Excited State Destri - De Vega Equation for Sine-Gordon and Restricted Sine-Gordon Models

TL;DR

This work extends the Destri–De Vega framework to excited states of the Sine-Gordon model by formulating a continuum DdV equation for hole excitations over twisted vacua. Through quantum group reduction, these excited states are interpreted within Restricted Sine-Gordon theory as off-critical deformations of diagonal Kac-table primaries, with analytic UV limits yielding exact conformal dimensions $\Delta_{\pm}$ and spin. Numerical checks against TBA data and explicit UV analyses confirm a consistent picture across SG$_p$/RSG$_p$, including level crossings and fractional spins in twisted sectors. The results provide a powerful, exact description of scaling functions for a broad class of excited states and outline a path toward a complete spectrum by incorporating string excitations. This advances understanding of finite-volume dynamics in integrable quantum field theories and their conformal limits, with potential applications to perturbed minimal models and related monodromy analyses.

Abstract

We derive a generalization of the Destri - De Vega equation governing the scaling functions of some excited states in the Sine-Gordon theory. In particular configurations with an even number of holes and no strings are analyzed and their UV limits found to match some of the conformal dimensions of the corresponding compactified massless free boson. Quantum group reduction allows to interpret some of our results as scaling functions of excited states of Restricted Sine-Gordon theory, i.e. minimal models perturbed by phi_13 in their massive regime. In particular we are able to reconstruct the scaling functions of the off-critical deformations of all the scalar primary states on the diagonal of the Kac-table.

Figures (1)

• Figure 1: Energy versus $R$ of spin 0 pure hole excitations in the $\alpha=0$ sector of SG$_{p=3}$ model.