Graded S-Matrices, Generalised Gibbs Ensembles and Fractional-Spin CDD Deformations

TL;DR

This paper introduces a new class of 2D integrable quantum field theories with internal cyclic symmetry, extending factorised scattering to include fractional-spin charges, and explores their spectral and deformation properties.

Contribution

It constructs models with fractional-spin conserved charges, develops a graded Thermodynamic Bethe Ansatz, and connects these to CDD deformations and Hagedorn-like spectra.

Findings

Finite-volume spectra show infinite level crossings with increasing coupling.

Functional relations match those from the ODE/IM correspondence.

Deformations lead to finite limiting temperatures in the spectrum.

Abstract

We introduce and study a class of two-dimensional integrable quantum field theories that carry an internal $\mathbb{Z}_n$ structure. These models extend factorised scattering beyond the conventional framework, featuring both the usual hierarchy of integer-spin conserved charges and an additional tower of fractional-spin ones. Our construction relies on a reparametrisation of rapidity space that lifts standard scattering amplitudes to a multiplet related by an internal cyclic symmetry. This construction is naturally embedded within a generalised Gibbs ensemble, which provides the natural framework for a consistent graded Thermodynamic Bethe Ansatz. This leads to new Y-systems encoding the graded spectrum. In a special case, these functional relations match those obtained via the ODE/IM correspondence from the monodromy analysis of the quantum cubic oscillator. Even in the simplest models, for one sign of the auxiliary temperature, the finite-volume ground-state energy spectrum undergoes an infinite sequence of level crossings as the coupling strength increases. A preliminary analysis also suggests that these theories exhibit structural connections with cyclic orbifolds. Within this setup, one can consistently include extra CDD factors that realise fractional-spin analogues of the $T\bar{T}$ deformation. In analytically tractable cases, a Hagedorn-like behaviour is observed for a sign of the flow parameter, and the deformed spectrum develops a finite limiting temperature.