Diffusion in generalized hydrodynamics and quasiparticle scattering

1) We believe that we have spelled out in the introduction which are the assumptions underlying our derivation, especially now in the resubmitted version. Please notice that ref.[86], which deals with free fermionic theories, is not in contradiction with our results as this reference pointed the existence of higher order corrections to Euler hydrodynamics that start at third order in derivative expansion for free theories (while the diffusion processes we are dealing here are at the second order and are absent in free theories)

2) See answer to referee 1. We have added a detailed discussion on gauge invariance and gauge covariance in Appendix C. We also improved clarity in the main text. Please do not confuse the microscopic diffusion, defined via the diffusive spreading of correlation functions, with the diffusion matrix.

3) Thank you for pointing this possibly confusing notation. We have optimised and uniformised our notation.

i) Even if some part of those chapters is partially spread out in the literature, we believe it is useful for the scientific community to have a consistent presentation as we have proposed. Chapters 1-4 contain a large quantity of unpublished material.

ii) By this statement we mean that the diffusion matrix is zero in free systems. This was recently put forward in ref.[69] and it is confirmed by our analysis. The example mentioned by referee 2 deals with open systems (which of course may show diffusive behaviours) while we are here dealing with closed systems. We actually wrote explicitly in the introduction that « confirming the general intuition that there is no diffusive dynamics in non-interacting systems [69,86] (except, potentially, with external disorder [87–90]) ».

iii) Thank you. We corrected it.

iv) We have added references to the previous proof, and give more detailed steps in the Appendix D. But since both referees are not happy with calling this a proof we changed the title of the appendix to “derivation” instead of proof.

v) Thank you for pointing out these misprints.

vi) It is true that the simple diffusion equation comes directly from the hydrodynamic assumption with the extra information that $v_\infty=0$ (by symmetry) in this particular case. However, we do provide a computation of the diffusion constant as a function of all external parameters (temperature, etc.) in the XXZ chain from the microscopic model and checked it numerically. This prediction contains new information: We show that in the limit of half filling (states with zero total magnetisation) the spin ballistic current (or alternatively, the spin Drude weight) is zero and the diffusion constant is not. We analytically and numerically evaluate the latter, and we compare with state-of-the-art DRMG simulations in that specific model.

vii) Thank you. We corrected it.

viii) Actually we already mentioned in Section 2 and 6 the possibility of super-diffusion, which will manifest itself through divergences in the Onsager matrix (for example in the Heisenberg XXX chain at finite temperature). We pointed this again in the introduction as this is an interesting point to investigate further. This is a subject of current research.

ix) Formulas 6.34 for the spins diffusion constant at half-filling (zero magnetic field) is a direct consequences of our main result, formula 4.22 for the Onsager matrix and its derivation from the latter is the main scope of section 6. Formula 4.22 is based on a conjecture on the poles of the form factors. We believe that this is very different from stating that 6.34 is a conjectured equation.

We thank the referee for his.her insightful comments which led, we believe, to clarifications of important aspects of the discussion that are now in the resubmitted version.


We have spelled out in the main text which are the assumptions underlying our derivation. In the introduction, we have clarified the main hypotheses for the paper: those underlying the hydrodynamic approach, and those used for the form factors. The former is a standard set of hypotheses dating from more than one century. The latter is grounded on many examples, but a general theory is still missing and worth investigating further (see the recent paper [137]).

“In (2.6) and (2.7) functions \cal F and \cal D are postulated allowing to express the expectation value of j from the expectation value of q and its derivative in case of inhomogeneous systems.”

it maybe worth clarifying that we actually compute the Onsager matrix, which code for the microscopic diffusive spreading of the charges correlation functions. The diffusion coefficient \mathfrak{D} is then reconstructed using the hydrodynamics assumptions. The validity of the hydrodynamic assumptions have been proved in mathematically rigorous way only for a very restricted set of models (which however include the classical hard rod model, see ref.[80]).

“From (2.7) the relation (B.3) between the <jq> and <qq> correlations is "derived". I do not trust this derivation as (B.2) is used with a generating function depending on some generating field beta(x) which may or may not depend on time. If it does not depend on time, then a time integral on the r.h.s. of (B.2) should appear. If the field beta(x) is time dependent, then relation (2.7) must have built in a dependence on this field and then additional terms appear in (B.3).
The next step in the computation of the diffusion matrix makes use of the partial differential equation for the <qq> correlator and explicit results for this correlator.”

Thank you for pointing out the lack of clarity here. The question of the referee concerns the dynamical variables used in hydro. We had discussed aspects of this in section 2, however we have tried to further clarify and added more explanations: sentences around the new equation 2.4, and explanations in the appendix B. The main point is that the dynamical variables of hydro are the conserved densities evaluated on a given time slice. The choice of reference time slice is arbitrary (similarly to, for instance, classical field theory). These dynamical variables evolve in time according to the hydrodynamic equations. Therefore, local changes of initial conditions of this dynamical system - local changes of $\bar{\mathfrak{q}}_i(x,0)$ - produce changes of averages evaluated at any positive time, because such averages are evaluated by time evolving from a perturbed initial condition. By statistical mechanics principle, a change of initial condition at some point $x$ is equivalent to the insertion of a local observable at $x$ and $t=0$. There is no time integration, as only the initial condition is changed; the time evolution is unchanged. The effect is of course nevertheless nontrivial on observables evaluated at all later times, and this is described by a correlation, as on the right hand side of B.3. We assume there is a change that exactly inserts local conserved densities; the associated variables are the local effective temperatures $\beta_i(x)$.

“It is, however, quite disturbing that the results for the diffusion matrix depend on the "gauge" of the local conserved operator. Adding a divergence field preserves the continuity equation, but changes the diffusion matrix. The authors state this honestly by "One must therefore choose a gauge in order to fix the diffusion matrix". And they do it by choosing a gauge rendering q and j invariant under PT transformation. They even prove that there is a unique gauge to this end. But they do not prove why this yields the physical diffusion matrix. I would find the procedure presented by the authors convincing if they could prove that the two differing diffusion matrices for the left hand side and the right hand side of (2.26) would result in compatible L-coefficients (by use of (2.14)).”

Thank you again for this good question. There was already a comment in particular about the independence of the L matrix upon choice of gauge, but it was not clear and not well stated. We have clarified the discussion in the main text, see the first paragraph of section 2.4, and we have added further details in appendix C, sub appendix C.1. The Drude weight $D_{ij}$ and the Onsager coefficients ${\cal L}_{ij}$ are indeed gauge invariant (and the latter code for the microscopic diffusive spreading of the correlation function). The coefficients ${\cal D}_{i}^{~j}$ is on the hand not invariant, but covariant, under gauge transformation, as it depends on the specific gauge for local conserved charges.

“The authors use the current formula (3.18) and comment after (3.20) that in Appendix D we also provide an alternative proof for this formula. I do not see any proof of (3.18) in the main body of the manuscript. So it is not appropriate to call Appendix D an "alternative" proof. (In addition I have to note that the "proof" of Appendix D is full of assumptions.)”

We are sorry about this confusion. What we meant was that there already are proofs of this formula in the literature, we have added references to these (Refs 49,105, 106). As for the proof in appendix D, too many steps were implicit. The main points are as follows. (1) Given the existence of a form factor expansion, the results from thermodynamic Bethe ansatz give us the one-particle form factors of densities and currents at equal particle and hole rapidities. Indeed, TBA gives eq 3.39, while the form factor expansion gives 3.40. Comparing, we get the density form factors 3.37. The continuity relation then gives the result for the currents. (2) We can then evaluate $B_{ij}$ using form factor expansion. At the same time, the average currents in homogeneous stationary states are from general principle linear functionals of the $h_i(\theta)$, thus expressible as integrals over $h_i(\theta)$ times some state-dependent function of rapidity. Thus the form-factor expression for $B_{ij}$ gives a first-order differential equation for this state-dependent function. The effective velocity times particle density is a solution to this equation. We assume uniqueness of the solution. This is the single nontrivial assumption.

"Many calculations depend on regularization schemes like (3.37), but I trust that the authors here apply (their own) state of the art results for the Lieb-Liniger gas"

We are sorry for the confusion but we stated that the final result for the diffusion matrix does not depend on the details of the regularisation scheme.
We stressed now below eq 3.30 that the only requirement is that the form factor expansion eq 3.30 exist with regularised integrations on the real axis and below eq 4.15 that eventually the final result does not depend on the regularisation.