Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modular Properties of Generalised Gibbs Ensembles

Max Downing, Faisal Karimi

TL;DR

This work analyzes the modular properties of generalized Gibbs ensembles in 2d CFTs by inserting higher-spin charges into the torus partition function $Z(R,L)$ and studying its modular transform. The authors derive an asymptotic transformed GGE expressed in terms of zero modes of all quasi-primary fields, and show that it can be re-exponentiated to a new GGE involving an extended set of charges, beyond the original KdV charges. Focusing on the Lee-Yang model, they reproduce these results via a Thermodynamic Bethe Ansatz (TBA) and uncover additional energies required for the exact modular transform, including noncommuting charges that appear in the transformed spectrum. They further demonstrate, with numerical TBA analysis, that the transformed GGE spectrum is fully captured by the TBA framework, while also uncovering non-asymptotic solutions with fractional scaling that must be included for completeness. The study provides a bridge between modular properties, defect Hilbert spaces, and integrable methods, and suggests extensions to other minimal models and W-algebras.

Abstract

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.

Modular Properties of Generalised Gibbs Ensembles

TL;DR

This work analyzes the modular properties of generalized Gibbs ensembles in 2d CFTs by inserting higher-spin charges into the torus partition function and studying its modular transform. The authors derive an asymptotic transformed GGE expressed in terms of zero modes of all quasi-primary fields, and show that it can be re-exponentiated to a new GGE involving an extended set of charges, beyond the original KdV charges. Focusing on the Lee-Yang model, they reproduce these results via a Thermodynamic Bethe Ansatz (TBA) and uncover additional energies required for the exact modular transform, including noncommuting charges that appear in the transformed spectrum. They further demonstrate, with numerical TBA analysis, that the transformed GGE spectrum is fully captured by the TBA framework, while also uncovering non-asymptotic solutions with fractional scaling that must be included for completeness. The study provides a bridge between modular properties, defect Hilbert spaces, and integrable methods, and suggests extensions to other minimal models and W-algebras.

Abstract

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.
Paper Structure (29 sections, 240 equations, 2 figures, 7 tables)

This paper contains 29 sections, 240 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Transformed GGEs and defects
  3. GGEs in a Generic 2d CFT
  4. Asymptotic Analysis of the GGE in the Lee-Yang Model
  5. Thermodynamic Bethe Ansatz for the transformed GGE
  6. TBA and mirror TBA
  7. TBA for the GGE
  8. Transformed TBA
  9. Asymptotic results from the TBA
  10. Numerical results
  11. Non-asymptotic solutions to the TBA
  12. Conclusions and Outlook
  13. Modular forms
  14. Construction of Charges
  15. Mapping between the cylinder and plane
  16. ...and 14 more sections

Figures (2)

  • Figure 1: Interpretation of the modular transformed GGE traces: on torus (I), the GGE is given by a defect inserted as an operator $\hat{D}$ in the trace; on torus (II) the defect is rotated and the transformed GGE is given by a trace over the Hilbert space $\mathcal{H}_D$ with a defect Hamiltonian $H_D(L)$ inserted in the trace.
  • Figure 2: Strip of width $L$ and length $R$. On the horizontal slice $\mathcal{B}$ we have the Hilbert space $\mathcal{H}_\mathcal{B}$ and on the vertical slice $\mathcal{C}$ we have the Hilbert space $\mathcal{H}_\mathcal{C}$.