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

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.

Figures (18)

• Figure 1: Ground state scaling function of the graded Ising field theory for sufficiently small values of $\alpha$. When $n=3$, only the trivial case $\alpha = 0$ flows to three independent copies of the Ising CFT. When $n=5$, no ultraviolet corrections are observed, yet a non-trivial structure persists at finite radius. Analogous behaviours are observed for all odd $n>5$.
• Figure 2: Normalised level sets of the function $1 + \exp(-e^{\vartheta} - \alpha e^{\vartheta/3})$ displayed over a $20 \times 20$ complex strip centred at the origin in the $\vartheta$-plane for various $\alpha$ values. The dashed red horizontal lines indicate $\Im \vartheta\in \{0,\pm 2 \pi\}$. The plots for $k=1,2$ are formally obtained from this picture by translating the imaginary direction, $\Im\vartheta \mapsto \Im \vartheta \pm 2 \pi$. As $\alpha$ varies, the zeroes of the displayed function move in the complex plane; at the first critical value $\alpha \simeq 4.7218$, a pair of zeroes collides with the $\pm 2 \pi i$ lines, signalling a change in the analytic structure of the ground-state scaling function.
• Figure 3: Plots summarising the behaviour of the first few excited-state scaling functions that populate the spectrum as a function of $\alpha$. In Figure \ref{['fig:13a']}, the excited states that successively overtake the perturbative ground state as $\alpha$ is increased towards positive values are highlighted. The dotted grey line corresponds to the numerical evaluation of the ground-state scaling function. Figure \ref{['figddd:1b']} shows instead all excited states satisfying \ref{['QC']} for $j=0,1$.
• Figure 4: Ground-state energy of the deformed $\mathbb{Z}_3$-graded Ising model. Both branches $E^{\pm}$ are plotted as a function of $R$ for fixed $\beta = 1$ and $\mu = -1$. At the critical value $R_* = 1/8\pi$, the two branches merge, and the Casimir energy becomes complex. We also observe that, at the isolated point $R \simeq 0.05587$, reality of the ground state is restored. Qualitatively similar behaviours are observed in non-Hermitian, $\mathcal{PT}$-symmetric models of quantum mechanics -- see, for example, Figures 14 and 20 of Bender:1998gh.
• Figure 5: A diagrammatic representation of the two-particle S-matrix consistency conditions. Here, the vertical direction represents time: incoming particles flow upward into the scattering region, and outgoing particles emerge above it. In Figure \ref{['fig:braid']}, two particles $a$ and $b$ scatter into intermediate channels $e, f$, and then scatter again into $c, d$. The sum over all allowed intermediate species reproduces two straight worldlines. In other words, performing the exchange twice is equivalent to doing nothing, yielding \ref{['braid']}. In Figure \ref{['fig:cross']}, a two-particle scattering amplitude in the direct channel (or "$s$-channel") can be analytically continued into a process in the crossed one ("$t$-channel"). Moving an external leg from the incoming to the outgoing side corresponds to replacing the particle with its antiparticle and shifting the rapidity by $i\pi$, yielding \ref{['cs']}.