions for the conserved charges on the cylinder and use these to construct Generalised Gibbs Ensembles (GGEs) in the particular case of the W(1,2){W}(1,2)W(1,2) triplet model. We derive exact expressions for the modular SSS-transforms in each sector of the symplectic fermion (and so of the whole GGE) and further extract the expressions in the asymptotic regime where the chemical potentials go to zero. Subsets of the conserved charges are known to reproduce the KdV and Boussinesq hierarchies. For the case in which the charge is identified with the zero mode W0W_0W0​ of the W3W_3W3​ algebra, we obtain asymptotic behaviour in precise agreement with the conjecture proposed in our companion paper [1]; for the KdV subset we obtain results which exactly mirror the case for a single free fermion. Finally we identify the GGE with a translation invariant and purely transmitting defect for the symplectic fermion fields, and make some comments on the relation to other WnW_nWn​ algebras.
A bstract In this paper we consider the modular properties of generalised Gibbs ensembles in the Ising model, realised as a theory of one free massless fermion. The Gibbs ensembles are given by adding chemical potentials to chiral charges corresponding to the KdV conserved quantities. (They can also be thought of as simple models for extended characters for the W-algebras). The eigenvalues and Gibbs ensembles for the charges can be easily calculated exactly using their expression as bilinears in the fermion fields. We re-derive the constant term in the charges, previously found by zeta-function regularisation, from modular properties. We expand the Gibbs ensembles as a power series in the chemical potentials and find the modular properties of the corresponding expectation values of polynomials of KdV charges. This leads us to an asymptotic expansion of the Gibbs ensemble calculated in the opposite channel. We obtain the same asymptotic expansion using Dijkgraaf’s results for chiral partition functions. By considering the corresponding TBA calculation, we are led to a conjecture for the exact closed-form expression of the GGE in the opposite channel. This has the form of a trace over multiple copies of the fermion Fock space. We give analytic and numerical evidence supporting our conjecture.
We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-σσσ KdV charge inserted with chemical potential ασα_σασ​. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the c=1/2c=1/2c=1/2 free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-555 deformation for the first few charges evaluated on low-lying states.
Modular properties of W3 {\mathcal{W}}_3 W3​ generalised Gibbs ensembles
A bstract In this paper we make a proposal for the solution to a long-standing problem — the asymptotic expansions of the modular S -transform of a generalised Gibbs ensemble (GGE) in a theory with W3 {\mathcal{W}}_3 W3​ W 3 symmetry where the GGE includes the first non-trivial charge. Equivalently, we give a proposal for the modular S -transform of traces of arbitrary powers of the zero mode W 0 . We provide evidence in the form of exact results using Zhu’s recursion, results obtained using conjectured results for Verma modules, and exact results for the particular value c = –2. We expect these have generalisations to other symmetry algebras/hierarchies such as the Virasoro algebra/KdV charges, and to GGEs with arbitrary finite sets of charges.
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.
Abstract We revisit the question of whether and how the steady states arising after non-equilibrium time evolution in integrable models (and in particular in the XXZ spin chain) can be described by the so-called generalized Gibbs ensemble (GGE). Whereas it is known that the micro-canonical ensemble built on a complete set of charges correctly describes the long-time limit of local observables, it has been shown recently by Ilievski et al that the corresponding canonical ensemble is not well defined, and instead a different canonical ensemble was proposed in terms of particle occupation number operators. Here we provide an alternative construction by considering truncated GGEs (tGGEs) that include only a finite number of local and quasi-local conserved operators. It is shown that the tGGEs can approximate the steady states with arbitrary precision, i.e. all physical observables are exactly reproduced in the infinite truncation limit. We trace back the problems encountered in defining an untruncated GGE to the dependence of the associated Lagrange multipliers on the truncation index. Conversely, we show that this problem may be circumvented by considering a new set of (quasi)local charges which are linear combinations of the standard ones, and whose associated Lagrange multipliers are well-defined state functions. Our general arguments are applied to concrete quench situations in the XXZ chain, where the initial states are simple two-site or four-site product states. Depending on the quench we find that numerical results for the local correlators can be obtained with remarkable precision using truncated GGEs with only 10–100 charges.