Integrable quantum field theory with boundaries: the exact g-function

TL;DR

The paper addresses how to obtain an exact off-critical g-function for boundary integrable QFTs, introducing a cluster-expansion method aligned with TBA pseudoenergies.By applying the approach to the Ising model and the scaling Lee-Yang model, it derives an infrared expansion for ln g that matches known UV results and shows the limitations of previous proposals.The main achievement is an exact infrared expression for the Lee-Yang g-function, written as a dressed integral term plus an infinite series over multi-particle kernels, valid for theories with entirely diagonal scattering, and generalizable to N species.These results provide a principled route to exact boundary entropy calculations in massive, diagonal theories and suggest directions for extending the framework to more complex scattering structures.

Abstract

The g-function was introduced by Affleck and Ludwig in the context of critical quantum systems with boundaries. In the framework of the thermodynamic Bethe ansatz (TBA) method for relativistic scattering theories, all attempts to write an exact integral equation for the off-critical version of this quantity have, up to now, been unsuccesful. We tackle this problem by using an n-particle cluster expansion, close in spirit to form-factor calculations of correlators on the plane. The leading contribution already disagrees with all previous proposals, but a study of this and subsequent terms allows us to deduce an exact infrared expansion for g, written purely in terms of TBA pseudoenergies. Although we only treat the thermally-perturbed Ising and the scaling Lee-Yang models in detail, we propose a general formula for g which should be valid for any model with entirely diagonal scattering.