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NLIE formulations for the generalized Gibbs ensemble in the sine-Gordon model

Arpad Hegedus

TL;DR

This work develops two nonlinear integral equation (NLIE) frameworks, NLIE I and NLIE II, to describe sine-Gordon thermodynamics in a generalized Gibbs ensemble that includes higher-spin conserved charges. Both NLIEs are derived from the TBA/Y-system structure via T-Q relations in the repulsive regime and analytically continued to broader coupling ranges, with NLIE I offering partial access to the attractive regime. NLIE I is valid for 1 - 1/(2N-1) < p < ∞ and may access attraction only up to a depth determined by the largest spin included, while NLIE II remains valid for all p>1 with any number of charges. These NLIEs provide efficient routes to compute the GGE potential G and flux Φ, as well as expectation values of conserved densities, currents, and vertex operators ⟨e^{i m β_SG φ}⟩, enabling applications to non-equilibrium dynamics and generalized hydrodynamics in the sine-Gordon model.

Abstract

In this paper we propose two sets of nonlinear integral equations (NLIE) for describing the thermodynamics in the sine-Gordon model, when higher Lorentz spin conserved charges are also coupled to the Gibbs ensemble. We call them NLIE I and II. The derivation of the equations, is based on T-Q relations given by the equivalent thermodynamic Bethe ansatz (TBA) formulation of the problem in the repulsive regime. Though the equations are derived in the repulsive regime at discrete values of the coupling constant, a straightforward analytical continuation ensures their validity within the whole repulsive regime of the theory. For the NLIE I formulation, appropriate analytical continuation makes the penetration into the attractive regime also possible. However, the magnitude of this penetration is restricted by the spin of the largest spin conserved charge contained in the Gibbs ensemble. Within their range of validity, these NLIE formulations provide efficient theoretical frameworks for computing expectation values of conserved charge densities, their associated currents, and vertex operators and their descendants, with respect to the generalized Gibbs ensemble.

NLIE formulations for the generalized Gibbs ensemble in the sine-Gordon model

TL;DR

This work develops two nonlinear integral equation (NLIE) frameworks, NLIE I and NLIE II, to describe sine-Gordon thermodynamics in a generalized Gibbs ensemble that includes higher-spin conserved charges. Both NLIEs are derived from the TBA/Y-system structure via T-Q relations in the repulsive regime and analytically continued to broader coupling ranges, with NLIE I offering partial access to the attractive regime. NLIE I is valid for 1 - 1/(2N-1) < p < ∞ and may access attraction only up to a depth determined by the largest spin included, while NLIE II remains valid for all p>1 with any number of charges. These NLIEs provide efficient routes to compute the GGE potential G and flux Φ, as well as expectation values of conserved densities, currents, and vertex operators ⟨e^{i m β_SG φ}⟩, enabling applications to non-equilibrium dynamics and generalized hydrodynamics in the sine-Gordon model.

Abstract

In this paper we propose two sets of nonlinear integral equations (NLIE) for describing the thermodynamics in the sine-Gordon model, when higher Lorentz spin conserved charges are also coupled to the Gibbs ensemble. We call them NLIE I and II. The derivation of the equations, is based on T-Q relations given by the equivalent thermodynamic Bethe ansatz (TBA) formulation of the problem in the repulsive regime. Though the equations are derived in the repulsive regime at discrete values of the coupling constant, a straightforward analytical continuation ensures their validity within the whole repulsive regime of the theory. For the NLIE I formulation, appropriate analytical continuation makes the penetration into the attractive regime also possible. However, the magnitude of this penetration is restricted by the spin of the largest spin conserved charge contained in the Gibbs ensemble. Within their range of validity, these NLIE formulations provide efficient theoretical frameworks for computing expectation values of conserved charge densities, their associated currents, and vertex operators and their descendants, with respect to the generalized Gibbs ensemble.

Paper Structure

This paper contains 13 sections, 107 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The model and the TBA equations
  3. The derivation of the NLIEs
  4. Functional relations stemming from the TBA
  5. Analyticity strips
  6. Deriving NLIE I.
  7. Deriving NLIE II
  8. Discussion of the attractive regime
  9. Expectation values
  10. Expectation values of conserved charges and their associated currents
  11. Expectation values of local operators
  12. Summary and discussion
  13. Some numerical results

Figures (1)

  • Figure 1: The coupling constant dependence of the Gibbs potential around the free fermion point, when $\beta_1=3 \cdot 10^{-1}, {\beta}_{-1}= 10^{-1}, \beta_3=5/6,\, {\beta}_{-3}=1/6$.