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

TL;DR

This paper explores the modular properties of generalized Gibbs ensembles in a free fermion model, deriving new results on their asymptotic behavior and proposing a conjecture for their exact form in the opposite channel.

Contribution

It introduces a novel analysis of modular transformations of GGEs with KdV charges in free fermion theories, including a conjecture for their exact form and asymptotic expansions.

Findings

Derived the constant term in KdV charges from modular properties.

Established the modular behavior of polynomial expectation values of KdV charges.

Proposed a conjecture for the exact GGE in the opposite channel, supported by analytic and numerical evidence.

Abstract

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 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.