Emergent Hydrodynamics in a 1D Bose-Fermi Mixture

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INTRODUCTION OF MY LECTURE scale, all the densities and currents of local observables are the functions of ξ = x/t. Here the evolution can be Nonequilibrium dynamics of one dimensional (1D) intuitively understood by the transport of the quasipar- integrable systems is very insightful in understanding ticles in integrable systems. Within the light core the transport of many-body systems. Significantly differ- quasiparticles can arrive and leave that totally depends ent behaviours are observed from that of the higher di- on their local velocities. Moreover, the GH method has mensional systems since the existence of many conserved been adapted to study the evolution of 1D systems con- quantities in 1D. For example, 1D integrable systems do fined by the external field [8,9], also see experiment [10]. not thermalize to a usual state of thermodynamic equi- Beyond the Euler type of hydrodynamics, diffusion has librium [1], where after a long time evolution there is a also been studied by the GH [11, 12]. maximal entropy state constrained by all the conserved On the other hand, at low temperatures, universal phe- quantities[2]. When we consider such a process of time nomena emerge in quantum many-body systems. In high evolution, it is extremely hard to solve the Schrodinger dimensions, the low energy physics can be described by equation of the system with large number particles. Landau’s Fermi liquids theory. However, for 1D sys- tems, the elementary excitations form collective motions of bosons. Thus the low energy physics of 1D systems is elegantly descried by the theory of Luttinger liquids [13– 15]. The essence of the Luttinger liquid is the Haldane’s macroscopic quantum hydrodynamics in which the ele- mentary excitations over the ground state of the 1D sys- tem can be treated as a free Gaussian field theory, i.e. a gas of noninteracting particles with a linear dispersion. Thanking the linear dispersions, the low energy physics of the 1D systems with spin internal degrees of freedom is universally described by the Luttinger liquid theory of spin charge separated degrees of freedom. The Luttinger liquid theory is particularly important in the unified de- scription of the properties in thermodynamic equilibrium states of 1D systems. It turns out that this theory can be applied to study the transport problems in 1D [16]. Based on this theory, within the light core, the transport FIG. 1: The light-core diagram of the of the energy current of the 1D systems at low temperatures is essentially cap- of the 1D BF mixtures at low temperatures. The initial state is set up as two semi-infinitely long 1D tubes joined together tured by the behaviors of the free collective excitations, with different temperatures. see Fig.1. However, in the region ξ ± v ∼ O(TL(R)), the nonlinear excitations should be also considered, where v Recently, a generalized Hydrodynamics (GH) approach is the velocity of collective excitations, and TL(R) are the [3,4] was introduced to study the evolution of macro- temperatures of the initial states of the left (right) halve. scopical systems in 1D. For integrable systems, there are However, the study of quantum hydrodynamics of the 1D infinite many conserved quantities in the thermodynamic systems with spin internal degrees of freedom is rather limit. With the help of local equilibrium approximation, preliminary [7]. whole states of the system are determined by the density In contrast to the spin charge separated Luttinger liq- distributions of the conserved quantities. These density uids in interacting Fermi gas [15, 17], the 1D degenerate distributions of all conserved quantities and the their con- Bose-Fermi (BF) mixtures display two decoupled Lut- tinuity equations are capable of determining the evolu- tinger liquids in bosonic and fermionic degrees of free- tions of the systems. Due to the existence of conserved dom at low temperatures [18–22]. In the former, the quantities, the steady state may admit nonzero station- effective antiferromagnetic spin-spin interaction leads to ary charge currents, indicating the presence of ballistic the Luttinger liquid of spinons, whereas in the latter the transport of integrable models. This elegant GH method Luttinger liquids separately emerge in the fermionic and has been applied to the study of the time evolution of two bosonic degrees of freedom. semi-infinite chains joined together with different tem- In this lecture, I will give an elementary introduction peratures [3–7]. The result shows that in a large time of the GH method and its applications in 1D integrable 2 models. In particular, I will present our recent study on the low-temperature transport of one-dimensional fermi the transport properties of the 1D Bose-Fermi mixtures gases. Physical Review B, 99(1):014305, 2019. with delta-function interactions by means of the GH ap- [8] Benjamin Doyon and Takato Yoshimura. A note on gen- proach. Building on the Bethe anssatz solution, I rig- eralized hydrodynamics: inhomogeneous fields and other concepts. SciPost Physics, 2(2):014, 2017. orously prove the existence of conserved charges in both [9] Jean-S´ebastienCaux, Benjamin Doyon, J´erˆomeDubail, the bosonic and fermionic degrees of freedom, providing Robert Konik, and Takato Yoshimura. Hydrodynamics the Euler type of hydrodynamics. The velocities of el- of the interacting bose gas in the quantum newton cradle ementary excitations in the two degrees of freedom are setup. arXiv preprint arXiv:1711.00873, 2017. significantly different due to different quantum statisti- [10] Max Schemmer, Isabelle Bouchoule, Benjamin Doyon, cal effects. The charge-charge separation emerges in the and J´erˆomeDubail. Generalized hydrodynamics on an steady state of the systems within the light core. The dis- atom chip. Physical Review Letters, 122(9):090601, 2019. tributions of the densities and currents of local conserved [11] Jacopo De Nardis, Denis Bernard, and Benjamin Doyon. Hydrodynamic diffusion in integrable systems. Physical quantities in bosonic and fermionic degrees of freedom review letters, 121(16):160603, 2018. show the ballistic transport of the quasiparticles with two [12] Jacopo De Nardis, Denis Bernard, and Benjamin Doyon. different velocities of the excitations in the two degrees Diffusion in generalized hydrodynamics and quasiparticle of freedom. The obtained results reveal subtle differences scattering. arXiv preprint arXiv:1812.00767, 2018. in densities and currents of bosons and fermions, showing [13] F. D. M. Haldane. Effective harmonic-fluid approach to interaction and quantum statistic effects in transport. low-energy properties of one-dimensional quantum fluids. To students: please read the key reference [3] in ad- Phys. Rev. Lett., 47:1840–1843, Dec 1981. [14] F. D. M. Haldane. ’luttinger liquid theory’ of one- vance. dimensionalquantum fluids: I. properties of the luttinger model and their extension to the general 1d interacting spinless fermi gas. J. Phys. C: Solid State Physics, 14. [15] Thierry Giamarchi. Quantum physics in one dimension. [16] Bruno Bertini, Lorenzo Piroli, and Pasquale Calabrese. [1] Toshiya Kinoshita, Trevor Wenger, and David S Weiss. A Universal broadening of the light cone in low-temperature quantum newton’s cradle. Nature, 440(7086):900, 2006. transport. Physical review letters, 120(17):176801, 2018. [2] Marcos Rigol, Vanja Dunjko, Vladimir Yurovsky, and [17] J. Y. Lee, X. W. Guan, K. Sakai, and M. T. Batche- Maxim Olshanii. Relaxation in a completely integrable lor. Thermodynamics, spin-charge separation, and cor- many-body quantum system: an ab initio study of the relation functions of spin-1/2 fermions with repulsive in- dynamics of the highly excited states of 1d lattice hard- teraction. Phys. Rev. B, 85:085414, Feb 2012. core bosons. Physical review letters, 98(5):050405, 2007. [18] C. K. Lai and C. N. Yang. Ground-state energy of a [3] Olalla A Castro-Alvaredo, Benjamin Doyon, and Takato mixture of fermions and bosons in one dimension with a Yoshimura. Emergent hydrodynamics in integrable quan- repulsive δ-function interaction. Phys. Rev. A, 3:393–399, tum systems out of equilibrium. Physical Review X, Jan 1971. 6(4):041065, 2016. [19] M. T. Batchelor, M. Bortz, X. W. Guan, and N. Oelk- [4] Bruno Bertini, Mario Collura, Jacopo De Nardis, and ers. Exact results for the one-dimensional mixed boson- Maurizio Fagotti. Transport in out-of-equilibrium x x z fermion interacting gas. Phys. Rev. A, 72:061603, Dec chains: Exact profiles of charges and currents. Physical 2005. review letters, 117(20):207201, 2016. [20] Holger Frahm and Guillaume Palacios. Correlation func- [5] Denis Bernard and Benjamin Doyon. Conformal field tions of one-dimensional bose-fermi mixtures. Phys. Rev. theory out of equilibrium: a review. Journal of Statisti- A, 72:061604, Dec 2005. cal Mechanics: Theory and Experiment, 2016(6):064005, [21] Adilet Imambekov and Eugene Demler. Exactly solvable 2016. case of a one-dimensional bose–fermi mixture. Physical [6] Benjamin Doyon, J´erˆomeDubail, Robert Konik, and Review A, 73(2):021602, 2006. Takato Yoshimura. Large-scale description of interacting [22] Adilet Imambekov and Eugene Demler. Applications of one-dimensional bose gases: generalized hydrodynamics exact solution for strongly interacting one-dimensional supersedes conventional hydrodynamics. Physical review bose–fermi mixture: Low-temperature correlation func- letters, 119(19):195301, 2017. tions, density profiles, and collective modes. Annals of [7] M´arton Mesty´an, Bruno Bertini, Lorenzo Piroli, and Physics, 321(10):2390–2437, 2006. Pasquale Calabrese. Spin-charge separation effects in Selected for a Viewpoint in Physics PHYSICAL REVIEW X 6, 041065 (2016)

Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium

Olalla A. Castro-Alvaredo,1 Benjamin Doyon,2 and Takato Yoshimura2 1Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, United Kingdom 2Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom (Received 12 July 2016; revised manuscript received 22 September 2016; published 27 December 2016) Understanding the general principles underlying strongly interacting quantum states out of equilibrium is one of the most important tasks of current theoretical physics. With experiments accessing the intricate dynamics of many-body quantum systems, it is paramount to develop powerful methods that encode the emergent physics. Up to now, the strong dichotomy observed between integrable and nonintegrable evolutions made an overarching theory difficult to build, especially for transport phenomena where space- time profiles are drastically different. We present a novel framework for studying transport in integrable systems: hydrodynamics with infinitely many conservation laws. This bridges the conceptual gap between integrable and nonintegrable quantum dynamics, and gives powerful tools for accurate studies of space- time profiles. We apply it to the description of energy transport between heat baths, and provide a full description of the current-carrying nonequilibrium steady state and the transition regions in a family of models including the Lieb-Liniger model of interacting Bose gases, realized in experiments.

DOI: 10.1103/PhysRevX.6.041065 Subject Areas: Nonlinear Dynamics, Quantum Physics, Statistical Physics

I. INTRODUCTION study of prethermalization or prerelaxation under small integrability breaking [22,28–30], the elusive quantum Many-body quantum systems out of equilibrium give Kolmogorov-Arnold-Moser theorem [31,32], the develop- rise to some of the most important challenges of modern ment of perturbation theory for nonequilibrium states, and physics [1]. They have received a lot of attention recently, the exact treatment of nonequilibrium steady states and of with experiments on quantum heat flows [2,3], generalized nonhomogeneous quantum dynamics in unitary interacting thermalization [4,5], and light-cone effects [6]. The leading integrable models remain difficult problems. principle underlying nonequilibrium dynamics is that of In this paper, using the recent advances on generalized local transport carried by conserved currents. Deeper thermalization and developing further aspects of integrabil- understanding can be gained from studying nonequili- ity, we propose a solution to such problems by deriving a brium, current-carrying steady states, especially those general theory of hydrodynamics with infinitely many emerging from unitary dynamics [7]. This principle gives conservation laws. The theory, applicable to a large integra- rise to two seemingly disconnected paradigms for many- bility class, is derived solely from the fundamental tenet of body quantum dynamics. On the one hand, taking into emerging hydrodynamic: local entropy maximization (often account only few conservation laws, emergent hydrody- referred to as local thermodynamic equilibrium) [33–37]. namics [8–12] offers a powerful description where the Focusing on quantum field theory (QFT) in one space physics of fluids dominates [13–18]. On the other hand, in dimension, we then study a family of models that include integrable systems, the infinite number of conservation the paradigmatic Lieb-Liniger (LL) model [38] for interact- laws is known to lead to generalized thermalization [19–21] ing Bose gases, explicitly realized in experiments [4,5,39– (there are many fundamental works on the subject; see the 41]. We concentrate on far-from-equilibrium states driven by review [22]), and the presence of quasilocal charges has heat baths in the partitioning protocol [7,26,27,42] (see been shown to influence transport [23,24] (see the review Fig. 1). We provide currents and full space-time profiles [25]). However, except at criticality [26,27] (see the review which are in principle experimentally accessible, beyond [28]), no general many-body emergent dynamics has been linear response and for arbitrary interaction strengths. We proposed in the integrable case; with the available frame- make contact with the physics of rarefaction waves, and with works, these two paradigms seem difficult to bridge. The the concept of quasiparticle underlying integrable dynamics.

Published by the American Physical Society under the terms of II. SETUP the Creative Commons Attribution 3.0 License. Further distri- bution of this work must maintain attribution to the author(s) and Let two semi-infinite halves (which we refer to as the the published article’s title, journal citation, and DOI. left and right reservoirs) of a homogeneous, short-range

2160-3308=16=6(4)=041065(17) 041065-1 Published by the American Physical Society CASTRO-ALVAREDO, DOYON, and YOSHIMURA PHYS. REV. X 6, 041065 (2016)

where c is the central charge of the conformal field theory (CFT) (below, we set kB ¼ ℏ ¼ 1). This result arises from the independent thermalization of emerging left- and right-moving energy carriers (chiral separation). It was numerically verified [43] and agrees with recent heat-flow experiments [2]. It was generalized using hydrodynamic methods to higher-dimensional critical points [13,14,17,18] and to deviations from criticality [15,16,18]. Under con- ditions that are fulfilled in universal near-critical regions, FIG. 1. The partitioning protocol. With ballistic transport, a inequalities that generalize Eq. (2) can be derived [28,44] current emerges after a transient period. Dotted lines represent (here with unit Lieb-Robinson velocity [45]): different values of ξ ¼ x=t. If a maximal velocity exists (e.g., due to the Lieb-Robinson bound), initial reservoirs are unaffected eL − eR kL − kR ξ 0 ≥ jsta ≥ beyond it (light-cone effect). The steady state lies at ¼ . 2 2 ; ð3Þ one-dimensional quantum system be independently ther- where eL;R and kL;R are, respectively, the energy densities malized, say, at temperatures TL and TR. Let this initial and the pressures in the left and right reservoirs [46]. Many further results exist in free-particle models (see state h iini be evolved unitarily with the Hamiltonian H representing the full homogeneous system. One may then Ref. [28] and references therein), where independent investigate the steady state that occurs at large times (see, thermalization of right and left movers still holds. In e.g., Ref. [28]), contrast, however, only conjectures and approximations are available for interacting integrable models [47–49].In Osta ≔ iHt −iHt limhe Oe iini;Olocal observable: ð1Þ addition, a striking dichotomy is observed between inte- t→∞ grable situations and hydrodynamic-based results: for If the limit exists, it is a maximal-entropy steady state instance, conformal hydrodynamics is expected to emerge involving, in principle, all (quasi)local conserved charges in strong-coupling CFT [13,14], leading to shock struc- of the dynamics H [see Eq. (4)]. Generically, the dynamics tures, but generically fails in free-particle conformal mod- only admits a single conserved quantity, H itself: this els [50], where transition regions are smooth. This points to means that, due to diffusive processes, ordinary Gibbs the stark effect of integrability on nonequilibrium quantum thermalization occurs. However, when conserved charges dynamics, still insufficiently understood with available exist that are odd under time reversal, the steady state may techniques. admit nonzero stationary currents. This indicates the presence of ballistic transport, and the emergence of a III. EMERGING HYDRODYNAMICS current-carrying state that is far from equilibrium (breaking IN QUANTUM SYSTEMS time-reversal symmetry). This is the partitioning protocol Let us recall the basic concepts underlying the hydro- for building nonequilibrium steady states. See Fig. 1. dynamic description of many-body quantum systems, and The study of such nonequilibrium steady states has its use in the setup described above (similar concepts exist received a large amount of attention recently (see in many-body classical systems). Ref. [28] and references therein). They form a uniquely Let Q , i ∈ f1; 2; …;Ng, be local conserved quantities in interesting set of states: they are simple enough to be i involution. These are integrals of local densities q ðx; tÞ, theoretically described, yet encode nontrivial aspects of i and the conservation laws take the form ∂ q ðx; tÞþ nonequilibrium physics. They naturally occur in the uni- t i ∂ j ðx; tÞ¼0, where j are the associated local currents. versal region near criticality described by QFT, where x i i A Gibbs ensemble is a maximal-entropy ensemble under ballistic transport emerges thanks to continuous translation conditions fixing all averaged local conserved densities. It invariance, and in integrable systems, where it often arises is described by a density matrix thanks to the infinite family of conservation laws. P P The works in Refs. [26,27] open the door to the study of − β Q − β Q ρ i i i i i i nonequilibrium steady states at strong-coupling critical ¼ e =Tr½e ; ð4Þ points with unit dynamical exponent, obtaining, in par- β ticular, the full universal time evolution. The steady state where i are the associated potentials. For instance, Q1 is β was found to be homogeneous within a light cone, with the taken as the Hamiltonian, and 1 is the inverse temperature. β β β … β energy current being We denote ¼ð 1; 2; ; NÞ the vector representing this state, and h iβ the averages. π 2 ckB 2 2 Clearly, the Gibbs averages of local densities q q jsta ¼ ðT − T Þ; ð2Þ i ¼h iiβ 12ℏ L R (these are independent of space and time by homogeneity

041065-2 EMERGENT HYDRODYNAMICS IN INTEGRABLE QUANTUM … PHYS. REV. X 6, 041065 (2016) Z Z x2 t2 and stationarity) may be seen as defining a map from states q −q j −j 0 β ↦ q dx½ ðx;t2Þ ðx;t1Þþ dt½ ðx2;tÞ ðx1;tÞ ¼ ; to averages, . This is expected to be a bijection: the x1 t1 set of averages fully determines the set of potentials. j ð7Þ Therefore, the current averages i ¼hjiiβ are functions of the density averages: q j where ðx; tÞ¼hqiβðx;tÞ and ðx; tÞ¼hjiβðx;tÞ. Here, inte- grals may be taken to cover a macroscopic number of fluid j ¼ FðqÞ: ð5Þ cells: these become macroscopic conservation equations. Macroscopic conservation equations can be rewritten in These are the equations of state, and are model dependent. differential form, with differentials representing small The averages q can be generated by differentiation of the variations amongst fluid cells: (specific, dimensionless) free energy fβ. Similarly, one can show [28] (see Appendix A) that there exists a function gβ ∂tqðx; tÞþ∂xjðx; tÞ¼0: ð8Þ that, likewise, generates the currents, These are the pure hydrodynamic (Euler-type) equations, q ¼ ∇βfβ; j ¼ ∇βgβ: ð6Þ representing the slow, large-scale quantum dynamics of conserved densities and currents flowing amongst neigh- A hydrodynamics description of quantum dynamics is boring cells. expected to emerge at large space-time scales. This has been The problem of emergence of hydrodynamics in many- exploited, in the present setup, in Refs. [13–18].The body systems is one of the most important unsolved emergence of hydrodynamics is solely based on the problems of modern mathematical physics. Although there assumption of local entropy maximization (or local thermo- are few proofs of emergence of hydrodynamics, there is dynamic equilibrium) [51].Technically,thisisthe strong evidence for the validity of emerging Euler equa- assumption that averages of local quantities hOðx; tÞi tend tions in many situations; see Refs. [33–37] and the recent uniformly enough, at large times, to averages evaluated in paper Ref. [52] for a study of emerging Euler equations in local Gibbs ensembles hOiβðx;tÞ with space-time dependent classical anharmonic chains. Combined with the equations of state Eq. (5), Euler potentials βðx; tÞ. Physically, this is a consequence of equations (8) give separation of scales, as follows (see, for instance, Ref. [37]). Assume that, after some time, physical properties vary ∂ q (q )∂ q 0 only on space-time scales that are much larger than t ðx; tÞþJ ðx; tÞ x ðx; tÞ¼ ; ð9Þ microscopic scales. This may be referred to as the “local ” relaxation time. From that time on, microscopic processes where JðqÞ ≔ ∇qj is an N by N matrix, the Jacobian such as particle collisions or intersite interactions give rise matrix of the transformation from densities to currents: to fast, local relaxation: the reaching of a (approximate) steady state on space-time scales small compared to JðqÞ ¼ ∂F ðqÞ=∂q : ð10Þ variations but large enough for thermodynamics to be ij i j applicable. By Boltzmann’s phase-space argument, these local steady states are obtained from entropy maximization, Equation (9) is the emergent pure hydrodynamic equation and as usual maximization is under the conditions provided in quasilinear (or characteristic) form [12]. The complete by conservation laws (properties of the microscopic model dependence, including all quantum effects, is dynamics). That is, on each space-time “fluid cell” a encoded, besides the number N of conserved quantities, Gibbs state is (very nearly) reached. Neighboring Gibbs in the Jacobian JðqÞ. states are different, but their variations are small. This is The density averages q, like the potentials β, correspond local entropy maximization. to a set of state coordinates. One may choose any other set Assume local entropy maximization. On each fluid cell, of state coordinates n, with q ¼ F qðnÞ and j ¼ F jðnÞ. the Gibbs state is initially characterized by the values A similar equation is obtained, of the conserved densities at the local-relaxation time. The large-scale dynamics is thereon obtained from con- ∂ nðx; tÞþJ(nðx; tÞ)∂ nðx; tÞ¼0; ð11Þ servation laws, as follows. ConsiderR microscopic conser- t x x2 2 − 1 vationR in integral form, x1 dx½qiðx; t Þ qiðx; t Þþ t2 −1 − 0 where J n ∇nq ∇nj. Observe that J n and J q t1 dt½jiðx2;tÞ jiðx1;tÞ ¼ . Since averages of densities ð Þ¼ð Þ ð Þ ð Þ and currents, after the local relaxation time, take the form are related to each other by a similarity transformation: −1 n ∇ j q q ∇ j hqiðx; tÞi¼hqiiβðx;tÞ and hjiðx; tÞi¼hjiiβðx;tÞ uniformly Jð Þ¼ð n Þ Jð Þjq¼F ðnÞ n . Therefore, the spec- enough, we have trum of JðnÞ is independent of the choice of coordinates,

041065-3 CASTRO-ALVAREDO, DOYON, and YOSHIMURA PHYS. REV. X 6, 041065 (2016) and is a fundamental property of the model. We denote this velocities in the state characterized by the averages q— eff n 1 … spectrum by fvi ð Þ;i¼ ; ;Ng. form a finite, discrete set for finite N. Choosing coordinates n that diagonalize JðnÞ, one Solutions to Eqs. (13) and (14) are typically composed of obtains regions of constant q values separated by transition regions [12]. Transition regions may be of two types: either shocks ∂ n eff(n )∂ n 0 t iðx; tÞþvi ðx; tÞ x iðx; tÞ¼ : ð12Þ (weak solutions), where q values display finite jumps, or rarefaction waves, where they form a smooth solution to These express the vanishing of the convective derivatives, Eq. (13). Rarefaction waves, the most natural type of n representing the constancy of each fluid mode iðx; tÞ on solution, cannot, generically, cover the full space between eff(n ) fluid cells. The eigenvalues vi ðx; tÞ are, therefore, two reservoirs. Indeed, Eq. (13) specifies that the curve interpreted as the propagation velocities of these normal traced by the solution in the q plane must at all points be modes. The normal modes interact with each other only tangent to an eigenvector of JðqÞ. Since eigenvectors—and through the propagation velocities, which is generically a available propagation velocities—form a discrete set, function of all state coordinates. smooth variations of q along the curve imply a unique Let us now apply the above to the solution of the choice of eigenvector at each point (except possibly at partitioning problem. For clarity of the following discus- points where eigenvalues cross). Thus, the curve is com- sion, we come back to the q coordinates (but it is easy to pletely determined by its initial point, and cannot join two generalize to any coordinates n). Consider the large-scale arbitrary reservoir values. That is, in ordinary pure hydro- limit ðx; tÞ ↦ ðax; atÞ;a→ ∞. Because Eq. (12) is invari- dynamics, shocks are often required. ant under this scaling, in the limit, if it exists, the solution is also invariant. Thus, we may assume self-similar solutions βðx; tÞ¼βðξÞ, where ξ ¼ x=t, and Eq. (12) becomes an IV. HYDRODYNAMICS WITH INFINITELY eigenvalue equation: MANY CURRENTS In integrable systems, there are infinitely many local ½JðqÞ − ξ1∂ξq ¼ 0: ð13Þ conservation laws. In fact, this space is enlarged to that of “pseudolocal” conservation laws, where the densities The initial condition is determined by the state at the local qiðx; tÞ and currents jiðx; tÞ are supported on extended relaxation time (at which the fluid-dynamics description spacial regions with weight decaying fast enough away starts to be valid). This state is unknown, as it depends on from x. This enlargement plays an important role in the full quantum dynamics, but its asymptotic at large jxj is nonequilibrium quantum dynamics [20,21,23–25]. Under identical to that of the original state. In the large-scale maximal-entropy principles, Gibbs states are then replaced solution, the initial condition t → 0þ is implemented as by generalized Gibbs ensembles (GGE) [19,21,22]: for- asymptotic conditions as ξ → ∞. Therefore, it depends mally, the limit N → ∞ of the density matrix Eq. (4), only on the asymptotic form of the initial state, and we involving all basis elements in the space of conserved impose pseudolocal charges. We choose Q1 ¼ H (the Hamiltonian) and Q2 ¼ P (the momentum operator). lim qðξÞ¼ lim hqðx; 0Þi : ð14Þ Under the influence of infinitely many conservation ξ→ ∞ → ∞ ini x laws, the picture of local entropy maximization is still In the present setup, these involve Gibbs states at potentials expected to hold: all physical principles underlying it stay unchanged, the only difference being the use of GGEs βR;L: instead of Gibbs ensembles. This, along with the emer- 0 gence of self-similar solutions in the partitioning protocol, lim hqðx; Þiini ¼hqiβR;L ; ð15Þ x→∞ are our working hypotheses; see Appendix B for a discussion. The emergence of a generalized type of and the steady-state averages are given by hydrodynamics was proven in the classical hard-rod prob- lem [37,53], whose relation with the present quantum qsta ≔ qðξ ¼ 0Þ; jsta ≔ jðξ ¼ 0Þ: ð16Þ problem we will study in a future work. The emergence of self-similar solutions was observed numerically in the The solution to the eigenvalue equation (13) and initial quantum XXZ chain in Ref. [54]. In free-particle quantum conditions Eq. (14) provides the exact large-scale asymp- models, hydrodynamic ideas and related semiclassical totic form of the full quantum solution, along any ray x ¼ approximations, as well as ray-dependent local entropy ξt (see Fig. 1). The eigenvalue equation (13) represents the maximization, were studied in various works; see small changes of averages along various rays, due to the Refs. [55–60]. exchange of conserved charges amongst fluid cells. The set Looking for a full solution to the infinity of Eqs. (9), of eigenvalues of JðqÞ—the available propagation (13), and (14), an appropriate choice of state variables is

041065-4 EMERGENT HYDRODYNAMICS IN INTEGRABLE QUANTUM … PHYS. REV. X 6, 041065 (2016) Z crucial. A powerful way is to recast them into the dγ hdrðθÞ¼hðθÞþ φðθ − γÞnðγÞhdrðγÞ: ð20Þ quasiparticle language underlying the thermodynamic 2π Bethe ansatz (TBA) method for integrable systems [61]. Using this language, we derive the exact GGE equations of The potentials β can be recovered: the occupation state and the ensuing generalized hydrodynamics equation. Pnumber is related to theP one-particle eigenvalue wðθÞ¼ β θ β We determine the exact normal modes and propagation i ihið Þ of the charge i iQi in the GGE Eq. (4) via the velocities, and obtain full ray-dependent solutions. so-called pseudoenergy ϵwðθÞ [61,62]: 1 A. GGE equations of state nðθÞ¼ ; 1 þ exp½ϵwðθÞ We assume that the spectrum of stable quasiparticles is Z dγ −ϵ γ composed of a single quasiparticle species of mass m (see ϵ θ θ − φ θ − γ 1 wð Þ wð Þ¼wð Þ 2π ð Þ logð þ e Þ: ð21Þ Appendix C for a many-particle generalization). The dispersion relation is encoded via a parametrization EðθÞ, The above ingredients give exact average densities as pðθÞ of the energy and momentum: in the relativistic case, θ functions of GGE states. However, they do not provide is the rapidity, EðθÞ ≔ m coshðθÞ, pðθÞ ≔ m sinhðθÞ,andin expressions for average currents as functions of state coor- the Galilean case, θ is the velocity, EðθÞ ≔ mθ2=2, θ ≔ θ dinates, and for equations of states. Hence, they are not pð Þ m . In integrable models, scattering is elastic and sufficient in order to develop generalized hydrodynamics. factorizes into two-particle processes, and a model is fully We solve this problem by obtaining the following specified by giving the elastic two-particle scattering ampli- expressions: tude Sðθ1 − θ2Þ. The differential scattering phase is defined Z Z φ θ − θ θ θ dpðθÞ dEðθÞ as ð Þ¼ id log Sð Þ=d . We denote by hið Þ the one- q θ dr θ j θ dr θ i ¼ 2π nð Þhi ð Þ; i ¼ 2π nð Þhi ð Þ; particle eigenvalue of the conserved charge Qi; in particular, h1ðθÞ¼EðθÞ and h2ðθÞ¼pðθÞ. ð22Þ Let us first recall the basic ingredients of TBA. Three related quantities play important roles: the quasiparticle where hdrðθÞ is the dressed one-particle eigenvalue. These ρ θ ρ θ i density pð Þ, the state density sð Þ, and the quasiparticle expressions emphasize the role of relativistic or Galilean θ ≔ ρ θ ρ θ occupation number nð Þ pð Þ= sð Þ. The functions symmetry: the sole difference between GGE averages of ρpðθÞ and nðθÞ are two different sets of state coordinates; charge densities and currents is the integration measure, each can be used to fully characterize the GGE. The former determined by the dispersion relation. specifies all average densities in a simple way: The first equation in Eq. (22) is well known and is a Z consequence of Eqs. (17) and (19). In integral-operator notation [with measure dθ=ð2πÞ], the dressing operation is qi ¼ dθρpðθÞhiðθÞ: ð17Þ hdr ¼ð1 − φN Þ−1h; ð23Þ This can, in fact, be seen as a definition of ρ ðθÞ. Here and p N 2π θ δ θ − α φ below, integrations are over R. where is diagonal with kernel nð Þ ð Þ, and has φ θ − α As a consequence of interactions, quasiparticle and state kernel ð Þ. Therefore, introducing the symmetric U N 1 − φN −1 densities are related to each other. Using the Bethe ansatz, operatorR ¼ ð Þ and the bilinear form one finds the following constitutive relation [61]: a · b ¼ dθ=ð2πÞaðθÞbðθÞ,wehave

Z 0 0 qi ¼ hi · Up ¼ p · Uhi; ð24Þ 0 2πρsðθÞ¼p ðθÞþ dαφðθ − αÞρpðαÞ; ð18Þ which leads to the first equation of Eq. (22). The second equation in Eq. (22) is new. It may be where p0ðθÞ¼dpðθÞ=dθ. This relation gives rise to a proven, in the relativistic case, using relativistic crossing nonlinear relation between the state coordinates ρpðθÞ and θ symmetry, and then obtained by the nonrelativistic limit in nð Þ. The transformation from the former to the latter is the Galilean case. In the relativistic case, crossing sym- direct from the above definitions. In the opposite direction, metry says that local currents j , in the cross channel, are the transformation is effected by i local densities qi; therefore, the current expression in Eq. (22) is obtained from that of the density under an 2πρ θ θ 0 dr θ pð Þ¼nð Þðp Þ ð Þ; ð19Þ appropriate exchange of energy and momentum. Let C be the crossing transformation ðx; tÞ ↦ ðit; −ixÞ, implemented where the “dressing” operation h ↦ hdr is defined by the on rapidities by θ ↦ iπ=2 − θ. Note that it squares to the solution to the linear integral equation: identity C2 ¼ 1. Let us denote by q½h and j½h the density and

041065-5 CASTRO-ALVAREDO, DOYON, and YOSHIMURA PHYS. REV. X 6, 041065 (2016) current operators, respectively, associated to a one-particle for this integrable model. The proof of Eq. (28) is obtained eigenvalue hðθÞ. Then, the statement that local currents ji,in by isolating nðθÞ in both Eqs. (19) and (26), in the −1 0 −1 0 the cross channel, are local densities qi translates into forms 2πðN − φÞρp ¼ p and 2πðN − φÞρc ¼ E , and Cðj½hÞ ¼ iq½hC, where hCðθÞ¼hðiπ=2 − θÞ. Let us also equating the resulting expressions. O O denote by h iw the average of observables in the state Finally, recalling Eq. (26), the left-hand side of Eq. (28) θ C O O eff characterized by wð Þ. Then, h ð Þiw ¼h iwC , where is v ðθÞ. Simple manipulations of Eq. (28) then give a C θ π 2 − θ C(C ) eff w ð Þ¼wði = Þ. Using hj½hiw ¼h ðj½hÞ iw ¼ linear integral equation for the effective velocity v ðθÞ in c q q ihq½h iwC and the expression for i ¼ ½hi in Eq. (22), terms of quasiparticle densities: j j we obtain the expression for i ¼ ½hi. An alternative Z proof, using the machinery of integrable systems, is pre- φðθ − αÞρ ðαÞ veffðθÞ¼vgrðθÞþ dα p ½veffðαÞ − veffðθÞ; sented in Appendix D. p0ðθÞ Expressions (22) have interesting consequences. First, j U 0 0 θ θ θ ð29Þ using i ¼ hi · E , where E ð Þ¼dEð Þ=d , in Eq. (22), the average current may also be written in terms of a current where vgrðθÞ¼E0ðθÞ=p0ðθÞ is the group velocity. In this spectral density ρcðθÞ: Z form, the equations of state of integrable systems are seen as equations specifying an effective velocity of quasipar- j θρ θ θ i ¼ d cð Þhið Þ; ð25Þ ticles, as a modification of the group velocity. We note that the effective velocity we derive here agrees which takes the forms with that proposed in Ref. [63]. [Note that in Ref. [63] veffðθÞ is written in a form similar to, but different from, Eq. (27), 0 dr eff 2πρcðθÞ¼nðθÞðE Þ ðθÞ¼2πv ðθÞρpðθÞ: ð26Þ using a different definition of dressing.] This is interesting, as our derivation is based on comparing current spectral density Here, veffðθÞ is the effective velocity, defined by to quasiparticle density, while the concept proposed in Ref. [63] is based on stationary-phase arguments. ðE0ÞdrðθÞ veffðθÞ ≔ : ð27Þ ðp0ÞdrðθÞ B. Generalized hydrodynamics The effective velocity depends on the state via the occu- The basic equation of generalized pure hydrodynamics is pation number entering the dressing operation, and brings derived from Eq. (8) along with the quasiparticle expres- out the quasiparticle interpretation of the current expres- sions Eqs. (17) and (25). The fact that the space of pseudolocal charges is complete [21] suggests that these sion: since ρ ðθÞ¼veffðθÞρ ðθÞ, quasiparticles are seen as c p hold for a complete set of functions h ðθÞ, and, thus (here moving at effective velocities veff θ , influenced by the i ð Þ and below we suppress explicit x, t dependences for state in which they move. lightness of notation), Second, one may extract explicit GGE equations of state from expressions (22). The equations of states are neces- ∂ ρ θ ∂ ρ θ 0 sary and sufficient relations between densities and currents, t pð Þþ x cð Þ¼ : ð30Þ guaranteeing the existence of nðθÞ such that both relations θ q j in Eq. (22) hold for all hið Þ. Assume that i and i are Using the equations of state Eq. (28), this is an integro- averages in a state, not necessarily a GGE. In complete differential system for the space-time-dependent state ρ θ generality, both are linear functionals of hðθÞ; hence, we characterized by the particle densities pð Þ. may still write Eqs. (17) and (25) for some quasiparticle Alternatively, using the effective-velocity formulation density ρpðθÞ and current spectral density ρcðθÞ. GGE Eqs. (26) and (29), Eq. (30) may be written as equations of states can, therefore, be written as relations ρ θ ρ θ eff between pð Þ and cð Þ, necessary and sufficient for the ∂tρpðθÞþ∂x½v ðθÞρpðθÞ ¼ 0: ð31Þ existence of nðθÞ such that Eqs. (22) hold. One can show that these relations are This is the conservation form of generalized hydrodynam- R ics. It is a density-type conservation equation, and it ρ θ 0 θ αφ θ − α ρ α cð Þ E ð Þþ d ð Þ cð Þ identifies ρ θ as a conserved fluid density. R : pð Þ ¼ 0 ð28Þ ρ θ ρpðθÞ p ðθÞþ dαφðθ − αÞρpðαÞ The state coordinates pð Þ are, however, not the most convenient. We show that the occupation numbers nðθÞ These relations are independent of the state: they hold in diagonalize the Jacobian JðnÞ in the quasilinear form any GGE, in the model described by the differential Eq. (11): the space-time-dependent occupation number scattering phase φðθ − αÞ. They characterize the set of nðθÞ satisfies the following integro-differential system, doublets of functions ðρp; ρcÞ describing available GGEs the vanishing of the convective derivative of nðθÞ:

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eff ∂tnðθÞþv ðθÞ∂xnðθÞ¼0: ð32Þ represent the asymptotic baths as per Eq. (14). They are obtained using Eq. (21) with w ¼ wL;RðθÞ the one-particle Here, Eq. (27) may be used to express the effective velocity eigenvalues characterizing the GGE of the left and right in terms of nðθÞ. Hence, nðθÞ are the normal modes of asymptotic reservoirs; for instance, with reservoirs at — L;R θ −1 θ generalized hydrodynamics, and further, the eigenvalues temperatures TL;R,wehavew ð Þ¼TL;REð Þ. the propagation velocities—are exactly the effective veloc- Indeed, the solution Eq. (35) of the scaled problem holds ities veffðθÞ. since veffðθÞ is monotonic and covers the full range of θ The proof of Eq. (32) is as follows. Using the integral- (which is ½−1; 1 in the relativistic case and R in the 0 0 operator relations 2πρp ¼ Up and 2πρc ¼ UE ,we Galilean case); therefore, there is a unique solution to 0 0 eff have ð∂tUÞp þð∂xUÞE ¼ 0. Taking derivatives, ∂x;tU ¼ v ðθÞ¼ξ, and, thus, a unique jump, and θ⋆ is monotonic −1 −1 ξ ð1 − N φÞ ð∂x;tN Þð1 − φN Þ , and we obtain with ; hence, the asymptotic conditions are correctly implemented. ∂ N ð1 − φN Þ−1p0 þ ∂ N ð1 − φN Þ−1E0 ¼ 0; ð33Þ The system of integral equations (22), (20), (35), t x and (36) can be solved numerically using Mathematica, yielding extremely accurate results. Integral equations in which gives Eq. (32) using Eq. (23). Eqs. (21) and (20) can be solved iteratively, a procedure Observe that, using Eqs. (31) and (32), it is simple to ρ θ that converges fast [61]. The hydrodynamic solution is show that the state density sð Þ as well as the hole density obtained by first constructing the thermal occupation ρ θ ≔ ρ θ − ρ θ also satisfy the same density-type hð Þ sð Þ pð Þ numbers nL;RðθÞ [Eq. (21)]. Then, the nonequilibrium conservation equation (31) (this was noted in Ref. [64]). occupation number is evaluated by solving the system Further, as a consequence, the entropy density [61], equations (35) and (36): one first chooses θ⋆ ¼ η in order to construct nðθÞ, and then evaluates pdrðθÞ. The zero of s θ ≔ ρ θ log ρ θ − ρ θ log ρ θ − ρ θ log ρ θ ; ξ ð Þ sð Þ sð Þ pð Þ pð Þ hð Þ hð Þ dr dr pξ ðθÞ is numerically found—we observe that pξ ðθÞ ð34Þ always has a single zero. The process is repeated until the zero is stable—we observe that this is a convergent also satisfies this conservation equation, ∂ s θ t ð Þþ procedure. Finally, the nonequilibrium occupation number ∂ eff θ θ 0 x½v ð Þsð Þ ¼ . Conservation of entropy density is a is used in Eqs. (22) and (20). The solving time increases fundamental property of perfect fluids, as no viscosity effects slowly with the numerical precision demanded; thus, this are taken into account. allows arbitrary-precision results. In the large-scale limit, the equation for the ray- The solution we present may be interpreted as a single dependent (ξ-dependent) occupation number n θ ð Þ space-covering rarefaction wave, in the sense that it is a simplifies to solution to the eigenvalue equation (13) where all physical q j observables i, i are continuous and interpolate between eff ½v ðθÞ − ξ∂ξnðθÞ¼0: the two reservoirs. With relativistic dispersion relation, the solution is smooth within the light cone, beyond which the This is the eigenvalue equation (13) in the occupation- states are constant and equal to the initial baths’ states; number coordinates (which diagonalize the Jacobian), and while in the Galilean case, the solution is generically its solution gives qðξÞ and jðξÞ via Eqs. (22) and (20). smooth on the whole space. In this solution, every normal One can show that the solution for the nonequilibrium, mode nðθÞ, seen as a function of ξ for fixed θ,is ray-dependent occupation number nðθÞ is the discontinu- discontinuous exactly at its propagation velocity. Every ous function normal mode therefore displays a “contact discontinuity” (a discontinuity without entropy production) [12]. Hence, the L R nðθÞ¼n ðθÞΘðθ − θ⋆Þþn ðθÞΘðθ⋆ − θÞ; ð35Þ rarefaction wave may be seen as being composed of infinitely many contact discontinuities. In contrast to the where Θð Þ is Heavyside’s step function. The position of finite-dimensional case, this single rarefaction wave can the discontinuity θ⋆ depends on ξ and is self-consistently account for generic reservoirs, and no shocks need to eff determined by v ðθ⋆Þ¼ξ; equivalently, it is the zero of develop. This is because in the infinite-dimensional case, the dressed, boosted momentum pξðθÞ ≔ pðθ − ηÞ, where the eigenvalues of JðnÞ form a continuum: all propagation ξ ¼ tanh η (relativistic case) or ξ ¼ η (Galilean case), velocities veffðθÞ are available as conserved charges guar- antee a large number of stable excitations, providing an dr pξ ðθ⋆Þ¼0: ð36Þ additional continuous parameter tuning the smooth state trajectory and guaranteeing its correct asymptotic-reservoir The GGE occupation numbers nL;RðθÞ entering Eq. (35) values. Since weak solutions (shocks) are not necessary to guarantee that the asymptotic conditions on ξ correctly connect the asymptotic reservoirs, they do not appear.

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V. ANALYSIS AND DISCUSSION 0.35 Concentrating on pure thermal transport, we analyze the 0.30 above general system of equations for two related models: 0.25 the relativistic integrable sinh-Gordon model and its non- 0.20 relativistic limit [65,66], the (repulsive) Lieb-Liniger 0.15 model. We also verify that our hydrodynamic equations 0.10 reproduce the known results for the case of free particles. 0.05 0.00 1.0 0.5 0.0 0.5 1.0 A. Relativistic sinh-Gordon model One of the simplest integrable relativistic QFTs with FIG. 2. The functions jðξÞ (dots) and eðξÞ (squares) for nontrivial interactions is the sinh-Gordon model. It is βL ¼ 1 and βR ¼ 30 in the sinh-Gordon model. defined by the Lagrangian [67,68]

2 1 2 m Fig. 4, the relative deviation of the steady-state current from L ∂μϕ − βϕ ; shG ¼ 2 ð Þ β2 coshð Þ ð37Þ its bounds [Eq. (3)] is shown. The bounds are extremely tight, pointing to the strength of this constraint and where ϕ is the sinh-Gordon field and m is the mass of the confirming that the proposed solution is correct. The single particle in the spectrum. The model is integrable and, bounds are indeed so tight that it is difficult to distinguish therefore, the only nontrivial scattering matrix is that some points in parts of Fig. 4. To better appreciate this, we associated to two-particle scattering. It is given by [69–71] present the numerical values of the points displayed in β2 Fig. 4 (divided by L) in Table I. tanh 1 ðθ − iπBÞ The numerical data are obtained by solving the integral SðθÞ¼ 2 2 : ð38Þ tanh 1 ðθ þ iπBÞ equations recursively until convergence is reached. Sources 2 2 of error are the discretization and finite range of θ for The parameter B ∈ ½0; 2 is the effective coupling constant, numerical integration. Adjusting the number of divisions which is related to the coupling constant β in the and the range, we estimate the error to be less than 0.1%. Lagrangian by B. Lieb-Liniger model 2β2 β λ 0 Bð Þ¼ 2 ; ð39Þ The Lieb-Liniger model, in the repulsive regime ( > ), 8π þ β can be regarded as a nonrelativistic limit of the sinh-Gordon model, as shown in Refs. [65,66]. The Hamiltonian of the under CFT normalization [72]. The S matrix is obviously model is given by invariant under the transformation B → 2 − B, a symmetry that is also referred to as weak-strong coupling duality, as Z 1 it corresponds to B β → B β−1 in Eq. (39). The point ∂ ψ †∂ ψ λψ †ψ †ψψ ð Þ ð Þ HLL ¼ dx x x þ : ð41Þ B ¼ 1 is known as the self-dual point. At the self-dual point 2m the TBA differential scattering phase is simply 2 0.25 φ θ − d θ shGð Þ¼ i log Sð Þ¼ : ð40Þ dθ cosh θ 0.20 Contrary to the Lieb-Liniger model, which we discuss 0.15 L later, the general features of any quantities of interest in the 2 0.10 sinh-Gordon model are very similar for any values of the coupling B. For this reason, in this paper we concentrate 0.05 our analysis solely on the self-dual point in the under- 0.00 standing that similar results hold for other values of B. 0.0 0.2 0.4 0.6 0.8 We evaluate the energy density e ≔ q1, energy current j ≔ j1, and pressure k ≔ j2. Typical profiles are shown β2 j ξ 0 β 30β FIG. 3. The functions L ð > Þ for R ¼ L and in Figs. 2 and 3. Figure 2 shows smooth interpolation −p βL ¼ 10 , with p ¼ 0 (stars), 1 (triangles), 2 (inverted trian- within the light cone between the asymptotic baths at gles), 3 (squares), and 4 (circles). The continuous bold line ξ −1 ξ 1 β2 j ξ π 1 − 1 ¼ and ¼ (the speed of light is set to 1). Figure 3 represents the conformal value L ð Þ¼12 ð 900Þ which, as shows how, as temperatures rise, the current approaches the expected, is reached at high temperatures. Dashed curves are plateau [Eq. (2)] predicted by CFT [26,27]. Further, in interpolations.

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R 30 L A uniform chemical potential μ is introduced, associated to 0.25 the conserved charge Q0 that counts the number of quasi- particles [with h0ðθÞ¼1]. The energy current is chosen to be the current associated to the charge H − μQ0, 0.20

j ≔ j1 − μj0 ðLL modelÞ: ð45Þ 0.15 Below, we present some numerical results for several values of the coupling λ and for m ¼ 1. 0.10 10 5 10 4 0.001 0.01 0.1 1 Current profiles obtained for λ ¼ 3 and various values of L μ are displayed in Fig. 5. The main difference between the relativistic and nonrelativistic cases is the lack, in the latter, FIG. 4. Verification of the inequalities Eq. (3) in the sinh- β2 jsta of any sharp light-cone effect. Nevertheless, at low temper- Gordon model. Displayed are the functions L (circles), 2 2 atures T ≪ μ, Luttinger liquid physics emerges [73], β ðeL − eRÞ=2 (triangles), and β ðkL − kRÞ=2 (squares). L;R L L including an emerging light cone due to the Fermi velocity. This can be seen in Fig. 5: a plateau forms whose height is This is obtained from the Hamiltonian of the sinh-Gordon again in agreement with the general CFT result Eq. (2). The model by a double limit, plateau lies between nearly symmetric values ξ=vF ≈ 1 pffiffiffi fixed by the Fermi velocity vF. Thermal occupation L;R c → ∞; β → 0; βc ¼ 4 λ; ð42Þ numbers n ðθÞ are very sharplypffiffiffiffiffiffiffiffiffiffiffiffi supported between θL;R θL;R ≳ 2μ Fermi points F , with =m, and the Fermi where c is the speed of light [which is implicit in Eq. (37)]. velocity, which depends on ξ very weakly, is the effective (This is the only equation in the present paper where the eff θR velocity v ð FÞ associated to the lowest temperature c c speed of light appears explicitly. Everywhere else denotes (TR

θ − 2λi θ L 1, R 5, 3 SLLð Þ¼ ; ð43Þ θ þ 2λi 0.25 and corresponding differential scattering phase 0.20 4λ 0.15 φ ðθÞ¼ : ð44Þ LL θ2 þ 4λ2 0.10 0.05

TABLE I. The functions jsta, ½ðeL − eRÞ=2, and 0.00 L R ½ðk − k Þ=2 over a wide range of values of βL. The bounds 2 1 0 1 2 [Eq. (3)] are always met. vF β jsta eL − eR 2 kL − kR 2 L ½ð Þ= ½ð Þ= FIG. 5. Energy current in the Lieb-Liniger model for low 10−5 2.5661 × 109 2.5701 × 109 2.5624 × 109 temperatures λ ¼ 3 and chemical potentials μ ¼ 3 (circles), μ 6 μ 10 10−4 2.5450 × 107 2.5522 × 107 2.5386 × 107 ¼ (squares), and ¼ (triangles). The CFT value π 1 − 1 10−3 250 665.6 252 117.9 249 421.1 12 ð 25Þ (bold horizontal line) is reached for high values of −2 μ ξ 10 2424.9 2461.8 2396.4 . By plotting the currents against =vF, we observe the collapse μ 10−1 22.0 23.3 21.1 of the various curves, which becomes better as increases. The ξ ∈ −1 1 1 0.126 0.181 0.101 regions where plateaus emerge are roughly =vF ½ ; with vF ≈ 2.5, 2.9, 3.6.

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L 1, R 5, 1, 0.05 This corresponds to a free Fermion, in agreement with the 0.14 expected Tonks-Girardeau physics occurring in the regime pffiffiffi 0.12 γ ≳ maxð1; τÞ [75].Forξ ≈ 0 and μβL;R ≫ 1, it is easy to 0.10 π β−2 − β−2 show that the integral above gives 12 ð L R Þ, so that we 0.08 recover the CFT result for the current with c ¼ 1 (Dirac 0.06 fermion). Figure 7 shows a comparison between numerical 0.04 values of the current for λ ¼ 50 and the formula above. 0.02 Let us now consider the particle current. Naturally, in the 0.00 LL model, equilibrium states at higher temperatures have 4 2 0 2 4 lower particle densities. Therefore, although the energy current flows from the left to the right in the present setup FIG. 6. Energy current in the Lieb-Liniger model for low (with TL >TR), the initial particle density imbalance temperatures, small coupling, and negative chemical potential would naively suggest a particle flow from the right (higher (circles). The dashed curve represents the current [Eq. (46)] for density) to the left (lower density). The opposite occurs: the same temperatures and chemical potential. Figure 9 shows that the particle current is positive, hence, flows form the left to the right. This means that the fluid with this differential scattering phase, admit no solution for flow produced by the temperature difference drags particles the pseudoenergy if μ > 0, but for μ < 0, they can be with enough force to counteract the particle imbalance and solved exactly and reproduce the free boson solution (for bring particles towards the higher-density bath. The fact which μ > 0 would make no physical sense). In particular, that heat carries particles along its motion is a thermoelec- the energy current takes the free boson form, tric effect. It has been experimentally demonstrated in a Z Z quasi-two-dimensional fermionic cold atoms channel [3], 1 ∞ θ 1 ∞ θ and theoretically shown in CFT in dimensions higher than 1 j ξ θ − θ lim ð Þ¼ 2 d θ 2 d θ ; ð46Þ [17]. It is nontrivial in integrable models, as the large λ→0 β α e − 1 β α e − 1 R R L L amount of conservation laws allows for independent 2 currents to coexist, and our result gives the first theoretical where α ¼ β ½ðξ =2Þ − μ. In Fig. 6, we compare L;R L;R prediction of this effect in the integrable one-dimensional numerical values for λ ¼ 0.05 and μ ¼ −1 to this analytical Bose gas. expression. The agreement is very good, confirming that a An additional consequence of the thermoelectric effect is free boson theory is smoothly recovered in this limit. With that the particle density q0ðξÞ shows particle accumulation μ > 0,asλ becomes small, the TBA equations gradually around v and depletion around −v (see Fig. 8). For break down. How this occurs is subtle, and will be F F instance, the start of the dip can be explained by the fact discussed in Ref. [74]. that, in any local spacial region originally in the left The qualitative change in behavior of the TBA solutions reservoir, the first particles to start moving towards the as λ → 0 is related to the two distinct regimes observed at small values of λ [75]. Consider the dimensionless coupling L 1, R 5, 6, 50 γ ≔ 2mλ=q0 (where we recall that q0 is the gas density, which may be taken in the initial baths, for instance) and 0.25 τ ≔ 2 q2 “ the reduced temperature mT= 0. The decoherent 0.20 regime,” with large phase and density fluctuations, occurs pffiffiffi for γ ≲ minðτ2; τÞ. In this regime, ideal Bose gas physics 0.15 is recovered, and we indeed verify that the inequality is 0.10 satisfied in the parameter space where good agreement with Eq. (46) is observed (small λ, negative μ). On the other 0.05 hand, the “Gross-Pitaevskii regime” occurs for τ2 ≲ γ ≲ 1, 0.00 a quasicondensate with large phase fluctuations but sup- 2 1 0 1 2 pressed density fluctuations. It is such quasicondensate 2 physics that strongly affects TBA solutions as λ → 0 with μ > 0. FIG. 7. Energy current in the Lieb-Liniger model for low μ 6 The other interesting limit is limλ→∞φ ðθÞ¼0. In this temperatures, large coupling, and chemical potential ¼ LL α 0 case, we can also find an analytical expression for the (circles).ffiffiffiffiffi Local stationary points occur at L;R ¼ , that is, ξ p2μ 3 46 current: ¼ ¼ . (the Fermi velocity). The dashed curve represents the current [Eq. (47)] for the same temperatures and Z Z 1 ∞ θ 1 ∞ θ chemical potential, whose profile is not dissimilar to the plots j ξ θ − θ lim ð Þ¼ 2 d θ 2 d θ : ð47Þ shown in Fig. 5. As before, the bold horizontal line is the CFT λ→∞ β α e 1 β α e 1 π 1 R R þ L L þ value 12 ð1 − 25Þ. The agreement is extremely good.

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L 1, R 5, 6, 3 R 3; L 0.1 1.90 1.0 1.89 0.5 1.88

0 1.87 0.0 q 1.86 0.5 1.85 1.84 1.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 4 2 0 2 4 vF ξ 0 FIG. 8. A characteristic profile of the Lieb-Liniger particle FIG. 10. Effective velocity in the sinh-Gordon model for ¼ . Displayed are the effective velocity veff ðθÞ (blue line) and the density for TL;R ≪ μ, λ ¼ 3, and μ ¼ 6. The local maxima and bare relativistic velocity tanh θ (red line). minima are located around ξ ¼vF. The dashed curve is an interpolation. the initial GGEs, in a way that depends on the rapidity. It right are those on the right of the region, escaping and thus connects with the picture, proposed in Refs. [26,47], depleting it. Since time evolution at fixed position is according to which, in the steady state (ξ ¼ 0), quasipar- obtained by scanning Fig. 8 from left to right, this explains ticles traveling towards the right (left) are thermalized the initial dip on the left. This depleting effect continues as according to the left (right) reservoir. However, in the long as the outgoing current on the right of the region is present solution, what determines the traveling direction is higher then the incoming current on its left—that is, until the effective velocity in the steady state: quasiparticles with the region lies in the steady state. However, as time goes on, positive (negative) dressed velocities, reaching the point the effective local temperature decreases, and this tends to x ¼ 0 at large times, will have traveled mostly towards the increase the particle density. This effect eventually over- right (left) (after a relatively small transient). In the sinh- takes the depleting effect, accounting for the rebounce to Gordon model with TL >TR, the effective velocity the higher steady-state value. The behavior of the current behaves as in Fig. 10. We observe that it is greater than θ j0 in Fig. 9 is then a consequence of the continuity the bare velocity tanh for small or negative rapidities, and equation ξ∂ξq0 ¼ ∂ξj0. smaller for large positive rapidities. This is in agreement This is a nonuniversal effect, not present in the density with the intuition according to which the quasiparticles are q1ðξÞ − μq0ðξÞ controlled by low-energy processes, where effectively carried by the flow, which transports them the physics of chiral separation dominates and monotonic towards the right, for small enough rapidities, but slowed “ ” transition regions occur. down by dominant friction effects of thermal fluctua- tions at large rapidities. A similar effect occur in the Lieb-Liniger model. C. General features The generalized hydrodynamic result differs from pre- The form of the nonequilibrium occupation number vious proposals in interacting integrable models [47–49] indicates that quasiparticles are thermalized according to (while all results agree in noninteracting cases). The original proposal [47] was later shown [44] to break the second inequality in Eq. (3), while the second proposal L 1, R 5, 6, 3 [48], based on similar ideas, gave slight disagreements with 0.08 numerical simulations. The conjecture [49], which corre- sponds to taking θ⋆ ¼ 0 in our framework, seems to give 0.06 good agreement with numerical simulations. This may be θ 0 0 due to the fact that taking ⋆ ¼ , gives very small errors in j 0.04 wide temperature ranges, of the order of 0.5%–1% (we 0.02 confirm this numerically).

0.00 1.5 1.0 0.5 0.0 0.5 1.0 1.5 VI. CONCLUSIONS v F In this paper, we develop a hydrodynamic theory for FIG. 9. A characteristic profile of the Lieb-Liniger particle infinitely many conservation laws and apply it to the study current for TL;R ≪ μ, λ ¼ 3, and μ ¼ 6. The local maxima and of heat flows in experimentally relevant integrable models. minima are located around ξ ¼vF. The dashed curve is an It would be interesting to study further the nonequilibrium interpolation. physics of the Lieb-Liniger model, including the effects of

041065-11 CASTRO-ALVAREDO, DOYON, and YOSHIMURA PHYS. REV. X 6, 041065 (2016) Z Z the Gross-Pitaevskii quasicondensate and transport dxhq ðxÞj ð0Þic ¼ dxhj ð0Þq ðxÞic between different regimes. The emerging physical picture m n n m Z and solution we give can be applied to any Bethe ansatz 0 c integrable model, where infinitely many conservation laws ¼ dxhjnðxÞqmð Þi exist and a quasiparticle description is available. This Z includes quantum chains (see Ref. [64]), as continuity of c ¼ − dxxh∂xjnðxÞqmð0Þi space on which the microscopic theory lies is not needed Z for emerging hydrodynamics. It also includes relativistic ∂ 0 c models with nondiagonal scattering, such as the sine- ¼ dxxh tqnðxÞqmð Þi Gordon model, where, for instance, our TBA construction Z − ∂ 0 c may be generalized along the lines of the famous approach ¼ dxxhqnðxÞ tqmð Þi of Destri and De Vega [76,77]. Of course, the hydro- Z dynamic ideas do not require a quasiparticle description, ∂ c ¼ dxxhqnðxÞ yjmðyÞi jy¼0 and it might be possible to develop generalized hydro- Z dynamics using a variety of techniques from integrability. − ∂ 0 c We note that it is remarkable that independent quasiparticle ¼ dxx xhqnðxÞjmð Þi mode thermalization agrees, in integrable models, with Z c local entropy maximization. The dynamical equations ¼ dxhqnðxÞjmð0Þi : ðA1Þ derived can be used to describe more general situations in ultracold gases, such as the release from a trap (see, e.g., R In the first line, we use the fact that dxq ðxÞ is a Ref. [78]). This new theory and its extensions, including m conserved quantity and thus commutes with the density viscosity effects and forcing, should also allow for efficient ρ studies of integrability breaking and related problems in matrix ; in the second line, we use space translation any dimensionality, as well as for exact descriptions of invariance; in the third, integration by parts and the fast- dynamics in smooth trapping potentials [4] at arbitrary enough vanishing of correlation functions; in the fourth, coupling strength. current conservation; in the fifth, time-translation invari- ance; in the sixth, current conservation; in the seventh, space-translation invariance; and in the eighth, integration ACKNOWLEDGMENTS by parts. Therefore, We thank Denis Bernard, M. Joe Bhaseen, Jean- ∂ ∂ Sébastien Caux, Fabian Essler, Mauricio Fagotti, and j j n ¼ m; ðA2Þ Eric Lutz for discussions, and especially Bruno Bertini ∂βm ∂βn and Lorenzo Piroli for sharing their preliminary ideas and Pasquale Calabrese for encouraging us to pursue this and, thus, research. O. A. C.-A. and B. D. are grateful to SISSA, ∂ Trieste, Italy, for support during a visit where this work j m ¼ gβ; ðA3Þ started. T. Y. thanks the Takenaka Scholarship Foundation ∂βm for financial support. Open access for this article was funded by King’s College London. showing Eq. (6). In the TBA context, we note that expressions (22) show Note added.—Recently, similar dynamical equations as the existence of appropriate free energies fw and gw those derived here were independently obtained in the generating densities and currents, respectively, as in integrable XXZ Heisenberg chain by assuming, in addition Eq. (6). Indeed, they may be rewritten as to local entropy maximization, an underlying kinetic theory Z δf [64]. Solutions to these equations of the same type as those q ¼ dθh ðθÞ w ; considered here were constructed and confirmed by i i δwðθÞ Z numerical simulations. dpðαÞ −ϵ α − 1 wð Þ fw ¼ 2π logð þ e ÞðA4Þ APPENDIX A: CURRENT GENERATORS and Let h iβ be the state given by Eq. (4), and a x b y c ≔ a x b y − a x b y the con- Z h ð Þ ð Þi h ð Þ ð Þiβ h ð Þiβh ð Þiβ δg j ¼ dθh ðθÞ w ; nected correlation functions. These are time independent i i δwðθÞ and functions of the difference x-y only. Let us assume that Z dEðαÞ −ϵ α connected correlation functions of conserved densities and wð Þ gw ¼ − logð1 þ e Þ: ðA5Þ currents vanish faster than the inverse distance jx-yj. Then, 2π

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It then follows that that functional wðθÞ derivatives of these in the internal space. Let the spectrum be composed of l free energies give the quasiparticle and current densities, particles, of masses ma, a ¼ 1; …; l, and assume that their ρpðθÞ¼δfw=δwðθÞ and ρcðθÞ¼δgw=δwðθÞ. scattering is diagonal. In this case, the TBA equations can still be applied [61,79]: the differential scattering phase is φ θ − γ APPENDIX B: EMERGENCE OF GENERALIZED replaced by a matrix of functions abð Þ, and the one- θ; HYDRODYDNAMICS particle eigenvalue of Qi is denoted by hið aÞ. The solution qðξÞ, jðξÞ of the generalized hydrodynamic The only principle at the basis of hydrodynamics, and of problem is the derivation we provide, is that of the emergence of local Z generalized thermalization (local entropy maximization). X dp θ; a q ξ ð Þ θ; dr θ; Technically, this is the assumption that averages of local ið Þ¼ nð aÞhi ð aÞ; O 2π quantities h ðx; tÞi tend uniformly enough, at large x and t, a Z to averages evaluated in GGEs (infinite-volume, maximal- X dE θ; a j ξ ð Þ θ; dr θ; entropy states, under conditions on infinitely many con- ið Þ¼ 2π nð aÞhi ð aÞ; ðC1Þ servation laws), with space-time-dependent potentials. This a assumption is sufficient to derive the explicit dynamics for where pðθ; aÞ¼m sinh θ, Eðθ; aÞ¼m cosh θ, and all single-point averages of local conserved densities and a a currents: no ad hoc kinetic principle is needed. Z dγ X In the case of infinitely many conservation laws, one dr θ; θ; φ θ − γ γ; dr γ; hi ð aÞ¼hið aÞþ 2π a;bð Þnð bÞhi ð bÞ: delicate point is the consideration of quasilocal densities b and currents, which are involved in generalized thermal- ðC2Þ ization. Such a quantity is not supported on a finite region, but on an extended region, with a weight (as measured by, The nonequilibrium occupation number nðθ; aÞ is given by for instance, the overlap with any other local observable) the discontinuous function that decays away from a point. However, since hydro- dynamics is concerned with large-scale space-time regions θ; L θ; Θ(θ − θ ) R θ; Θ(θ − θ) (the fluid cells), it is natural to consider them on the same nð aÞ¼n ð aÞ ⋆ðaÞ þ n ð aÞ ⋆ðaÞ ; footing. This is implicitly done in the derivation we present ðC3Þ in this paper by assuming a completeness property of conservation laws. where each particle a is associated to a different dis- Another delicate point concerns the definition of GGEs. continuity at position θ⋆ðaÞ. These positions are self- In finite systems, such states depend on the boundary consistently determined by the zeros of the dressed, conditions imposed, and, in general, these boundary con- boosted momenta of particles a; with pξðθ; aÞ ≔ ditions may still have an effect in the infinite-volume limit. ma sinhðθ − ηÞ (relativistic) or maθ (nonrelativistic): For instance, walls simply preclude any nonzero potential associated to the momentum operator, as they break dr p (θ⋆ðaÞ; a) ¼ 0;a¼ 1; …; l: ðC4Þ translation invariance. Nevertheless, given a set of allowed ξ conserved charges, at large volumes, boundary conditions Again, the thermal occupation numbers nL;Rðθ; aÞ entering have little effect on local averages of conserved currents ξ and densities (as they do not affect specific free energies). Eq. (C3) guarantee that the asymptotic conditions on Further, periodic boundary conditions, at the basis of the correctly represent the asymptotic baths, as per Eq. (14). TBA formalism, appear to provide the maximal set of They are obtained using the TBA equations in terms of the ϵ θ; conserved charges. It is, in fact, possible to construct GGEs pseudoenergies wð aÞ [61,79], directly in infinite volumes [21]. We expect local thermal- 1 ization, and the full set of available conserved charges, to be L;R θ; n ð aÞ¼1 ϵ θ; ; correctly described by such constructions, and we expect þ exp½ wL;R ð aÞ these to agree with the TBA formalism we use here. ϵwðθ; aÞ¼wðθ; aÞ− We finally mention that the classical hard-rod problem, Z X dγ −ϵ γ; proven to give rise to a form of hydrodynamics [37,53], has − φ θ − γ 1 wð bÞ 2π a;bð Þ logð þ e Þ: strong connections with the integrable systems we inves- b tigate here, which we will investigate in a future work. ðC5Þ P APPENDIX C: MANY-PARTICLE SPECTRUM L;R θ; βL;R θ; Here, w ð aÞ¼ i i hPið aÞ are the one-particle βL;R The theory we develop here is directly applicable to any eigenvalues of the charge i i Qi characterizing the integrable model whose two-particle scattering is diagonal GGE of the left and right asymptotic reservoirs.

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APPENDIX D: CURRENT AVERAGES θ⃖O θ~ ≔ c O; θ~ h j j ic F2lð Þ An alternative proof of Eq. (22) may be given using the ⃖ ≔ FP lim F2lðO; θ~ þ iπ þ δ~; θÞ; ðD5Þ δ →0 technology of integrable systems, which has the advantage k of generalizing to flows generated by any conserved charge θ⃖O θ~ ≔ s O; θ~ instead of just the Hamiltonian. For completeness, we h j j is F2lð Þ present here the main arguments. The idea is to prove ⃖ j ≔ limF2lðO; θ~ þ iπ þ δ; θÞ; ðD6Þ expression (22) for current averages i given the expres- δ→0 sion for density averages qi. This is akin to extending the LeClair-Mussardo (LM) formula [80] so that it incorporates where FP means “finite part” [81], δ~ ¼ðδ1; …; δlÞ, and the the infinite number of conserved charges, and applying it to FF FlðO; θ~Þ is defined by the current with the aid of form factors (FFs). We use the notation introduced in Ref. [81]. Following the derivation FlðO; θ~Þ¼hvacjOð0Þjθ~i: ðD7Þ in Ref. [81], we generalize the LM formula for a one-point O function of a generic local operator ðx; tÞ: Notice that with a limit such as in Eq. (D5), where the δ parameters k differ in each component, different orders of X∞ 1 Yl θ limits lead to different results which may be singular; this is d k ⃖ ~ hOðx; tÞi ¼ nðθkÞ hθjOð0Þjθi ; ðD1Þ l! 2π c because when δk → 0, the FF [Eq. (D5)] becomes singular l 0 1 ¼ k¼ due to kinematic poles. It is in order to circumvent this ambiguity that one defines connected and symmetric FFs. ⃖ where jθ~i¼jθ1; …; θli (and hθj¼hθl; …; θ1j is its The connected FF is a finite part, which simply prescribes Hermitian conjugate) and diagonal matrix elements to set to zero terms with singularities in δk [81], whereas the (DMEs) in the sum are connected (the meaning of being symmetric FF is defined by sending all parameters to zero “connected” is described below). Here, nðθÞ is the same simultaneously. occupation number as that involved in Eqs. (20), (21), and It was pointed out in Ref. [83] that any multiparticle (22). It is then immediate to see that an expression for the symmetric FF can be written solely in terms of the q θ density average i with the one-particle eigenvalue hið Þ connected FFs. For instance, for a two-particle state, the c proved by Saleur [82] is modified to connected FF F4ðO; θ1; θ2Þ and the symmetric FF s F4ðO; θ1; θ2Þ satisfy ∞ l X Y dθ q m k n θ c O; θ θ s O; θ θ − φ θ O; θ i ¼ 2π ð kÞ F4ð 1; 2Þ¼F4ð 1; 2Þ ð 12ÞF2ð 1Þ l¼0 k¼1 − φðθ21ÞF2ðO; θ2Þ; ðD8Þ × φðθ12Þ…φðθl−1;lÞhiðθ1Þ cosh θl; ðD2Þ c s where F2ðO; θÞ¼F ðO; θÞ¼F ðO; θÞ (in the case of a φ θ φ θ − θ 2 2 where ð ijÞ¼ ð i jÞ. Observe that this is indeed in single parameter δ1, there is no singularity, hence, no agreement with the expression in Eq. (22). The expression ambiguity). Applying this relation to ji, we have (D2) can be derived using the DMEs of qiðx; tÞ given by c s F4ðji; θ1; θ2Þ¼F4ðji; θ1; θ2Þ − φðθ12ÞF2ðji; θ1Þ ⃖ hθjq jθ~i ¼ mφðθ12Þφðθ23Þ…φðθl−1 lÞ i c ; − φðθ21ÞF2ðji; θ2Þ: ðD9Þ

× hiðθ1Þ cosh θl þ permutations: ðD3Þ This can be expressed in terms of FFs of the density qi thanks to the conservation law ∂tqi þ ∂xji ¼ 0, which Similarly, once we evaluate DMEs for the current jiðx; tÞ, j entails we can construct its average i. The expression in Eq. (22), that we want to show, will then follow if the DMEs of the P l sinh θ s ; θ~ Pk¼1 k s ; θ~ currents are obtained from those of the densities by the F2lðji Þ¼ l F2lðqi Þ: ðD10Þ 1 cosh θ replacement of cosh θn with sinh θn: k¼ k Hence, putting Eq. (D10) into Eq. (D9) yields θ⃖ θ~ φ θ φ θ φ θ h jjij ic ¼ m ð 12Þ ð 23Þ ð l−1;lÞ c F ðj ; θ1; θ2Þ¼mφðθ12Þh ðθ1Þ sinh θ2 þfθ1 ↔ θ2g; × hiðθ1Þ sinh θl þ permutations: ðD4Þ 4 i i ðD11Þ Before embarking upon showing it, we elaborate on the definitions of connected and symmetric DMEs. Formally, which is consistent with Eq. (D4). It is readily seen that for they are given by, respectively [83], multiparticle states, similar arguments hold, and, thus, we

041065-14 EMERGENT HYDRODYNAMICS IN INTEGRABLE QUANTUM … PHYS. REV. X 6, 041065 (2016) obtain Eq. (D4). Finally, the generalized LM formula for [13] J. Bhaseen, B. Doyon, A. Lucas, and K. Schalm, Far from the current gives Equilibrium Energy Flow in Quantum Critical Systems, Nat. Phys. 11, 509 (2015). [14] H.-C. Chang, A. Karch, and A. Yarom, An Ansatz for One X∞ Yl dθ Dimensional Steady State Configurations, J. Stat. Mech. j ¼ m k nðθ Þ i 2π k (2014) P06018. l 0 1 ¼ k¼ [15] D. Bernard and B. Doyon, A Hydrodynamic Approach to × φðθ12Þφðθl−1;lÞhiðθ1Þ sinh θl: ðD12Þ Non-Equilibrium Conformal Field Theories, J. Stat. Mech. (2016) 033104. This exactly coincides with Eq. (22). Similar arguments [16] R. Pourhasan, Non-Equilibrium Steady State in the Hydro Regime, J. High Energy Phys. 02 (2016) 005. give rise to current averages associated to flows i½Qk;qiþ [17] A. Lucas, K. Schalm, B. Doyon, and M. J. 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