Exact $\phi_{1,3}$ boundary flows in the tricritical Ising model

TL;DR

This paper derives exact thermodynamic Bethe ansatz equations to analyze boundary renormalization group flows in the tricritical Ising model under a specific boundary perturbation, classifying excitations and mapping character changes.

Contribution

It provides a detailed lattice-based derivation of TBA equations for boundary flows in the tricritical Ising model, including classification of excitations and character mappings.

Findings

Derived exact TBA equations for boundary flows

Classified excitations by string content and quantum numbers

Mapped changes in Virasoro characters along the flows

Abstract

We consider the tricritical Ising model on a strip or cylinder under the integrable perturbation by the thermal $\phi_{1,3}$ boundary field. This perturbation induces five distinct renormalization group (RG) flows between Cardy type boundary conditions labelled by the Kac labels $(r,s)$. We study these boundary RG flows in detail for all excitations. Exact Thermodynamic Bethe Ansatz (TBA) equations are derived using the lattice approach by considering the continuum scaling limit of the $A_4$ lattice model with integrable boundary conditions. Fixing the bulk weights to their critical values, the integrable boundary weights admit a thermodynamic boundary field $\xi$ which induces the flow and, in the continuum scaling limit, plays the role of the perturbing boundary field $\phi_{1,3}$. The excitations are completely classified, in terms of string content, by $(m,n)$ systems and quantum numbers but the string content changes by either two or three well-defined mechanisms along the flow. We identify these mechanisms and obtain the induced maps between the relevant finitized Virasoro characters. We also solve the TBA equations numerically to determine the boundary flows for the leading excitations.