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.
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.
Using the theory of generalized hydrodynamics (GHD), we derive exact Euler-scale dynamical two-point correlation functions of conserved densities and currents in inhomogeneous, non-stationary states of many-body integrable systems with weak space-time variations. This extends previous works to inhomogeneous and non-stationary situations. Using GHD projection operators, we further derive formulae for Euler-scale two-point functions of arbitrary local fields, purely from the data of their homogeneous one-point functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuation-dissipation principle along with the exact solution by characteristics of GHD, and gives a recursive procedure able to generate n n -point correlation functions. Owing to the universality of GHD, the results are expected to apply to quantum and classical integrable field theory such as the sinh-Gordon model and the Lieb-Liniger model, spin chains such as the XXZ and Hubbard models, and solvable classical gases such as the hard rod gas and soliton gases. In particular, we find Leclair-Mussardo-type infinite form-factor series in integrable quantum field theory, and exact Euler-scale two-point functions of exponential fields in the sinh-Gordon model and of powers of the density field in the Lieb-Liniger model. We also analyse correlations in the partitioning protocol, extract large-time asymptotics, and, in free models, derive all Euler-scale n n -point functions.
Generalised hydrodynamics of 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 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 T T ¯ type. For the massive free fermion theory, we find an analytic expression for the effective inverse temperature after at TT T T ¯ perturbation in terms of the bare inverse temperature by making use of Lambert’s W function.
Abstract The staircase model (SM) is a simple generalization of the sinh-Gordon model, obtained by complexifying the coupling constant. This produces a new theory with many interesting features. Chief among them is the fact that scaling functions such as Zamolodchikov’s c -function display ‘roaming’ behaviour, that is, they visit the vicinity of an infinite number of conformal fixed points, the unitary minimal models of conformal field theory. This rich structure also makes the model an interesting candidate for study using the generalized hydrodynamic approach to integrable quantum field theory (IQFT). By studying hydrodynamic quantities such as the effective velocities of quasiparticles we can develop a more physical picture of interaction in the theory, both at and away from equilibrium. Indeed, we find that in the SM the effective velocity displays monotonicity features not found in any other IQFT. These admit a natural interpretation when investigated in the context of the massless theories known as M A k ( + ) models, which provide an effective description of massless flows between consecutive unitary minimal models. Consequently, we find that the effective velocity in the SM can be reconstructed by a ‘cut and paste’ procedure involving the equivalent functions associated to the massless excitations of suitable M A k ( + ) models. We also investigate the average currents and densities of higher spin conserved quantities in the partitioning protocol.
At thermal equilibrium, the concept of effective central charge for massive deformations of two-dimensional conformal field theories (CFT) is well understood, and can be defined by comparing the partition function of the massive to that of a CFT. This temperature-dependent effective charge interpolates monotonically between the central charge values corresponding to the IR and UV fixed points at low and high temperatures, respectively. We propose a non-equilibrium, time-dependent generalization of the effective central charge for integrable models after a quantum quench, ceff(t), obtained by comparing the return amplitude to that of a CFT quench. We study this proposal for a large mass quench of a free boson, where the charge is seen to interpolate between ceff=0 at t=0, and ceff∼1 at t→∞, as is expected. We use our effective charge to define an to quench protocol, where the charge evolves from ceff=1/2 at t=0, to ceff=7/10 at t→∞, the corresponding values of the first two unitary minimal CFT models. We then argue that the inverse Tricritical Ising to quench is impossible with our methods. These conclusions can be generalized for quenches between any two adjacent unitary minimal CFT models. We finally study a large mass quench into the model (sinh-Gordon with a particular complex coupling). At short times after the quench, the effective central charge increases in a discrete staircase structure, where the values of the charge at the steps can be computed in terms of the central charges of unitary minimal CFT models. When the initial state is a pure state, one always finds that ceff(t→∞)≥ceff(t=0), though ceff(t), generally oscillates at finite times. We explore how this constraint may be related to RG flow irreversibility.
Generalised Hydrodynamics of 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 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 type. For the massive free fermion theory, we find an analytic expression for the effective inverse temperature after at TTˉ perturbation in terms of the bare inverse temperature by making use of Lambert's W function.
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures Tl, Tr, and waiting for a long time. We evaluate the current J(Tl,Tr) using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov's method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium c-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the "additivity" property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl,Tr) is not of the form f(Tl)−f(Tr).