Deformations of Hamiltonian many-body systems by certain products of conserved currents, referred to as \(T\bar{T}\) -deformations, are known to preserve integrability. Generalised \(T\bar{T}\) -deformations, based on the complete space of pseudolocal currents, were suggested (Doyon et al. in Scipost Phys 13:072, 2022) to give rise to integrable systems with arbitrary two-body scattering shifts, going beyond those from known models. However, locality properties were not clear. In order to address this, we apply \(T\bar{T}\) -deformations to the simplest setup, classical Hamiltonian particle systems: we construct explicit generalised \(T\bar{T}\) -deformations of classical free particles. We show rigorously that they are Liouville integrable Hamiltonian systems with interactions that are of finite range, albeit momentum-dependent. We show elastic, factorised scattering, with a two-particle scattering shift that can be any continuously differentiable non-negative even function of momentum differences, fixed by the \(T\bar{T}\) -deformation function. We show that the scattering map (or “wave operator”) has a finite-range property allowing us to trace carriers of asymptotic momenta even at finite times—an important characteristics of many-body integrability. We evaluate the grand-canonical free energy in (generalised) Gibbs ensembles and prove the thermodynamic Bethe ansatz with Maxwell–Boltzmann statistics, including with space-varying potentials and in finite and infinite volumes. We give equations for the particles’ trajectories where time appears explicitly, generalising the contraction map of hard rod systems: the effect of generalised \(T\bar{T}\) -deformations is to modify the local metric perceived by each particle, adding extra space in a way that depends on their neighbours. The systems generalise the gas of interacting Bethe ansatz wave packets recently introduced in the Lieb–Liniger model. They form a new class of models that, we believe, most clearly make manifest the structures of many-body integrability.
