We introduce and study a novel class of classical integrable many-body systems obtained by generalized TT[over ¯] deformations of free particles. Deformation terms are bilinears in densities and currents for the continuum of charges counting asymptotic particles of different momenta. In these models, which we dub "semiclassical Bethe systems" for their link with the dynamics of Bethe ansatz wave packets, many-body scattering processes are factorized, and two-body scattering shifts can be set to an almost arbitrary function of momenta. The dynamics is local but inherently different from that of known classical integrable systems. At short scales, the geometry of the deformation is dynamically resolved: either particles are slowed down (more space available), or accelerated via a novel classical particle-pair creation and annihilation process (less space available). The thermodynamics both at finite and infinite volumes is described by the equations of (or akin to) the thermodynamic Bethe ansatz, and at large scales generalized hydrodynamics emerge.
We introduce and study a novel class of classical integrable many-body systems obtained by generalized TT[over ¯] deformations of free particles. Deformation terms are bilinears in densities and currents for the continuum of charges counting asymptotic particles of different momenta. In these models, which we dub "semiclassical Bethe systems" for their link with the dynamics of Bethe ansatz wave packets, many-body scattering processes are factorized, and two-body scattering shifts can be set to an almost arbitrary function of momenta. The dynamics is local but inherently different from that of known classical integrable systems. At short scales, the geometry of the deformation is dynamically resolved: either particles are slowed down (more space available), or accelerated via a novel classical particle-pair creation and annihilation process (less space available). The thermodynamics both at finite and infinite volumes is described by the equations of (or akin to) the thermodynamic Bethe ansatz, and at large scales generalized hydrodynamics emerge.
Generalised TTˉ-Deformations of Classical Free Particles
Abstract Deformations of Hamiltonian many-body systems by certain products of conserved currents, referred to as TTˉ T T ¯ -deformations, are known to preserve integrability. Generalised TTˉ 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 TTˉ T T ¯ -deformations to the simplest setup, classical Hamiltonian particle systems: we construct explicit generalised TTˉ 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 TTˉ 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 TTˉ 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.
Generalised TTˉ-deformations of classical free particles
Deformations of many-body Hamiltonians by certain products of conserved currents, referred to as TTˉ-deformations, are known to preserve integrability. Generalised TTˉ-deformations, based on the complete space of pseudolocal currents, were suggested [B. Doyon, J, Durnin, T. Yoshimura, Scipost Physics 13, 072 (2022)] to give rise to integrable systems with arbitrary two-body scattering shifts, going beyond those from known models or standard CDD factors. However, locality properties were not clear. We construct explicit generalised TTˉ-deformations of the system of classical free particles. We show rigorously that they are Liouville integrable Hamiltonian systems with finite-range interactions. 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 TTˉ-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 free energy 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 TTˉ-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.
The Space of Integrable Systems from Generalised TTˉ-Deformations
We introduce an extension of the generalised T\bar{T} T T ‾ -deformation described by Smirnov-Zamolodchikov, to include the complete set of extensive charges. We show that this gives deformations of S-matrices beyond CDD factors, generating arbitrary functional dependence on momenta. We further derive from basic principles of statistical mechanics the flow equations for the free energy and all free energy fluxes. From this follows, without invoking the microscopic Bethe ansatz or other methods from integrability, that the thermodynamics of the deformed models are described by the integral equations of the thermodynamic Bethe-Ansatz, and that the exact average currents take the form expected from generalised hydrodynamics, both in the classical and quantum realms.
TT deformations and the width of fundamental particles
Abstract We provide a simple geometric meaning for deformations of so-called TT T T ¯ type in relativistic and non-relativistic systems. Deformations by the cross products of energy and momentum currents in integrable quantum field theories are known to modify the thermodynamic Bethe ansatz equations by a “CDD factor”. In turn, CDD factors may be interpreted as additional, fixed shifts incurred in scattering processes: a finite width added to the fundamental particles (or, if negative, to the free space between them). We suggest that this physical effect is a universal way of understanding TT T T ¯ deformations, both in classical and quantum systems. We first show this in non-relativistic systems, with particle conservation and translation invariance, using the deformation formed out of the densities and currents of particles and momentum. This holds at the level of the equations of motion, and for any interaction potential, integrable or not. We then argue, and show by similar techniques in free relativistic particle systems, that TT T T ¯ deformations of relativistic systems produce the equivalent phenomenon, accounting for length contractions. We also show that, in both the relativistic and non-relativistic cases, the width of particles is equivalent to a state-dependent change of metric, where the distance function discounts the particles’ widths, or counts the additional free space. This generalises and explains the known field-dependent coordinate change describing TT T T ¯ deformations. The results connect such deformations with generalised hydrodynamics, where the relations between scattering shifts, widths of particles and state-dependent changes of metric have been established.
We obtain the exact generalized hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorized scattering of Toda particles, we obtain the generalized free energy and exact average currents and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and, in particular, the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorized scattering of Toda particles.
We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang δ-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.