The finite-size behaviours of the homogeneous sine-Gordon models are analysed in detail, using the thermodynamic Bethe ansatz. Crossovers are observed which allow scales associated with both stable and unstable quantum particles to be picked up. By introducing the concept of shielding, we show that these match precisely with the mass scales found classically, supporting the idea that the full set of unstable particle states persists even far from the semiclassical regime. General rules for the effective TBA systems governing individual crossovers are given, and we also comment on the Lagrangian treatment of the theories, novel issues which arise in the form-factor approach for theories with unstable particles, and the role of heterotic cosets in the staircase flows exhibited by the HSG models.
Dorey, P., & Miramontes, J. L. (2004). Mass scales and crossover phenomena in the homogeneous sine-Gordon models. Nuclear Physics B, 697(3), 405-461. https://doi.org/10.1016/j.nuclphysb.2004.07.019