Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2405.06564

Generalised Hydrodynamics of $\mathrm{T\bar{T}}$-Deformed Integrable Quantum Field Theories

Riccardo Travaglino, Michele Mazzoni, Olalla A. Castro-Alvaredo

Published May 10, 2024• 0 citations
arXivHTMLPDF

Abstract

In this paper we evaluate the averages of conserved densities and currents associated to charges of generic spin in (1+1)-dimensional massive integrable Quantum Field Theories perturbed by the irrelevant $\mathrm{T\bar{T}}$ operator. By making use of the Thermodynamic Bethe Ansatz approach and of the theory of Generalised Hydrodynamics, we study the non-equilibrium steady state averages of conserved densities and currents in a partitioning protocol. We show that in particular limits, averages can be evaluated exactly in terms of quantities known from the unperturbed theory. In the massless limit we recover known results for the energy and momentum currents and generalise those to any higher spin conserved quantities. We extend some of our results to perturbations of the generalised $\mathrm{T\bar{T}}$ type. For the massive free fermion theory, we find an analytic expression for the effective inverse temperature after at $\mathrm{T\bar{T}}$ perturbation in terms of the bare inverse temperature by making use of Lambert's $W$ function.

DOI
10.48550/arxiv.2405.06564

Related papers

Generalised hydrodynamics of $$ \textrm{T}\overline{\textrm{T}} $$-deformed integrable quantum field theories

Riccardo Travaglino, Michele Mazzoni, Olalla A. Castro-Alvaredo

A bstract In this paper we evaluate the averages of conserved densities and currents associated to charges of generic spin in (1+1)-dimensional massive integrable Quantum Field Theories perturbed by the irrelevant $$ \textrm{T}\overline{\textrm{T}} $$ T T ¯ operator. By making use of the Thermodynamic Bethe Ansatz approach and of the theory of Generalised Hydrodynamics, we study the non-equilibrium steady state averages of conserved densities and currents in a partitioning protocol. We show that, in particular limits, averages can be evaluated exactly in terms of quantities known from the unperturbed theory. In the massless limit we recover known results for the energy and momentum currents and generalise those to any higher spin conserved quantities. We extend some of our results to perturbations of the generalised $$ \textrm{T}\overline{\textrm{T}} $$ T T ¯ type. For the massive free fermion theory, we find an analytic expression for the effective inverse temperature after at $$ \textrm{T}\overline{\textrm{T}} $$ T T ¯ perturbation in terms of the bare inverse temperature by making use of Lambert’s W function.

Journal of High Energy Physics•Aug 13, 2024•2 citations

Expectation values of conserved charges in integrable quantum field theories out of thermal equilibrium

Michele Mazzoni, Riccardo Travaglino, Olalla A. Castro-Alvaredo

Abstract In this work we present a computation of the averages of conserved charge densities and currents of (1+1)-dimensional Integrable Quantum Field Theories in generalised Gibbs ensembles. Our approach is based on the quasi-particle description provided by the Thermodynamic Bethe Ansatz combined with the principles of Generalised Hydrodynamics, and we focus on Non-Equilibrium Steady State averages. When considering the ultraviolet (i.e. high temperature) limit of such averages, we recover the famous result by Bernard and Doyon (2012) for the energy current and density in Conformal Field Theories, and we extend it to conserved quantities with spin s > 1. We show that their averages are proportional to T L s + 1 ± T R s + 1 , with TL , TR the temperatures of two asymptotic thermal reservoirs. The same power law is obtained when considering some non-thermal generalised Gibbs states. In Conformal Field Theory, the power law is a consequence of the transformation properties of conserved charge operators, while the proportionality coefficient depends on the spin of the operator and on the central charge of the theory. We present an exact analytic expression for this coefficient in the case of a massive free fermion. At equilibrium, proportionality of spin-s density averages to T s + 1 can be thought of as a generalisation of Stefan–Boltzmann’s law, which states that the energy per unit surface area radiated by a black body scales as T 4.

arXiv:2402.14788•Journal of Physics A Mathematical and Theoretical•Feb 24, 2025•1 citation

Completing the Bootstrap Program for $\mathrm{T}\bar{\mathrm{T}}$-Deformed Massive Integrable Quantum Field Theories

Olalla A. Castro-Alvaredo, Stefano Negro, Fabio Sailis

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 $\mathrm{T}\bar{\mathrm{T}}$ - 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 $\mathrm{T}\bar{\mathrm{T}}$-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 $S$-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.

arXiv:2305.17068•May 26, 2023•1 citation

Hydrodynamics of massless integrable RG flows and a non-equilibrium c-theorem

D. X. Horváth

Abstract We study Euler scale hydrodynamics of massless integrable quantum field theories interpolating between two non-trivial renormalisation group fixed points after inhomogeneous quantum quenches. Using a partitioning protocol with left and right initial thermal states and the recently developed framework of generalised hydrodynamics, we focus on current and density profiles for the energy and momentum as a function of ξ = x/t, where both x and t are sent to infinity. Studying the first few members of the A n and D n massless flows we carry out a systematic treatment of these series and generalise our results to other unitary massless models. In our analysis we find that the profiles exhibit extended plateaux and that non-trivial bounds exist for the energy and momentum densities and currents in the non-equilibrium stationary state, i.e. when ξ = 0. To quantify the magnitude of currents and densities, dynamical central charges are defined and it is shown that the dynamical central charge for the energy current satisfies a certain monotonicity property. We discuss the connection of the Landauer-Büttiker formalism of transport with our results and show that this picture can account for some of the bounds for the currents and for the monotonicity of the dynamical central charge. These properties are shown to be present not only in massless flows but also in the massive sinh-Gordon model suggesting their general validity and the correctness of the Landauer-Büttiker interpretation of transport in integrable field theories. Our results thus imply the existence of a non-equilibrium c-theorem as well, at least in integrable models. Finally we also study the interesting low energy behaviour of the A2 model that corresponds to the massless flow from the tricritical to the critical Ising field theory.

arXiv:1905.08590•Journal of High Energy Physics•Oct 1, 2019•13 citations

The Space of Integrable Systems from Generalised $T\bar{T}$-Deformations

Benjamin Doyon, Joseph Durnin, Takato Yoshimura

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.

arXiv:2105.03326•SciPost Physics•Sep 28, 2022•22 citations

Completing the bootstrap program for T T ― -deformed massive integrable quantum field theories

Olalla A. Castro-Alvaredo, Stefano Negro, Fabio Sailis

Abstract 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 T T ― —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 bootstrap program. In this paper we present a systematic approach to computing matrix elements of operators in generalised T T ― -perturbed models, based on employing the standard form factor program. We show that for theories with diagonal scattering and certain types of fields the deformed form factors, 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.

Journal of Physics A Mathematical and Theoretical•Jun 3, 2024•6 citations
Generalised Hydrodynamics of TTˉ\mathrm{T\bar{T}}TTˉ-Deformed Integrable Quantum Field Theories
In this paper we evaluate the averages of conserved densities and currents associated to charges of generic spin in (1+1)-dimensional massive integrable Quantum Field Theories perturbed by the irrelevant TTˉ\mathrm{T\bar{T}}TTˉ operator. By making use of the Thermodynamic Bethe Ansatz approach and of the theory of Generalised Hydrodynamics, we study the non-equilibrium steady state averages of conserved densities and currents in a partitioning protocol. We show that in particular limits, averages can be evaluated exactly in terms of quantities known from the unperturbed theory. In the massless limit we recover known results for the energy and momentum currents and generalise those to any higher spin conserved quantities. We extend some of our results to perturbations of the generalised TTˉ\mathrm{T\bar{T}}TTˉ type. For the massive free fermion theory, we find an analytic expression for the effective inverse temperature after at TTˉ\mathrm{T\bar{T}}TTˉ perturbation in terms of the bare inverse temperature by making use of Lambert's WWW function.
Generalised hydrodynamics of TT‾ \textrm{T}\overline{\textrm{T}} TT-deformed integrable quantum field theories
A bstract In this paper we evaluate the averages of conserved densities and currents associated to charges of generic spin in (1+1)-dimensional massive integrable Quantum Field Theories perturbed by the irrelevant TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ operator. By making use of the Thermodynamic Bethe Ansatz approach and of the theory of Generalised Hydrodynamics, we study the non-equilibrium steady state averages of conserved densities and currents in a partitioning protocol. We show that, in particular limits, averages can be evaluated exactly in terms of quantities known from the unperturbed theory. In the massless limit we recover known results for the energy and momentum currents and generalise those to any higher spin conserved quantities. We extend some of our results to perturbations of the generalised TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ type. For the massive free fermion theory, we find an analytic expression for the effective inverse temperature after at TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ perturbation in terms of the bare inverse temperature by making use of Lambert’s W function.
Abstract In this work we present a computation of the averages of conserved charge densities and currents of (1+1)-dimensional Integrable Quantum Field Theories in generalised Gibbs ensembles. Our approach is based on the quasi-particle description provided by the Thermodynamic Bethe Ansatz combined with the principles of Generalised Hydrodynamics, and we focus on Non-Equilibrium Steady State averages. When considering the ultraviolet (i.e. high temperature) limit of such averages, we recover the famous result by Bernard and Doyon (2012) for the energy current and density in Conformal Field Theories, and we extend it to conserved quantities with spin s > 1. We show that their averages are proportional to T L s + 1 ± T R s + 1 , with TL , TR the temperatures of two asymptotic thermal reservoirs. The same power law is obtained when considering some non-thermal generalised Gibbs states. In Conformal Field Theory, the power law is a consequence of the transformation properties of conserved charge operators, while the proportionality coefficient depends on the spin of the operator and on the central charge of the theory. We present an exact analytic expression for this coefficient in the case of a massive free fermion. At equilibrium, proportionality of spin-s density averages to T s + 1 can be thought of as a generalisation of Stefan–Boltzmann’s law, which states that the energy per unit surface area radiated by a black body scales as T 4.
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.
Abstract We study Euler scale hydrodynamics of massless integrable quantum field theories interpolating between two non-trivial renormalisation group fixed points after inhomogeneous quantum quenches. Using a partitioning protocol with left and right initial thermal states and the recently developed framework of generalised hydrodynamics, we focus on current and density profiles for the energy and momentum as a function of ξ = x/t, where both x and t are sent to infinity. Studying the first few members of the A n and D n massless flows we carry out a systematic treatment of these series and generalise our results to other unitary massless models. In our analysis we find that the profiles exhibit extended plateaux and that non-trivial bounds exist for the energy and momentum densities and currents in the non-equilibrium stationary state, i.e. when ξ = 0. To quantify the magnitude of currents and densities, dynamical central charges are defined and it is shown that the dynamical central charge for the energy current satisfies a certain monotonicity property. We discuss the connection of the Landauer-Büttiker formalism of transport with our results and show that this picture can account for some of the bounds for the currents and for the monotonicity of the dynamical central charge. These properties are shown to be present not only in massless flows but also in the massive sinh-Gordon model suggesting their general validity and the correctness of the Landauer-Büttiker interpretation of transport in integrable field theories. Our results thus imply the existence of a non-equilibrium c-theorem as well, at least in integrable models. Finally we also study the interesting low energy behaviour of the A2 model that corresponds to the massless flow from the tricritical to the critical Ising field theory.
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.
Abstract 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 T T ― —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 bootstrap program. In this paper we present a systematic approach to computing matrix elements of operators in generalised T T ― -perturbed models, based on employing the standard form factor program. We show that for theories with diagonal scattering and certain types of fields the deformed form factors, 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.