Graded S-matrices, generalised Gibbs ensembles and fractional-spin CDD deformations

A bstract We introduce and study a class of two-dimensional integrable quantum field theories that carry an internal 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 $$ T\overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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.