Collision rate ansatz for quantum integrable systems

For quantum integrable systems we revisit the currents averaged with respect to a generalized Gibbs ensemble. In case the system has a self-conserved current, i.e. some current is actually conserved, the symmetry of the current-charge susceptibility matrix implies the conventional collision rate ansatz. The argument is carried out in detail for the Lieb-Liniger model and the Heisenberg XXZ chain. We also explain how from the existence of a boost operator a self-conserved current can be deduced.


Introduction
Hydrodynamics is a universal tool to describe the long-wavelength dynamics of manybody systems, both quantum and classical. The cornerstone of hydrodynamics is the assumption of local equilibrium and its stable propagation in spacetime. Thereby the complex dynamics of a many-body system is guided by interactions between conserved charges only [1]. As a consequence, the dynamics is determined by a coupled set of continuity equations for the average charge densities and currents. Such a system closes only if all local conservation laws are included. For a generic system one expects to have a few of them, hence the description only involves a few coupled hyperbolic conservation laws. But for integrable dynamics the conserved fields are labelled by a spectral parameter from the real line, or even larger sets, depending on the model. Such generalized hydrodynamics (GHD) is particularly useful for quantum integrable systems, for which, even numerically, tracing the late-time dynamics is notoriously difficult mainly due to the rapid increase of entanglement across distant spatial regions [2][3][4].
The hydrodynamics of integrable systems was accomplished in 2016 [5,6], giving rise to a flux of related studies [7][8][9][10][11][12][13][14][15][16][17][18][19][20], including the determination of Drude weights [21], Green-Kubo type formulas for the transport coefficients [22,23], and applications to classical integrable systems [24][25][26][27]. On the ballistic spacetime scale GHD turns out to have a particularly simple structure, since charge densities and currents evaluated with respect to a generalized Gibbs ensemble (GGE) [28] can be written in terms of the thermodynamic Bethe ansatz (TBA), which is already known as a systematic method in the study of thermodynamics of quantum integrable systems. Compared to statics the novel key element is the effective velocity v eff (θ) as a function of the rapidity θ. This quantity describes the velocity of quasiparticles at the hydrodynamic scale and thereby expresses current densities as a nonlinear functional of the charge densities. The functional form of v eff was first conjectured in [5,6]. In the former one finds a sketchy reasoning as well as an argument from the crossing symmetry in case of relativistic field theories with diagonal scattering. The effective velocity is written as the solution of a rate equation counting the number of collisions per unit time experienced by a single tracer quasiparticle in a fluid of quasiparticles distributed according to some GGE. In this article we use the notion collision rate ansatz, as reflecting the physics intuition behind the defining formula for v eff .
While the formula for v eff was rapidly adopted, satisfactory theoretical arguments have become available only recently: the form factor expansion is used to establish the collision rate ansatz for both diagonally-scattering relativistic field theories [29] and the XXZ spin-1 2 chain [30]. In [31] the collision rate ansatz is confirmed for the models solvable by nested Bethe ansatz by employing a relation derived from long-range deformations of the chain. The aim of this manuscript is to add a very different line of arguments for justifying the collision rate ansatz. In fact, our argument is more direct and resorts neither to form factor expansions nor deformations. Our method is based on the availability of a self-conserved current. By this we mean a current, which itself appears in the list of conserved charges. For the Lieb-Liniger model the particle current is momentum which is itself conserved. Also, as well-known, in the XXZ spin- 1 2 chain the energy current is selfconserved. However, for the Fermi-Hubbard model the energy current is not conserved [32] and possibly the model has no self-conserved current at all. More generally, the existence of a self-conserved current is ensured by the boost operator, which is the first moment of some conserved charge density [33,34]. Thus our result could be rephrased that the existence of a boost operator alone suffices to validate the collision rate ansatz.

Collision rate ansatz for integrable field theories
Integrable quantum systems have an extensive number of (quasi-)local conserved charges. To simplify, we shall focus on the case of single quasi-particle species with diagonal scatterings. The more complicated structure of the XXZ model will be discussed in Sect. 3. The charges are denoted by Q j = dx q j (x), j = 0, 1, ... , in particular [H, Q j ] = 0. In the generalized Gibbs ensemble (GGE) each one of them is controlled by a chemical potential, µ j , and the corresponding density matrix reads From the field theory under consideration one has given a dispersion relation E(θ) as a function of the rapidity θ with the momentum p(θ) = E ′ (θ). In fact, our the argument will be written out in detail for the Lieb-Liniger, a Galilei-invariant field theory, but with a notation which will make the application to other field theories straightforward. We recall that for the Lieb-Liniger model E(θ) = 1 2 θ 2 in units for which the bare particle mass m = 1. Furthermore given is the two-body scattering matrix S(θ, ϑ), in terms of which the two-particle differential scattering kernel is given by The free energy of the system can be computed from the TBA equations with h j (θ) the one-particle eigenvalue associated to the charge Q j , h j (θ) = θ j in our case. From the pseudo-energy ε one obtains the occupation function n(θ) = 1/(1 + e ε(θ) ) = ρ(θ)/ρ tot (θ) with ρ the density of particles and ρ tot the density of states, related through In terms of these quantities, the GGE average of a charge density, q[h j ] := q j (0) GGE , can be written as [36] q Here, for any function f (θ) we use the shorthand f = R dθf (θ). The dressing transformation is defined through In the context of GHD it was a major discovery that the current average also admits a similar TBA expression [5,6]. The microscopic current is defined through the continuity Then the time t = 0 total current is given by J j = dx j j (x, 0) and the corresponding GGE average equals with the effective velocity v eff given as solution of the rate equation Its physical interpretation has been mentioned already, but can now be stated more precisely. θ is the spectral parameter of the tracer quasi-particle, which is moving in a fluid characterized by the density ρ(ϑ). The bare velocity of the tracer particle is E ′ /p ′ , which is modified through collisions with fluid particles. Under integral the first factor is the jump size of either sign and the second factor is the number of collisions per unit time [10].
Our task is to establish the ansatz (8) on the basis of a given microscopic model, for which purpose a convenient form of the effective velocity is The first input to our proof are the charge-charge and current-charge susceptibility matrices which are defined by [21] with the superscript referring to connected correlation functions. The matrix C is symmetric by construction. Less obvious, but also B is symmetric. Making use of the conservation laws, spacetime stationarity, and clustering of connected correlation functions [23], one arrives at which implies the symmetry of B.
The second input is the existence of a self-conserved current. For the Lieb-Liniger model J 0 = Q 1 , hence J 0 , which is the total current associated to the particle number operator Q 0 = N, is self-conserved. As will be discussed, for other models there might be a different self-conserved current. The symmetry of B yields then the following nontrivial identity Next note that by linearity in h j the left identity of (7) still holds provided v eff (θ) is replaced by the yet unknown current densityv(θ). Therefore (13) becomes satisfied for all j. Since the space spanned by h j 's is complete one arrives at the pointwise identity To be shown isv = v eff . From differentiating the TBA equations with respect to µ 0 , µ 1 the relations hold and imply Using one arrives at Altogether we obtained (3) one infers that the pseudo-energy ε(θ) ≃ µ 0 for µ 0 → ∞, thus n(θ) = 1/(1 + e ε(θ) ) → 0. Since ρ tot (θ) is uniformly bounded in µ 0 , also ρ vanishes. We conclude that the free constant must be zero, establishinḡ The only property needed for the above argument is the existence of a self-conserved current. In Galilei-invariant theories with the particle number conservation, the number current equals the momentum and our requirement is satisfied. Thereby our argument can be extended to other Galilei-invariant theories such as the Gaudin-Yang model [39], which is solved by a nested Bethe ansatz. In relativistic field theories Lorentz invariance ensures a distinct self-conserved current. In this case, the energy current J 2 coincides with the momentum operator Q 1 , from which ∂ µ 1 q j = ∂ µ 2 j j follows. In single-species models with diagonal-scatterings, the dispersion relation has the relativistic form p(θ) = m sinh θ, E(θ) = m cosh θ, m the particle mass, and the task then boils down to show This can be confirmed in a similar fashion as above by noting that h dr 2 (θ) = E dr (θ) = 2πρ tot (θ). We therefore conclude Since h 2 (θ) = m cosh θ > 0, this time in the µ 2 → ∞ limit n(θ) → 0, hencev = v eff . As in the non-relativistic cases, the above argument for relativistic theories can be straightforwardly generalized to other relativistic models such as the sine-Gordon model and the O(N) non-linear sigma model, in which cases the energy current is quasi-conserved. However, the situation is different for integrable spin chains, which do not possess an evident continuous symmetry implying the existence of a self-conserved current. As discussed in Sect. 4, the boost operator could be useful tool in finding such a current.
3 Collision rate ansatz for the XXZ spin-1 2 chain For the XXZ spin-1 2 chain the energy current is self-conserved. Because of strings in the Bethe equations the structure of the charges is more involved than for Lieb-Liniger. Thus the model is an interesting test for our method.
The hamiltonian of the XXZ model reads where we set J = 1. The TBA structure of the chain strongly depends on the value of ∆ [40] and, as a consequence, the Drude weight changes sensitively with the isotropy parameter ∆ [6,37,38]. However the energy current is self-conserved for any value of ∆, which is the only requirement for our argument to work. For concreteness, we focus on the gapless regime here (|∆| < 1), but the gapped regime can be handled in a similar fashion. The structure of TBA for the gapless XXZ spin-1 2 chain can be arranged so as to become rather similar to that for the Lieb-Liniger model. This is achieved by choosing particular values of ∆, which are called roots of unity, withl the length of the continued fraction and some positive integers ν 1 , · · · , ν ℓ−1 ≥ 1, ν ℓ ≥ 2, ℓ = 1, ...,l, using Pringsheim's notation. The number of strings equals s = l ℓ=1 ν ℓ , hence finite for such a ∆. The set of string labels is denoted by S = {1, ...., s}. The resulting TBA equations now involve various types of strings [40]. Apart from the fact that there are more particle types (strings), as a further modification of the TBA equations, the overall sign of p ′ j (λ) depends on the type j ∈ S. This is a consequence of a reparametrization of rapidities so as to make the differential scattering kernel T jk (λ) symmetric, which in turn induces a change to the integration measure dλ → j σ j dλ [23]. Here σ j = sign(q j ), where q j is related to the parity of j-th string and depends on ∆ [40].
In the gapless phase of XXZ, ρ tot j (λ) is given by ρ tot j (λ) = σ j (p ′ j ) dr (λ)/(2π), where for any function f j (λ) the dressing transformation is defined by Note that by convention the sign of T is opposite to the one used in the previous section. It will be convenient to work with integral operators. They act on functions over R × S, which are equipped with the standard scalar product Then, employing integral operators, (25) becomes To be complete, in the gapless phase of the XXZ spin-1 2 chain the bare momentum p j (λ) and the differential scattering kernel T jk (λ − µ) are where n j and v j are the length and the parity of the j-th string. For instance, when ω = π/ν with 2 ≤ ν ∈ N, those n j , q j , v j 's are given by [40] n j = j, v j = 1, q j = ν − n j , j = 1, 2, . . . , ν − 1, At the roots of unity, there is a family of quasi-local conserved charges Q (s) n labeled by integers n ∈ N and half-integer s ∈ 1 2 N, which corresponds to the higher-spin representation of U q (sl (2)) [42]. The energy current is Q , hence conserved. Writing the one-particle eigenvalue of the charges as h (s) n,j (λ), the GGE average of charge and currents densities take a form similar to the continuum case, Now, let us consider the energy current , it follows that which is in agreement with Q Having these relations at our disposal, let us proceed to the proof. As in the field theory case, (13) and its consequence (15) are the key identities. The only difference to these identities is that the self-conserved current is J . Denoting the Lagrange multipliers associated to Q (1/2) 2 and Q (1/2) 1 by µ 2 and µ 1 , respectively, we notice first that which implies ∂ µ 2 (p ′ ) dr = ∂ µ 1 (E ′ ) dr . As a final step, which then yields ∂ µ 1 [ρ j (λ)(v j (λ) − v eff j (λ))] = 0. For given j the energy one-particle eigenvalue equals E j (λ) = − 1 2 (sin ω)p ′ j (λ) with the property that either E j (λ) > 0 or E j (λ) < 0. In the former case, we let µ 1 → ∞. From the TBA it follows that n j (λ) → 0 in this limit. In the latter case we let µ 1 → −∞ and, as before, conclude that n j (λ) → 0. Thus the free constant vanishes and ρ j (λ)(v j (λ) − v eff j (λ)) = 0 for all j, λ, implying the desired result,v j (λ) = v eff j (λ).

Boost operator in spin chains
As we saw in the previous section, the existence of a self-conserved current gives rise to the collision rate ansatz. One then might wonder how such a current can be obtained in general. Indeed, even without invoking integrability, the existence of a self-conserved current can be directly inferred from either Galilei or Lorentz symmetry of the quantum field theory under consideration. This is no longer true for spin chains where such a continuous symmetry is absent and a self-conserved current has to be found along an alternative route. An essential tool for this task turns out to be the boost operator. First we briefly recall its basic property in the context of XYZ spin- A tower of conserved charges can be systematically obtained by the row-to-row transfer matrix as hence Let us consider some operator O which is constructed from a local density o(j) through O = j∈Z o(j). Then the boost operator is defined through The boost operator associated to the Hamiltonian 1 K[H] = j∈Z jh(j) indeed generates a boost, which is evident from the commutation relation with the transfer matrix which in turn amounts to [33,34] [ The fact that the boost operator K[H] generates the conserved charges recursively bears momentous implications. We recall the continuity equation in spin chains i[H, q n (j)] = j n (j) − j n (j + 1).
Multiplying j to both sides and summing over j, we formally obtain As was remarked in [31] the relation (43) is only formal, and is in general plagued by the divergence stemming from the charge density with an infinitely large coefficient. Nevertheless such a divergence can always be circumvented by subtracting a conserved charge Q n with a correspondingly diverging prefactor.
In spin chains, it is conventional to choose Q 0 = N = n S z n and Q 1 = H. Then, choosing n = 1 in (41) and (43), we observe that J 1 = j∈Z j 1 (j) is a self-conserved current, i.e.
Note that the above construction of a self-conserved current suggests J 1 being actually the only self-conserved current under the Hamiltonian flow.
In fact, the recursive commutation relations (41) and (43) can be thought of as the lattice analogue of the Poincaré algebra. Indeed, in the continuum limit the XYZ spin chain becomes the relativistic massive Thirring/sine-Gordon model [34]. The upshot of this limit is that the first few commutation relations (41) reduce to the usual Poincaré algebra in (1+1)-dimension, which is closed in itself, Using (43) this implies J E = P , which is what one would expect from Lorentz invariance. Naturally one can further take the non-relativistic limit of the Poincaré algebra, which is nothing but the Galilean algebra. In particular, when the resulting theory has U(1)symmetry (e.g. conserves particle number), such as the Lieb-Liniger model, the Galilean algebra is centrally extended to the Bargmann algebra whose commutation relations read where N = Q 0 is the U(1) charge. This algebra then entails J 0 = P , which again is merely a consequence of Galilean invariance. So far we have demonstrated that the XYZ spin-1 2 chain, hence also the XXZ spin-1 2 chain, possesses a self-conserved current j 1 thanks to the boost operator. This is also true for other integrable spin chains, provided that there is a boost operator which satisfies (41) and (43). A natural question is then, whether there are integrable systems which for some reason fail to have a boost operator of the form (39)? The answer is yes, and a notable example is the Fermi-Hubbard model (FHM), for which the energy current is not conserved [32]. This is consistent with the fact that FHM does not have the standard boost operator, and the lack of it suggests that there could be no self-conserved current at all. In fact, at the root of the existence of such a boost operator is the lattice Lorentz invariance of the system whose algebra is given by the ladder commutation relations (41) [34]. The invariance under a lattice Lorentz boost manifests itself through the Rmatrix of the system being of the form R(λ, µ) = R(λ −µ), hence invariant under a boost. The lattice Lorentz invariance reduces to the standard continuum Lorentz invariance in the continuum limit. FHM does not allow such invariance, since the model does not admit any continuum limit under which Lorentz invariance is achieved. Indeed, the low-energy physics of FHM is not a Luttinger liquid, but instead charges and spin carry gapless excitations with different velocities, which implies that the physics depends on the frame. This being said, it is actually possible to define a slightly generalized boost operator in FHM, which still satisfies (41) [41]. As a caveat, the generalized boost operator is not exactly the same as (39) and the connection to the conservation laws (42) is lost. Therefore a self-conserved current in FHM, if it should exist, has to be looked for by other means.

Conclusions
In this article, we proved the collision rate ansatz for a wide class of quantum integrable systems, once the existence of a self-conserved current is ensured. Surprisingly, the existence of such a self-conserved current is directly linked to the boost operator, which is written as the first moment of some charge density. In fact, the construction of a selfconserved current can be immediately extended to the generalized currents describing the flow of other conserved charges. Generalized currents j n,m (j) associated to Q n are defined by the continuity equation [16,31] i[Q m , q n (j)] = j n,m (j) − j n,m (j + 1) for each flow generated by Q m . Of course, m = 1 corresponds to the standard Hamiltonian flow. We then find that along each m-th flow there is always a unique self-conserved total current j∈Z j 1,m (j) = J 1,m = Q m+1 . Indeed, such a boost operator has been used to implement long-range deformations of integrable spin chains, from which the finitevolume diagonal matrix elements of current operators were obtained [31]. It would be very interesting to figure out the connection between the use of the boost operator in our proof and the one in [31], with the hope to better understand the overarching role of the boost operator in GHD.
Finally, let us remark that the approach followed here is in spirit the same as the one for the classical Toda lattice [43]. In this model the stretch current equals the negative of the momentum, hence is indeed conserved. We expect that our approach will be applicable to a larger variety of integrable systems, both quantum and classical, provided that the system has a self-conserved current.

Ackowledgement
TY is grateful to Enej Ilievski for useful comments on the Fermi-Hubbard model, and in particular informing him a paper [41], in which the (generalized) boost operator in the model is discussed. HS thanks Tomohiro Sasamoto for his generous hospitality at Tokyo Institute of Technology.