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

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 \xi = x=t, where both x and t are sent to infinity. Studying the first few members of the An and Dn 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 \xi = 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-Buttiker 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-Buttiker 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.


Introduction
Understanding the out-of-equilibrium dynamics of isolated quantum many-body systems and giving a rigorous foundation of quantum statistical mechanics are one of the most challenging problems in contemporary physics. Thanks to the recent advances in laboratory techniques [1][2][3][4][5][6][7][8][9] the experimental realisability of closed quantum systems provides a direct insight into quantum statistical physics. As a result of intensive experimental and theoretical investigations, significant progress has been made in the study of non-equilibrium behaviour and recent investigations have led to a series of interesting discoveries. Perhaps the most unusual behaviour is related to integrable quantum systems and the experimental observation of the lack of thermalisation therein [1][2][3]10]. Because of their unusual properties, their experimental relevance and analytic tractability, non-equilibrium physics in integrable models continues to attract a lot of attention.
A paradigmatic setting for non-equilibrium dynamics is provided by so-called quantum quenches corresponding to a sudden change in the parameters of a closed quantum system [11,12]. One of the simplest example is that of a homogeneous system where time evolution after a quantum quench is expected to lead to relaxation to thermal equilibrium. However, in the case of integrable system thermalisation is absent and the steady state was proposed to be described by the generalised Gibbs ensemble (GGE) [13] which is supported by experimental and theoretical studies [4,[14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Nevertheless, important issues such as a theoretical description of the eventual time evolution, as well as the complete set of relevant conserved quantities necessary for the construction of the steady state ensemble are still open in general.
Inhomogeneous quenches, in general, pose a more difficult problem than homogeneous ones with fewer exact or approximate results [28][29][30][31][32][33][34][35][36]. One promising approach exploits hydrodynamical description assuming the separation of spatial and temporal scales and a local (GGE) equilibrium, and is supported both by numerical simulations and experimental observations [37][38][39][40][41]. Based on the previously mentioned assumptions and on the functional completeness of conserved quantities a hydrodynamical description of integrable systems named Generalised Hydrodynamics (GHD) was developed in [42][43][44]. In its simplest form the GHD describes the exact average densities and currents associated with conserved quantities at the Euler scale, that is when x, t → ∞ such that their ratio is fixed. The appearance of non-trivial physics in this limit is in accordance with the ballistic spreading of quasi-particles in integrable models and in many relevant situations the GHD predictions become valid after a relatively short transient time interval [37][38][39][40]. The GHD approach has been applied to various systems including spin chains and the Hubbard model [43,[45][46][47][48][49][50][51][52], classical gases and fields [53][54][55][56] and quantum gases and fields [38,42,57,58]. Interesting view points on the GHD approach are given in [59][60][61]. Besides the Euler scale description of current and density averages in various integrable models important new directions are understanding the fluctuations of the hydrodynamic quantities [62,63] and incorporating diffusive effects which requires going beyond the Euler scale in the GHD description [64][65][66].
The aim of the present paper is to introduce a renormalisation group perspective and connect the GHD description of the non-equilibrium dynamics of integrable quantum field theories to their formulation as relevant perturbations of conformal field theories (CFT) [67]. In the works [68][69][70][71][72][73][74] a rather extensive description of transport properties of outof-equilibrium CFTs was given. The non-equilibrium setting was provided by the (bi-) partitioning protocol, in which two semi-infinite and independent systems described by the same Hamiltonian are prepared in different initial states, typically in two thermal states with different temperatures. The left and right semi-infinite systems are joined together at a given time and the subsequent time-evolution is governed by the full and homogeneous Hamiltonian acting on the whole space. In the resulting dynamics for CFTs, two shock waves originating from the boundary point move with the speed of light to the left and right directions and in the expanding region between the waves a non-equilibrium steadystate (NESS) emerges. The NESS is obtained in the limit t → ∞ for finite x and therefore corresponds to ξ = 0, supporting non-trivial flows of currents while the asymptotic left (ξ < 0) and right (ξ > 0) regions act as effective heat reservoirs. In [69][70][71][72][73] the properties of the NESS, current and density averages, fluctuation spectrums for large deviations and correlation functions in the NESS were studied. In particular for CFTs the average energy density q e and energy current j e are given by where T l and T r are the initial left and right temperatures and c is the central charge of the CFT. Due to relativistic invariance j e equals the momentum density q p and due to conformal invariance q e equals the momentum current j p (and also the pressure). These results are also valid for systems at or sufficiently close to their critical points but a certain neighbourhood of the critical point or more precisely a certain type of irrelevant perturbation of the fixed point CFT was also studied in [72]. In general, the vicinity of critical points are described by (relevant) perturbations of the critical theory resulting in either a massive quantum field theory (QFT) or a crossover to an infrared (long-distance) fixed point. The high energy behaviour and consequently the high temperature transport properties of such a system are captured by the fixed point CFT, but the intermediate and low energy physics is generally not known. If the perturbed CFT is integrable, which is quite often the case of interest, the GHD approach can be used to explore the transport properties at all scales of the field theory.
Some relevant perturbations of CFTs give rise to massless QFTs, which host massless excitations. For these interpolating renormalisation group flows describing crossover behaviour the high-and low-energy properties are dictated by CFTs. The massless models are not conformal invariant theories and transition from the ultra-violet (UV) CFT to the infra-red (IR) occurs at the characteristic energy scale M called the crossover scale.
Integrable massless flows have a long history and many of them have a detailed description based on the standard techniques of integrability such as S-matrix bootstrap, thermodynamic Bethe Ansatz (TBA) and form factor bootstrap. In this paper we primarily focus on the so-called A n and D n massless flows. The A n models interpolate between multi-critical Ising field theories [75][76][77] corresponding to the conformal minimal models M n+2 → M n+1 whereas the D n flows describe the crossover between the Z n parafermion model and M n+1 [78]. The S-matrices of these models and the corresponding TBA equations are well-known [75][76][77][78][79] together with certain form factors [80][81][82] and correlation functions [80] for the A 2 case, and it is worth noting the interesting connection of these flows to roaming trajectories and staircase models [83][84][85][86][87].
As these theories interpolate between a UV and an IR conformal field theory, it is natural to investigate the crossover in terms of transport behaviour. This is what the present work aims to accomplish by applying the GHD approach to the partitioning protocol and focusing on the interpolation between the behaviour of the fixed point CFTs describing the endpoints of the RG flow.
The paper is organised as follows. In section 2 the description of homogeneous and inhomogeneous macro-states in terms of the TBA type equations is discussed for integrable models and the main ideas of the GHD approach are summarised. In section 3 some elements of the thermodynamics of the A n and D n flows are discussed that are useful to understand the inhomogeneous problem. Section 4 is devoted to hydrodynamics of the A 2 model. Besides reviewing its interesting low temperature limit, we identify features that are later shown to be generally present in massless models and introduce the notion of dynamical central charges. In section 5 the discussion is generalised to higher members of the A n and D n hierarchy of massless flows. In section 6 we establish a connection with the Landauer-Büttiker (LB) theory of transport and discuss its implications for the dynamical central charges in RG flows. In this section the transport of massive sinh-Gordon integrable QFT is also studied and compared to the LB results, and we show that all the results suggest an out-of-equilibrium version of the c-theorem [88,89], at least for relativistic integrable QFTs. Finally we conclude in section 7. Some details are relegated to appendices: in appendix A, the low-temperature expansion of the TBA and GHD equations are performed for the A 2 model, while appendix B discusses the hydrodynamics of the W 3 5 → W 3 4 massless flow.
2 Macro-states in integrable models and the GHD In this section we briefly review how macro-states such as homogeneous thermal or inhomogeneous locally quasi-stationary states (LQSS) can be characterised in integrable models and discuss the main ideas behind the GHD approach. For most integrable models, the spectrum of energy levels can be described in terms of stable quasi-particles excitations. Averages of local operators in macro states can be constructed by representing the corresponding density matrices in terms of continuous densities associated with the distribution of these stable quasi-particles. In particular the so-called root density ρ(ϑ) gives the number of quasiparticles in the range [ϑ, ϑ + ∆ϑ], which is Lρ(ϑ)∆ϑ, if L is the system size and the rapidity ϑ parametrizes the energy and momentum of the quasi-particles. In the case of additional quantum numbers or different species of quasi-particles, multiple root densities labeled by the appropriate quantum numbers are required.
Similarly to the root densities one can introduce the hole densities ρ h (ϑ) as well, which are associated with unoccupied one-particle energy levels. The root and hole densities are not independent quantities due to the interaction between the quasiparticles. Extending our description to models with n quasiparticle species and with diagonal scattering, the root and hole densities ρ j and ρ j,h corresponding to the different species satisfy the Bethe Ansatz equations In (2.1), p j (ϑ) denotes the derivative of the one-particle momentum of the jth species with respect to the rapidity ϑ, ϕ jk is specified by the two-particle scattering matrix S jk as 2) and the symbol denotes convolution: It is convenient to introduce the filling functions n j (ϑ) and the pseudo-energies ε j (ϑ) via with j = 1, ..., n and also the dressing operation for a set of functions f j (ϑ) with j = 1, ..., n by the solution of the integral equation Given the root and hole densities or the filling functions, the densities of various extensive quantities can be easily calculated in the macro-state. For conserved charges Q i with oneparticle eigenvalue h (i) j (ϑ) with respect to the jth particle species, that is j (ϑ)|ϑ j , the corresponding density q i can be expressed equivalently as Similarly to the quantum mechanical operator q i , which is the density of the charge Q i , the average of the corresponding current j i in the macro-state can be expressed in terms of the filling functions or root densities. Before quoting the expressions we first introduce the effective velocity v eff While the velocity of a single particle can be defined as the dressing in (2.7) accounts for the effect of other quasi-particles in the macro-state specified by the root densities. Due to scattering processes, the single-particle velocity is modified according to (2.7). With the effective velocity, the averages j i of j i can be written as (2.8) The expressions were first proposed in [42] and later verified for relativistic QFT in [90]. These equations are in accordance with the ballistic transport of conserved quantities in integrable models. Finally, we mention that the entropy density s of the macro-state can be written as (2.9)

Thermodynamic Bethe Ansatz description for thermal and GGE states
The root densities or the filling functions of homogeneous and global thermal and GGE states can be obtained by solving the Thermodynamic Bethe Ansatz (TBA) equations where the driving term reads if the state to describe is a GGE state with density matrix with generalised chemical potentials β i associated with each conserved charge. For the particular case of thermal states, w j is merely where T is the temperature. The corresponding free-energy density or generalised free- (2.14) From the thermal free energy density the effective central charge can be obtained, which plays an important role in our subsequent considerations. Due to the c-theorem [89],c(T ) increases monotonously with the temperature 1 and signals the amount of the effective degrees of freedom in the field theory. In the T → ∞ limit its value is determined by the UV limiting conformal field theory, in particular where c U V is the central charge of the UV conformal field theory if it is unitary. In the T → 0 limit,c is zero in massive models but equals the central charge of the IR conformal field theory for massless flows lim T →0c (T ) = 0 massive case c IR massless case . (2.17)

GHD and the partitioning protocol
The main purpose of our paper is to study transport properties of massless integrable models in inhomogeneous initial states. To treat inhomogeneous situations, it is convenient to apply a hydrodynamic approach relying on the separation of space and time scales and on the assumption of local equilibration. The large-scale behaviour of inhomogeneous systems can be described by a space-time dependent GGE and consequently the large-scale expectation values of local operators can be obtained as where Similarly to the homogeneous case, (x, t)-dependent root densities ρ j (ϑ, x, t) and filling functions n j (ϑ, x, t) can be introduced to describe the LQSS. Exploiting eqs. (2.19) and (2.20), the continuity equation of the conserved quantities ∂ t q i + ∂ x j i = 0 transforms into where 1 In its most commonly formulated version, the effective central chargec entering the c-theorem is a function of the distance R instead of the temperature T .
In integrable models, assuming a sufficient functional completeness of the conserved charges, the continuity equation for the LQSS averages (2.21) can be recast in many different forms including the space-time dependent root densities or filling functions. For our purposes, the most direct rewriting reads [42] ∂ t n j (ϑ, where v eff j is defined by eq. (2.7) and the argument in the bracket stresses that v eff j is a complicated functional of the set of filling functions n j which now depend on space and time besides the rapidity. The equation (2.23) is in complete agreement with the ballistic spreading of quasi-particles in integrable models. The effect of the interactions is incorporated in the effective velocity of the quasi-particles. For the case of the partitioning protocol corresponding to an initial density matrix which is different on the left and right halves of the system, eq. (2.23) can be solved in a particularly simple way. For the particular case of thermal states considered in this paper, the initial density matrix reads where the Hamiltonians H l and H r act in the left and right half-spaces. The two halves are joined together at time t = 0 and subsequent time evolution is governed by the homogeneous Hamiltonian acting on the whole space. To be precise, some boundary conditions have to be prescribed for H r and H l at the position x = 0 before t = 0; nevertheless, it is expected that their effect becomes negligible at the Euler scale. Consequently, the initial condition for n j (ϑ, x, 0) can be written as where Θ H is the Heaviside function and n (r) j and n (l) j are the filling functions corresponding to the right and left density matrices describing homogeneous thermal (or GGE in general) states with temperatures T r and T l . To obtain a solution of (2.23) with an initial condition for n j compatible with (2.26) it is exploited that both the differential equations (2.23) and the initial condition (2.26) are invariant under the the reparametrisation x, t → λx, λt. As a consequence the solution of (2.23) depends only on the ratio x/t which we denote by ξ and call a ray. The corresponding ray-dependent continuity equations read and and the implicit solution of the latter is given by Its interpretation in terms of the ballistic spreading of quasi-particles is natural; quasiparticles that contribute at the ray ξ come from the left if their effective velocity is larger than ξ and they come from the right side if their effective velocity is slower than ξ; the effective velocity depends on all the other particles due to the elastic scattering between them. The self-consistent numerical solution of the above equation is usually straightforward to obtain by iteration. In particular when v eff j (ϑ, ξ) is a monotonously increasing function of ϑ for all ξ, the solution can be rewritten as for all j and v eff j , which is a functional of all n j (ϑ, ξ), is determined by (2.7). As discussed in section 3, the effective velocities of magnonic particles can be monotonously decreasing or non-monotonous functions of the rapidity too. In such a case (2.30) may include more terms, and more than a singe ϑ j is necessary to use to describe the jumps in the filling functions. The values of the ϑ (m) j parameters are still to be determined by (2.7) for all m in a self-consistent manner.
In massless models the effective velocities of the quasi-particles (either the later introduced right-and left-moving particles or the magnons are regarded) usually do not cover the full In such a case the solution of (2.28) for a ray ξ ∈ [−1, v min j ] is n (l) j (ϑ) since there are only faster than ξ particles, which must come from the left and similarly for a ray ξ ∈ and n (r) j (ϑ) are the filling functions of the homogeneous left/right thermal states or GGE). This situation is similar to what was discussed in [91], which focused on the transport of non-linear Luttinger liquids.
Once n j (ϑ, ξ) are determined, the ray-dependent averages of the densities and currents of conserved charges can be straightforwardly calculated using (2.5), (2.6) and (2.8).

Thermodynamics of the A n and D n massless flows
In this section we review on the finite temperature description of integrable massless flows using the TBA. The finite temperature filling functions are essential inputs for the hydrodynamics of the partitioning protocol, moreover many peculiar features of the emerging hydrodynamics can be understood by analysing the homogeneous, finite temperature case.
The TBA equations for massless flows associated with the A n and D n series can be written [79] as G jk is the adjacency matrix of the A n and D n Dynkin diagrams (c.f. figure 1) and the source terms read The nodes in the diagrams correspond to different particle species. The scattering between these species is non-trivial only for the neighbouring nodes. According to eq. (3.3) only two nodes in the Dynkin diagrams couple to non-vanishing driving terms. The source terms M 2T e ±ϑ correspond to right-and left-moving i.e. (RM) and (LM) particles, whose one-particle energy and momentum are M 2 e ±ϑ and ± M 2 e ±ϑ respectively. The energy scale M is the crossover scale, which separates the low and high energy regimes dominated by the UV and IR limiting CFTs. The other nodes in the Dynkin diagrams correspond to magnons, which describe internal degrees of freedom of the quasi-particle excitations. It is important to note that although the magnonic excitations themselves may be regarded as quasi-particles, they have zero one-particle eigenvalues with respect to the conserved charge operators, hence quantities such as the energy or momentum are carried by only the RM and LM species.
The eventual identification of the models described by the TBA equations is a nontrivial task. The A n flows interpolate between multi-critical Ising field theories [75][76][77] according to M n+2 → M n+1 with .
These integrable RG trajectories are the φ 1,3 perturbations with scaling dimension ∆ = 1 − 2 n+3 of the UV conformal theory M n+2 [75][76][77]. For the A n massless models, both c U V and c IR tends to 1 as n → ∞ and the difference between the UV and IR central charges vanishes in this limit.
The D n flows describe the crossover between the Z n parafermion model and M n+1 [78] with and are obtained by adding the perturbing operator For thermal states the filling function of the RM and LM particles are of a very peculiar form in all massless integrable models: they are kinks related by n RM (ϑ) = n LM (−ϑ). Focusing on n RM for ϑ → −∞ its value is a constant and for ϑ → ∞ it goes to zero as exp (−M (exp ϑ)/(2T )). This behaviour is illustrated on figure 2 for the case of the D 4 massless model.  Although the bare velocities of the RM and LM particles are ±1 in units of the speed of light, in highly-excited states such as thermal states the effective velocities are different from ±1 due to interactions which results in non-trivial kinetics in the GHD setting. The TBA description of the D 4 flow also includes two magnonic particles, whose effective velocities can even be non-monotonous functions of the rapidity as demonstrated by figure 3. Finally, to conclude this section subfigure (b) in figure 3 shows the c-function as a function − log T /M for the D 4 massless model.

Hydrodynamics of the tricritical to critical Ising flow
In this section we focus on the Euler scale hydrodynamics of the A 2 flow after a bipartite quench. The A 2 model, whose UV and IR limiting theories are the tricritical and critical Ising CFTs [75,76], is the simplest massless flow with a RM and LM particle in the TBA equations. By solving first the corresponding TBA equations (3.1) with (3.3) for the left and right filling functions and then the final form of the GHD equations (2.30) and (2.31), the ray-dependent density and current profiles are easy to obtain from eqs. (2.6) and (2.8). We start our analysis by first discussing some peculiar features of these profiles such as the existence of extended plateaux and the bounds on the currents and densities which turn out to be quite generic for all massless models studied in this work. Figure 4 shows the ray-dependent energy and momentum densities and currents and one peculiar feature of these quantities are the extended plateaux in the currents and densities. As discussed in the introduction, the Euler scale behaviour of these quantities in CFTs are described by exactly flat plateaux and discontinuous jumps at ξ = ±1. Since both at low and high energies, i.e. low and high temperatures the limiting theories of massless models are CFTs, one should not be surprised about the emergence of similar plateaux in the massless flows. It is slightly more interesting that the plateaux still exist outside the conformal regimes where the left and right temperatures are close to the inverse of the crossover scale M −1 , corresponding to the massless flow being far away from the conformal limits. The existence of the plateaux in this regime can be understood from the TBA equations using that in thermal states the filling functions n RM and n LM are kinks. Whereas these kinks possess some structure, they can be roughly regarded as Heaviside theta functions. The transition from a non-zero value to zero takes place at a temperature dependent rapidity, which we can denote as ϑ  .3)). In the bipartite quench protocol the left and right filling functions are joined together according to (2.30) and the transition from the right filling function to the left in n RM (ϑ, ξ) = n RM (ϑ, ϑ RM ) and n LM (ϑ, ξ) = n LM (ϑ, ϑ LM ) takes place at a rapidity ϑ RM and ϑ LM respectively. When the rapidities ϑ RM and ϑ LM sweep through the real interval and the filling functions are approximated by theta functions, n RM (ϑ, ϑ RM ) and n LM (ϑ, ϑ LM ) are either the left or right filling functions apart from the case when ϑ RM ∈ [ϑ Our argument seems to be invalid when the difference between the left and right temperatures is large. In such a case, however, the high energy particles almost exclusively originate from the left side and hence the UV limiting CFT dominates the dynamics as supported by table 1 discussed shortly. In summary, irrespectively of the magnitude of the left and right temperatures, the plateaux in the energy and momentum density and current profiles are a generic feature the Euler scale hydrodynamics of the A 2 massless flow. As demonstrated in the next section, the extended plateaux also occur in other massless integrable theories. To discuss a second characteristic feature, we first define a dynamical central charge inspired by (1.1) by writing

Plateaux in the profiles and the dynamical central charges
The dynamical central chargec je (together with its counterparts for the cumulants of the transferred energy) was already defined and used in [92] for non-equilibrium situations. Calculating the energy current (or momentum density) at the ray ξ = 0 ,c je can easily be obtained knowing the left and right temperatures. Whereas in CFTs the energy current j e (ξ) is constant if ξ ⊂ (−1, 1), for the massless flows j e (ξ) usually exhibits a non-trivial ξ-dependence, therefore we must specify at which ray ξ j e is to take in order to definec je .
It seems natural to choose ξ at which j e is maximal, which occurs at ξ = 0 as a consequence of the continuity equation (2.27). Moreover, taking the limit t → ∞ at any fixed position x gives the NESS, which also corresponds to ξ = 0.
In table 1 the values ofc je are collected for various left and right temperatures together with the effective central charges for the left and right thermal states. Based on the data we propose the following conjecture for the dynamical central charge in bipartite quenches with left and right initial thermal states: This is an interesting property of the dynamical central charge which (4.2) also implies upper and lower bounds on the maximum of the ray-dependent energy current or equivalently the current in the NESS, and an upper bound for the energy current at any ray. In [93] a lower bound for the steady-state energy current was given, which reads in our case where the averages of the momentum current are taken in the left and right thermal states. When a lower bound on j e (0) is considered, this inequality (4.3) turns out to be more restrictive than (4.2) in almost all cases, which is demonstrated by to compare the dynamical central charge with. A trivial consequence of (4.2) is the bound where the bounds are now independent of the temperatures, and the lower bound is less strict than the one obtained in [93]. This means that a simple estimate for j e (0) is given by and hence the the whole Euler scale energy current is always bounded by πc U V 12 T 2 l − T 2 r . The dynamical central chargec je has another nice property; similarly to the effective central charge, this quantity is a monotonously increasing function of the energy scale. In more precise terms (4.8) A monotonic behaviour ofc je was first pointed out in [92]. However, the validity of the approach used to determine the NESS in that particular work was not justified by later comparison with the GHD [42], although the difference between the numerical values was small. Our analysis relying on the GHD approach now gives solid evidence for the monotonicity ofc je in terms of (4.7) and (4.8).
Based on eq. (1.1), nevertheless, an effective central charge can be defined not only from j e , but also from q e and j p , which are not equal due to the lack of exact conformal symmetry in massless models. Again, it seems reasonable to chose their value at the ray ξ = 0, which corresponds to the NESS, but it is important to keep in mind that for these quantities their maximum does not occur at ξ = 0. Defining the corresponding dynamical central charges denoted them byc qe andc jp their numerical values can be determined as seen in table 1. As indicated by table 1 similar observation can be made to the case ofc je , which we summarise as follows: forc jp c IR ≤c jp (T l , T r ) ≤c je (T l , T r ) (4.9) holds, from which it also follows that On the contrary forc qe we find that c IR ≤c(max (T l , T r )) ≤c qe (T l , T r ) c U V . In this formula the symbol before c U V seems very unnatural first. Even though the problematic numerical values forc qe (i.e. those slightly large than 0.7) seem to be stable against varying the parameters in the numerical solutions (such as the number of iterations or discretisation points etc.) we cannot exclude the possibility of numerical errors and therefore leave this problem open. Nevertheless it is important to remember that q e has no global extremum at ξ = 0 and in the rest of the paper we present an interesting argument stating that unlike for currents such an anomalous behaviour is not prohibited by physical principles for densities (apart from the T (l) , T (r) → 0 and T (l) , T (r) → ∞ limits where the CFT description becomes exact).
In the next section it is also demonstrated that the conjectures (4.2), (4.9) and (4.11) and the monotonicity property (4.7), (4.8) forc je remain valid for the other flows of the A n and D n family together with the extended plateaux in the current and density profiles. . The appearance of the flat regions in the density and current profiles is related the fact that at not too high temperatures these intervals do not overlap. This means that, according to our discussion at the end of section 2.2, which are independent of ξ and consequently the current and density profiles have no always holds. At very low temperatures it is also possible to treat the problem by analytical means at least in two different ways. An obvious approach is the low temperature expansion of the thermal TBA equations first. If one is interested in the flat regions of the profiles the values of the densities and currents are straightforwardly obtained using again a low-temperature expansion in (2.6), (2.8) and (2.5) since (4.12) can be exploited.
Another approach is describing the low energy behaviour of the A 2 massless flow as an irrelevant perturbation of the IR CFT by´d 2 x TT [94,95]. In particular for the A 2 , model the low-energy effective Lagrangian reads [80,94] L ef f = ψ∂ψ +ψ∂ψ − 4 M 2 (ψ∂ψ) ψ∂ψ + . . . , (4.13) where ψ∂ψ +ψ∂ψ is the Lagrangian of the critical Ising field theory with c = 1 2 and (4.14) The bipartite quench with left and right thermal heat reservoirs was analysed in [72] for perturbed CFTs with the action In particular, the average of the energy and momentum densities and currents are modified by the perturbation [72] from eq. (1.1) to (4.17) Substituting − 16 M 2 for g according to eqs. (4.16) and (4.17) and trading the small parameter to T 2 , the results obtained describe the low temperature Euler scale behaviour of the A 2 flow: An alternative derivation of these results is given in appendix A by performing the low temperature expansion of the TBA equations (2.5), (2.6), (2.8) and (2.10). Finally, it is important to note that for the particular case of the A 2 flow no shocks develop due to integrability contrary to the general case of the TT perturbation [72].

Hydrodynamics of the A n and D n flows
Now we turn to studying higher members of the A n and also D n type massless models, which correspond to the Dynkin diagrams in figure 1.
In figure 5 the density and current profiles for the A 3 flow are displayed for various left and right initial temperatures. The model can be regarded as the RG trajectory from the tetra-critical to the tricritical Ising field theory with c U V = 0.8 and c IR = 0.7, and for this model the TBA equations also include a magnonic particle. Figure 5 shows a behaviour similar to the case of the A 2 flow: irrespectively of the left and right temperatures the profiles include extended plateaux and in fact the flat regions are even broader than in the previous case. For the A 2 model an explanation for the plateaux was given based on the high-and low-temperature conformal behaviour and also on the qualitative behaviour of GHD and TBA equations. To give an explanation for the plateaux for this case and eventually for all higher members of the A n flow it is sufficient to note that the difference between the IR and UV central charges decreases as n increases (3.4). This means that the UV and IR CFTs only slightly differ from each other as long as the number of effective degrees of freedom or the transport properties determined by the central charge are concerned. Thus the properties of the interpolating flow are expected to be similar to the CFT case including the appearance of the plateaux. In this respect, the A 2 case is the one expected to show the largest departure from the CFT behaviour since it is for this case that the difference between the UV and IR central charges is the largest.   It is worth noting that unlike for the A 2 for the A 3 flow (and also for higher members as one expects) the exactly flat constant regions in the profiles do not appear. Although at low temperatures the range of effective velocities for RM and LM particles do not overlap and consequently their filling functions become independent of ξ, the range of the effective velocity for the magnons spans this non-overlapping region and n M (ϑ, ξ) has real raydependence. Even though the contribution to the densities and currents originates from only the RM and LM particles, the ray-dependence of n M (ϑ, ξ) is transmitted to the densities and currents due to the dressing equation (2.5). This accounts for a weak ray-dependence in the profiles in the non-overlapping region in the range of the RM and LM effective velocities.
At sufficiently small temperatures the effective velocities for magnons also do not overlap with that of the RM or LM particles. It is a general observation the even in this case the range of the magnonic effective velocity (at least for one magnonic species) always touches the lower and upper endpoints of the range of the RM and LM effective velocities.  In table 2, the effective and dynamical central charges defined in eq. (4.1) are collected for different left and right temperatures in the A 3 interpolating flow, and the data confirm that the conjecture (4.2) proposed for the A 2 massless flow remains valid in the present case. Therefore it is plausible to assume that together with the appearance of the plateaux in the profile, the conjectures (4.2), (4.9) and (4.11) for the dynamical central charges and the bounds for the currents are a general property of the A n models in the thermal partitioning protocol.
In fact the broad plateaux in the profiles and the conjectures (4.2)-(4.11) seem to be a generic property of not only the A n massless models, but also of the D n series and eventually of all unitary massless integrable flows. This is confirmed by studying the first two members of of the D n series. What concerns the D 3 flow it is to mention that its TBA system is equivalent with that of the A 3 model. Though the operator content and consequently the models themselves are different, the energy and momentum densities and currents are given by the same equations and figure 5 and table 2 describes both the A 3 and D 3 flows. Another notable remark is that the Z 3 parafermion model, i.e. the UV limiting theory of the D 3 model is the critical three-state Potts model.    For the D n series the n → ∞ limit the difference of the UV and IR central charges is now 1 (3.5) and arguing that the UV and IR CFTs are 'similar' is not possible. Nevertheless, the presence of the extended plateaux and validity of the conjectures (4.2)-(4.11) in the D 4 model besides the D 3 case suggest that these features emerge in any member of the D n flows.
Whereas tables 2 and 3 suggest that the approximate inequality in (4.11) is rather a strict one, studying the hydrodynamics of massless perturbation of the W 3 5 CFT (carried out in appendix B) slightly larger than c U V values forc qe can be seen again. Our analysis is therefore not conclusive in the question whether c U V ≥c qe or c U V c qe holds for finite and non-zero left and right temperatures but we revisit this issue in the next section.
T l /M T r /Mc(T l )c(T r )c je (T l , T r )c qe (T l , T r )c jp (T l , T r )c lb je (T l , T r ) 0.

The Landauer-Büttiker picture and its implications on transport
In this section, we would like to discuss a simple physical picture that gives a natural interpretation for the observed bounds for the dynamical central charges and provides a possible explanation for the anomalous behaviour ofc qe . This picture is based on the Landauer-Büttiker (LB) formalism [96][97][98][99] of electronic systems, which claims that in the simplest case the total electric conductance for 1D systems with ballistic electron propagation is the number of open channels for the electrons times the conductance quantum 2e 2 /h. The prototypical case where this formalism is used is a clean quantum wire supporting coherent propagation of electronic excitations attached to a source and a drain with an applied voltage emitting and absorbing electrons according to Fermi-Dirac statistics. It is easy to use the LB formalism to describe thermal transport when the temperatures of the drain and the source are different. For the energy current of relativistic and spinless fermions with mass m elementary calculations give 2) whose m → 0 limit, which is the conformal limit, yields i.e. the CFT result. Whereas this result is obtained for the case of non-interacting quasiparticles and it is not obvious how the LB formalism could be applied for general CFTs, the connection between the LB formalism and the CFT treatment in the free case inspires the following interpretation for the CFT current formula: the central charge simply counts the number of open channels for the energy transport. This statement is in agreement with usual interpretation of the central charge counting the number of effective degrees of freedom in the theory. If one accepts this picture linking the central charge and the number of open channels for the energy transport, the existence of both an upper and a lower bound for the dynamical central chargec je in massless flows comes naturally. The number of effective degrees of freedom in massless models cannot be larger than c U V and smaller than c IR and consequently the number of open channels available for energy transport must be bounded by c U V and c IR too. Clearly, the same argument holds also for the current of the momentum.
Even if the transport is ballistic and an upper bound for the conductance is expected, the accumulation of charge is not prohibited, at least by our simple picture. In our analysis the problematic quantity with respect to an upper bound was the average energy density at ξ = 0, which is a density of a conserved charge and hence is compatible with the previous statements. Due to relativistic invariance j e = q p , which explains why anomalous behaviour could be detected only for q e .
The formulae (6.1) and (6.2) were derived for massive relativistic fermions, which means that c IR equals zero. Analysing the behaviour of the dynamical central chargec je for the energy current (6.1), it is easy to check that it is indeed bounded by 0 and 1/2 in accordance with the above interpretation and that c IR = 0. Due to the exact expression forc je , nevertheless, its monotonicity properties can also be explicitly investigated and (4.7) and (4.8) are satisfied. For this property, which is therefore valid for massless flows as well as relativistic fermions in the LB picture, the interpretation is also at hand. Increasing the temperature in any of the initial subsystems results in higher energy densities and a larger number of open channels available for the energy transport since in unitary theories the number of effective degrees of freedom is increased with increased energy density. Whereas it remains an assumption that the number of open channels for transport is a monotonous function of the number of effective degrees of freedom as it happens for free fermions, thinking in terms of the stable quasi-particles of integrable systems with ballistic propagation makes this link very plausible.
Having seen how the LB picture can explain some of the observed bounds for the dynamical central charges together with the monotonicity ofc je in massless integrable models and massive free fermions, it is natural to speculate about the case of massive integrable and eventually general 1+1D QFTs. For this reason we also analyse the dynamical central charges in the sinh-Gordon model using the GHD and the partitioning protocol again. Without explaining too many details about the model (in connection with GHD this model was investigated in [42] reviewing its definition) we merely quote the corresponding TBA equation containing one single particle species: with ϕ shG (ϑ) = − 4 cosh(ϑ) sin( Bπ 2 ) cos(Bπ) − cosh(2ϑ) , (6.5) where m is particle mass and the parameter B takes values from [0, 2] and is related to the strength of the interaction.    Table 4 shows the data for the effective and dynamical central charges for the sinh-Gordon model and the data are in perfect agreement with the predictions of the LB picture. All the dynamical central charges are bounded by c IR from below and for the currents (and in this case also for q e ) the data are consistent with an upper bound given by c U V , and the monotonicity property ofc je is also satisfied. Moreover, the conjectures (4.2), (4.9) and (4.11) proposed for the massless models also hold in this case together with (4.3), whose validity is independent of the massless or massive nature of the theory.
Although a systematic treatment of the hydrodynamics in massive integrable QFTs is out of the scope of this paper, based on the above results it is plausible to expect that the bounds for the dynamical central charges of currents given by c IR and c U V , the monotonicity ofc je and the conjectures (4.2), (4.9) and (4.11) are valid in any relativistic and unitary integrable QFT and the interpretation inspired by the LB picture is correct. The monotonic behaviour ofc je is an especially interesting finding, since it corresponds to an out-of-equilibrium version of the well-known c-theorem.
In fact, considering a generic unitary and relativistic 1+1 D QFT with homogeneous bulk action it is reasonable to assume that the dynamical central charges of the energy and momentum current are still bounded by c U V from above and c IR from below. In non-integrable cases there are usually no additional conserved quantities to the energy and the momentum and the ray-dependent profiles for the currents and densities of these conserved quantities can exhibit discontinuities at the Euler scale. Nevertheless the NESS corresponding to ξ = 0 at the Euler scale is expected to be well defined and therefore the dynamical central charges can be introduced similarly to the integrable case. Whether the strict monotonicity property ofc je alias a non-equilibrium c-theorem holds in non-integrable cases is a more subtle issue. Whereas the c-function [89] in eq. (2.15) was defined via the TBA and hence exists only for Bethe Ansatz integrable models, a c-function with the same monotonicity property was first defined from the two-point function of the stress-energy tensor [88]. This function can be ascribed to the same interpretation as the TBA c-function i.e. counting the number of effective degrees of freedom but it can be defined for any 1+1 D perturbed CFT and accordingly the equilibrium c-theorem is valid in any field theory of this type. Although its monotonicity suggests a monotonic behaviour forc je , due to the inelastic scattering between particles, the possible particle production processes and the absence of ballistic propagation it is difficult to give evidence and to reduce the uncertainty left. We therefore leave this question open, which we hope to be the subject of further studies.

Conclusions
In this paper we studied the Euler scale hydrodynamics of massless integrable quantum field theories (QFTs) in the partitioning protocol using the recently developed framework of generalised hydrodynamics (GHD) [42][43][44]. The peculiarity of these models is that they interpolate between two non-trivial renormalisation group fixed points corresponding to conformal field theories (CFTs) with a crossover from one CFT to another characterised by an energy scale called crossover scale. In particular we calculated the Euler scale current and density profiles for the energy and momentum after joining the semi-infinite left and right halves of the massless systems initially prepared to different thermal states. Focusing on the first few members of the A n and D n massless flows, we carried out a systematic treatment of the A n and D n series.
Our analysis identified some general characteristic features regarding the transport properties of these massless flows. Irrespectively of the magnitude of the left and right temperatures with respect to the crossover scale of the massless theory, the density and current profiles exhibit extended plateaux in the non-trivial region ξ ∈ [−1, 1], where ξ stands for x/t sending x and t to infinity. This behaviour is similar to the case of CFTs, whose profiles contain constant regions separated by discontinuous jumps at ξ = ±1 in the Euler limit [69,70], but integrability prevents the formation of such discontinuities and shock-waves in massless flows. The presence of discontinuities in the CFT profiles and their absence in the integrable massless case might seem surprising given that rational CFTs are eventually integrable models having infinitely many local conserved quantities. The main difference between these theories accounting for the different behaviour is the existence of particles with non-trivial dispersion in integrable cases, whereas in CFTs the propagation of modes is dispersionless, which can thus accumulate to form shock-waves.
We analysed the magnitude of the currents and densities and constructed a series of bounds for them at ray ξ = 0, which corresponds to the non-equilibrium stationary state (NESS) that is when t → ∞ but x remains finite. Simple upper and lower bounds for the energy current can be given using the CFT result (1.1) with central charge of the UV and IR limiting CFTs respectively and the initial left and right temperatures. We also showed the the dynamical central chargec je defined in eq. (4.1) inspired by writing the maximum of the energy current in form of the CFT energy current is bounded from below by not only the central charge of the IR limiting CFT but by the effective central charges (2.15) of the massless model corresponding to the left and right temperatures. In [93] a different lower bound was given for the energy current in the NESS or for the corresponding dynamical central charge. Our data are consistent with this lower bound, which in almost all cases turned out to be even more restrictive than the one resulting from the effective central charges. It is worth mentioning that the value of the energy current in the NESS is equal to its absolute maximum, therefore the upper bound for j e (0) is an upper bound for j e at any ray. We also verified, that the dynamical central chargec je satisfies a particular monotonicity property first discussed in [92], which resembles the usual effective central charge.
Similarly to the energy current we defined dynamical central chargesc qe andc jp for the energy density q e and for the momentum current j p using their values in the NESS and corresponding CFT formula (1.1). Forc jp a lower bound is given by c IR and an upper bound byc je whereas forc qe only a lower bound could be given byc je , as its value can occasionally be larger than c U V .
All these observations are based on the explicit investigation of the A 2 , A 3 D 3 and D 4 models and on the massless flow from the W 3 5 to the W 3 4 CFTs and their validity is very plausible for any member of the A n and D n flow and eventually in any unitary massless models. The emergence of the extended plateaux is easy to understand in the low-and hightemperature limits as in these cases the CFT description of the massless model becomes valid and for intermediate temperatures a simple argument was given based on the TBA equations and neglecting magnonic particles in section 4.
Regarding the peculiar properties of the currents and densities we discussed the Landauer-Büttiker formalism of ballistic transport and its applicability to the free fermion CFT in the partitioning protocol. This connection suggest that the dynamical central charges for the currents count the number of open channels for transport and also accounts for the extremal upper and lower bounds given by c IR and c U V . This picture, however, does not prohibit the accumulation of charge and can be compatible with the observed behaviour ofc qe . Using this picture it is also possible to argue for the monotonicity property ofc je . These predictions do not depend on the massless and massive nature of the integrable theory and were indeed found valid for the massive sinh-Gordon model together with the other bounds discussed in the previous paragraphs. The bounds, and the predictions coming from the Landauer-Büttiker picture such as the extremal bounds and the monotonicity of c je corresponding to a non-equilibrium c-theorem are therefore likely to be valid in any 1+1 D relativistic integrable QFT. A further notable implication of the argument based upon the Landauer-Büttiker formalism is that for any low dimensional perturbed conformal field theory universal upper and lower bounds exist for the energy and momentum currents in the NESS. These bounds are given by c U V and c IR i.e. the central charge of the UV limiting CFT and the IR limiting CFT when c IR = 0. The c IR = 0 case corresponds to massive theories, whose low-energy properties are not described by a CFT. This statement should also hold in near critical systems as long as the temperatures are not too large to spoil the effective field theory description.
Finally, for the particular case of the A 2 flow, which corresponds to the massless flow from the tricritical to the critical Ising model, interesting low-temperature transport properties were observed: in this case exactly constant regions appear in the profiles although at the same time they become smooth functions contrary to the CFT limit. The origin of this phenomenon is related to the existence of additional (temperature-dependent) bounds for the effective velocities of massless particles besides the speed of light, and the exactly flat regions emerge in any massless integrable models without magnons as demonstrated by the W 3 5 → W 3 4 flow in appendix B. For the particular case of the A 2 whose low-energy limit can be described by a TT perturbation of the IR CFT, we checked our results against the prediction of [72].
Our work leaves many interesting open directions for further exploration. The current formulation of GHD uses the language of TBA but for many integrable models the nonlinear integral equation (NLIE) [100][101][102][103][104][105][106][107] provides an alternative way to extract thermal (or equivalently finite volume) properties. It is therefore tempting to try to generalise the NLIE to be able to describe inhomogeneous situations. The applicability of a generalised NLIE would be especially valuable for models with a nested Bethe Ansatz. An important example is the sine-Gordon theory for general coupling, whose TBA system has infinitely many magnons, whereas the its NLIE description is much simpler. As for many massless models, such as the A n series, the NLIEs are known they provide an ideal playground to develop an 'inhomogeneous NLIE' because of their simple transport properties to crosscheck the NLIE results with.
The NLIE would be also helpful when analysing the flows associated with the exceptional Lie algebras as their TBA system contains several magnons and the numerical solution of the GHD equations can become very tedious. Another interesting potential application is to non-unitary flows such as the T n series of models. For many non-unitary flows that the effective central charge has a non-monotonic behaviour [103], which may have interesting implications for the transport properties of the system.
Finally, it would be interesting to study if the Landauer-Büttiker picture can explain the other observed bounds for the dynamical central charges and to explore if some of its implications together with the out-of-equilibrium version of the c-theorem remain valid in non-integrable relativistic and low dimensional quantum field theories.  T l /M T r /Mc(T l )c(T r )c je (T l , T r )c qe (T l , T r )c jp (T l , T r )c lb je (T l , T r ) 0.