TBA and TCSA with boundaries and excited states
Patrick Dorey, Andrew Pocklington, Roberto Tateo, Gerard Watts
TL;DR
This work analyzes the finite-interval spectrum of the boundary scaling Lee-Yang model by integrating boundary truncated conformal space (BTCSA) and boundary thermodynamic Bethe ansatz (BTBA) methods. It identifies two integrable boundary conditions, ${\rm 11}$ with reflection factor $R_{(1)}(\theta)$ and $\Phi$ described by a non-minimal one-parameter family $R_b(\theta)$, and shows how ultraviolet boundary data map to infrared scattering data, including a precise relation $h(b)$ between the boundary parameter and the reflection factor. Ground-state BTBA requires analytic continuation and yields a consistent $c(r)$ with bulk and boundary contributions fixed by cancellation, while excited states are treated via generalized BTBA and validated against BTCSA, revealing boundary flows for massless bulks and boundary-bound-state formation for massive bulks. The study also extends to other boundary combinations, demonstrating that the UV central charge and spectral structure can be recovered with modified BTBA formalisms, thereby providing a robust framework for relating boundary conformal data, reflection factors, and finite-volume spectra in integrable boundary quantum field theories.
Abstract
We study the spectrum of the scaling Lee-Yang model on a finite interval from two points of view: via a generalisation of the truncated conformal space approach to systems with boundaries, and via the boundary thermodynamic Bethe ansatz. This allows reflection factors to be matched with specific boundary conditions, and leads us to propose a new (and non-minimal) family of reflection factors to describe the one relevant boundary perturbation in the model. The equations proposed previously for the ground state on an interval must be revised in certain regimes, and we find the necessary modifications by analytic continuation. We also propose new equations to describe excited states, and check all equations against boundary truncated conformal space data. Access to the finite-size spectrum enables us to observe boundary flows when the bulk remains massless, and the formation of boundary bound states when the bulk is massive.