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Journal of High Energy Physics

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

Nicolò Brizio, Tommaso Morone, Nicolò Primi, Roberto Tateo

Published Mar 13, 2026• 0 citations
Publisher

Abstract

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 $$ T\overline{T} $$ T 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.

DOI
10.1007/jhep03(2026)137

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Journal of High Energy Physics•Sep 8, 2023•11 citations
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 TT‾T\overline{T}TT T 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.
We introduce and study a class of two-dimensional integrable quantum field theories that carry an internal Zn\mathbb{Z}_nZn​ 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 TTˉT\bar{T}TTˉ 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.
We introduce a systematic scheme for constructing the ODE/IM counterpart of generalised Gibbs ensembles, and apply it to the chiral sector of the Lee--Yang conformal field theory. We encode the chemical potentials in a deformation of the cubic oscillator, fixed order by order by requiring that the large-energy WKB period reproduces exactly the driving term of the corresponding thermodynamic Bethe ansatz equation; the subleading WKB periods then yield, as predictions, the deformed eigenvalues of the local conserved charges. We work out the spin-5 ensemble in detail for two states of the model: the ground state, in the module of the primary ϕϕϕ, dual to the deformed cubic oscillator, and the primary of the identity module, dual to the same oscillator supplemented by an apparent singularity, a double pole whose position is fixed by a trivial-monodromy condition. In both cases, the Stokes multiplier obeys the Lee--Yang Y-system, and the deformed eigenvalues match, to the orders checked, an independent conformal-field-theory computation. The Borel resummation of the resulting series for the ground-state energy reproduces the numerical TBA solution over many orders of magnitude in the chemical potential. The scheme relies only on general features of the ODE/IM correspondence and applies to generalised Gibbs ensembles in other integrable models.
We construct a set of conserved charges which are deformations of quantum KdV charges. They are associated with the modular transformation of a GGE with the spin-σσσ KdV charge inserted with chemical potential ασα_σασ​. The charges are formal power series in the chemical potential, with coefficients given by local integrals of motion. We define local recursion relations and equivalent flow equations, prove commutativity to all orders, and establish uniqueness of the currents modulo total derivatives for a fixed generating series. We show that the eigenvalues of these deformed charges can be obtained from the thermodynamic Bethe Ansatz in two models. In the c=1/2c=1/2c=1/2 free fermion theory, the deformed charges reduce to explicit linear combinations of undeformed KdV charges, and their ground-state eigenvalues agree with the TBA expansion to all orders. In the Lee-Yang model, we find numerical agreement through cubic order in the spin-555 deformation for the first few charges evaluated on low-lying states.
We consider the most general set of integrable deformations extending the TT[over ¯] deformation of two-dimensional relativistic QFTs. They are CDD deformations of the theory's factorized S matrix related to the higher-spin conserved charges. Using a mirror version of the generalized Gibbs ensemble, we write down the finite-volume expectation value of the higher-spin charges, and derive a generalized flow equation that every charge must obey under a generalized TT[over ¯] deformation. This also reproduces the known flow equations on the nose.
Form Factors and Correlation Functions of TT‾\mathrm{T}\overline{\mathrm{T}}TT-Deformed Integrable Quantum Field Theories
The study of TT‾\mathrm{T}\overline{\mathrm{T}}TT-perturbed quantum field theories is an active area of research with deep connections to fundamental aspects of the scattering theory of integrable quantum field theories, generalised Gibbs ensembles, and string theory. Many features of these theories, such as the peculiar behaviour of their ground state energy and the form of their scattering matrices, have been studied in the literature. However, so far, very few studies have approached these theories from the viewpoint of the form factor program. From the perspective of scattering theory, the effects of a TT‾\mathrm{T}\overline{\mathrm{T}}TT perturbation (and higher spin versions thereof) is encoded in a universal deformation of the two-body scattering matrix by a CDD factor. It is then natural to ask how these perturbations influence the form factor equations and, more generally, the form factor program. In this paper, we address this question for free theories, although some of our results extend more generally. We show that the form factor equations admit general solutions and how these can help us study the distinct behaviour of correlation functions at short distances in theories perturbed by irrelevant operators.
Form factors and correlation functions of TT‾ \textrm{T}\overline{\textrm{T}} TT-deformed integrable quantum field theories
Abstract The study of TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ -perturbed quantum field theories is an active area of research with deep connections to fundamental aspects of the scattering theory of integrable quantum field theories, generalised Gibbs ensembles, and string theory. Many features of these theories, such as the peculiar behaviour of their ground state energy and the form of their scattering matrices, have been studied in the literature. However, so far, very few studies have approached these theories from the viewpoint of the form factor program. From the perspective of scattering theory, the effects of a TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ perturbation (and higher spin versions thereof) is encoded in a universal deformation of the two-body scattering matrix by a CDD factor. It is then natural to ask how these perturbations influence the form factor equations and, more generally, the form factor program. In this paper, we address this question for free theories, although some of our results extend more generally. We show that the form factor equations admit general solutions and how these can help us study the distinct behaviour of correlation functions at short distances in theories perturbed by irrelevant operators.