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

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.