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arXiv:2312.14855

Generalised $T\bar{T}$-deformations of classical free particles

Benjamin Doyon, Friedrich Hübner, Takato Yoshimura

Published Dec 22, 2023• 0 citations
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Abstract

Deformations of many-body Hamiltonians 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 [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 $T\bar{T}$-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 $T\bar{T}$-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 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 $T\bar{T}$-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.

DOI
10.48550/arxiv.2312.14855

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Deformations of many-body Hamiltonians by certain products of conserved currents, referred to as TTˉT\bar{T}TTˉ-deformations, are known to preserve integrability. Generalised TTˉT\bar{T}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 TTˉT\bar{T}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 TTˉT\bar{T}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 TTˉT\bar{T}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.
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Abstract Deformations of Hamiltonian many-body systems by certain products of conserved currents, referred to as TTˉT\bar{T}TTˉ T T ¯ -deformations, are known to preserve integrability. Generalised TTˉT\bar{T}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 TTˉT\bar{T}TTˉ T T ¯ -deformations to the simplest setup, classical Hamiltonian particle systems: we construct explicit generalised TTˉT\bar{T}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 TTˉT\bar{T}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 TTˉT\bar{T}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.
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.
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The Space of Integrable Systems from Generalised TTˉT\bar{T}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.
Completing the Bootstrap Program for TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-Deformed Massive Integrable Quantum Field Theories
In recent years a considerable amount of attention has been devoted to the investigation of 2D quantum field theories perturbed by certain types of irrelevant operators. These are the composite field TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ - constructed out of the components of the stress-energy tensor - and its generalisations - built from higher-spin conserved currents. The effect of such perturbations on the infrared and ultraviolet properties of the theory has been extensively investigated. In the context of integrable quantum field theories, a fruitful perspective is that of factorised scattering theory. In fact, the above perturbations were shown to preserve integrability. The resulting deformed scattering matrices - extensively analysed with the thermodynamic Bethe ansatz - provide the first step in the development of a complete bootstrap program. In this letter we present a systematic approach to computing matrix elements of operators in generalised TTˉ\mathrm{T}\bar{\mathrm{T}}TTˉ-perturbed models, based on employing the standard form factor program. Our approach is very general and can be applied to all theories with diagonal scattering. We show that the deformed form factors, just as happens for the SSS-matrix, factorise into the product of the undeformed ones and of a perturbation- and theory-dependent term. From these solutions, correlation functions can be obtained and their asymptotic properties studied. Our results set the foundations of a new research program for massive integrable quantum field theory perturbed by irrelevant operators.
TT‾\mathrm{T}\overline{\mathrm{T}}TT-deformed 1d Bose gas
\mathrm{T}\bar{\mathrm{T}} T T ‾ deformation was originally proposed as an irrelevant solvable deformation for 2d relativistic quantum field theories (QFTs). The same family of deformations can also be defined for integrable quantum spin chains which was first studied in the context of integrability in AdS/CFT. In this paper, we construct such deformations for yet another type of models, which describe a collection of particles moving in 1d and interacting in an integrable manner. The prototype of such models is the Lieb-Liniger model. This shows that such deformations can be defined for a very wide range of systems. We study the finite volume spectrum and thermodynamics of the \mathrm{T}\overline{\mathrm{T}} T T ¯ -deformed Lieb-Liniger model. We find that for one sign of the deformation parameter (\lambda<0) ( λ < 0 ) , the deformed spectrum becomes complex when the volume of the system is smaller than certain critical value, signifying the break down of UV physics. For the other sign (\lambda>0) ( λ > 0 ) , there exists an upper bound for the temperature, similar to the Hagedorn behavior of the \mathrm{T}\overline{\mathrm{T}} T T ¯ deformed QFTs. Both behaviors can be attributed to the fact that \mathrm{T}\overline{\mathrm{T}} T T ¯ deformation changes the size the particles. We show that for \lambda>0 λ > 0 , the deformation increases the spaces between particles which effectively increases the volume of the system. For \lambda<0 λ < 0 , \mathrm{T}\overline{\mathrm{T}} T T ¯ deformation fattens point particles to finite size hard rods. This is similar to the observation that the action of \mathrm{T}\overline{\mathrm{T}} T T ¯ -deformed free boson is the Nambu-Goto action, which describes bosonic strings — also an extended object with finite size.