Abstract Deformations of Hamiltonian many-body systems by certain products of conserved currents, referred to as $$T\bar{T}$$ T T ¯ -deformations, are known to preserve integrability. Generalised $$T\bar{T}$$ T 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 $$T\bar{T}$$ T T ¯ -deformations to the simplest setup, classical Hamiltonian particle systems: we construct explicit generalised $$T\bar{T}$$ T 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 $$T\bar{T}$$ T 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 $$T\bar{T}$$ T 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.

Annales Henri PoincaréMar 24, 20260 citations

Physics Letters B

Generalised statistics of particles

Roberto Scipioni

Published May 1, 1994• 22 citations

Publisher

DOI: 10.1016/0370-2693(94)91528-8

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Abstract Deformations of Hamiltonian many-body systems by certain products of conserved currents, referred to as $T\bar{T}$ T T ¯ -deformations, are known to preserve integrability. Generalised $T\bar{T}$ T 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 $T\bar{T}$ T T ¯ -deformations to the simplest setup, classical Hamiltonian particle systems: we construct explicit generalised $T\bar{T}$ T 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 $T\bar{T}$ T 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 $T\bar{T}$ T 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.Abstract Deformations of Hamiltonian many-body systems by certain products of conserved currents, referred to as $$T\bar{T}$$ T T ¯ -deformations, are known to preserve integrability. Generalised $$T\bar{T}$$ T 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 $$T\bar{T}$$ T T ¯ -deformations to the simplest setup, classical Hamiltonian particle systems: we construct explicit generalised $$T\bar{T}$$ T 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 $$T\bar{T}$$ T 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 $$T\bar{T}$$ T 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.

Annales Henri PoincaréMar 24, 20260 citations

Generalised

T\bar{T}

-Deformations of Classical Free Particles

Abstract Deformations of Hamiltonian many-body systems by certain products of conserved currents, referred to as

T\bar{T}

T T ¯ -deformations, are known to preserve integrability. Generalised

T\bar{T}

T 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}

T T ¯ -deformations to the simplest setup, classical Hamiltonian particle systems: we construct explicit generalised

T\bar{T}

T 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}

T 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}

T 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.