\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.
\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.
T T ¯ TT -deformed 2D quantum field theories
It was noticed many years ago, in the framework of massless RG flows, that the irrelevant composite operator TT , built with the components of the energy-momentum tensor, enjoys very special properties in 2D quantum field theories, and can be regarded as a peculiar kind of integrable perturbation. Novel interesting features of this operator have recently emerged from the study of effective string theory models. In this paper we study further properties of this distinguished perturbation. We discuss how it affects the energy levels and one-point functions of a general 2D QFT in finite volume through a surprising relation with a simple hydrodynamic equation. In the case of the perturbation of CFTs, adapting a result by Lüscher and Weisz we give a compact expression for the partition function on a finite-length cylinder and make a connection with the exact g-function method. We argue that, at the classical level, the deformation naturally maps the action of N massless free bosons into the Nambu-Goto action in static gauge, in N + 2 target space dimensions, and we briefly discuss a possible interpretation of this result in the context of effective string models.
TT-deformation and long range spin chains
Abstract We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the TT T T ¯ -deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of N N = 4 super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the TT T T ¯ -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.
On the flow of states under TT
We study the TT deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the TT operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, “dressed” operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the TT deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.
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.
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.
Abstract We initiate the study of T T ¯ -like irrelevant solvable deformations in quantum field theory with boundaries and defects. For this purpose, we employ a general formalism developed in the context of spin chains, which allows us to derive both, the deformed bulk and boundary/defect scattering matrices of integrable models. Using the deformed scattering matrices, we derive the flow equation for the deformed finite volume spectrum, as well as the cylinder partition function and the exact g-function.