Diffusion in generalized hydrodynamics and quasiparticle scattering

1) We think that our claims and assumptions are now well spelled (we have made an explicit list of the two main assumptions in the introduction now). We shall repeat here once again our findings A— We have developed an hydrodynamic theory for the description of large scales interacting integrable models that account for diffusive terms. Hydrodynamics is a theory valid at large scales based on certain assumptions that we have provided here.

B— We have computed exactly the Onsager matrices of interacting integrable systems via a thermodynamic form factor expansion. The poles of the 2 particle-hole thermodynamic form factor have been conjectured via an educated guess that agrees many different checks.

2) One is indeed free to choose an arbitrary Gauge. Please see reply to 2nd referee.

1) Thank you, we have reformulated the sentence.

2) Thank you very much for pointing out the misprint.

The hydrodynamical approach holds for any choice of densities of charges. That is, the form of eq 2.10 is valid for any choice of gauge. However, the explicit function $\mathfrak{D}_i^{~j}(\bar{\mathfrak{q}})$ (and the form factors we have used to compute correlation functions) depends on the choice of gauge. Thus, one needs to fix one specific gauge in order to obtain a specific function (any gauge will do, but let us fix one). Under change of gauge, the function transforms according to specific rules, as expressed in appendix C.1. We here choose the PT symmetry for two reasons. The first is that it simplifies the relation between the Diffusion matrix and the Onsager matrix. By imposing PT symmetry we only need to compute the Onsager matrix and the susceptibility matrix in order to have the diffusion matrix. The second reason is that eventually we want to write the densities of the charges in terms of the densities of quasiparticle, eq 3.25. Since the rapidity parametrisation in densities of quasiparticles is usually chosen PT invariant (in analogy to GGE states), in order to do that it is simpler to impose PT symmetry. While this mapping is useful to carry on actual calculations, it does not mean that PT symmetry is fundamental for the hydrodynamical approach. We commented on this in the manuscript, footnote 3.  

3) We do not understand the reason of the referee’s confusion. The expectation value on the right hand side of B.3 is taken with respect to a density matrix of the form $\exp[\int dx \sum_j \beta_j(x) q_j(x)]$. That is, one has $<o(y,t)> = Z^{-1} Tr ( \exp[\int dx \sum_j \beta_j(x) q_j(x)] U_t(o(y)) )$ where $U_t$ is the time-evolution unitary operator (Heisenberg picture). Therefore differentiating the numerator with respect to $\beta_j(x)$ produces $Tr ( \exp[\int dx \sum_j \beta_j(x) q_j(x)] U_t(o(y)) q_j(x) )$, and the contribution of the denominator Z gives the connected average, the right hand side of B.3. One does not differentiate the evolution operator, only the initial distribution. Thus no integral over the time of the second operator appears. This is all commented below eq B.3. The generating functional does not involve integrals over tau of $\beta_j(x, \tau)$, as the perturbation is only in the initial state, there is no $\beta_j(x, \tau)$, there is only $\beta_j(x)$. After taking the derivative, one sets $\beta_j(x) = \beta_j$, independent of $x$, so that the correlation function becomes homogeneous and stationary (a homogeneous GGE): we are considering linear response correlations, namely small perturbation on top a homogenous GGE state. We would also like to mention that eq B.4, consequence of B.3, is relatively standard in the hydrodynamic theory.