Exact large-scale correlations in integrable systems out of equilibrium

1- New analytic findings for space-time averaged dynamical correlation functions in integrable systems out of equilibrium
2- The results sound general
3- There are explicit examples

1- The notations are not always standard, and readability is compromised by the large number of definitions
2- The analytic results are not checked against numerics

This paper addresses the problem of computing correlation functions in inhomogeneous states that time evolve under the Hamiltonian of an integrable system. The author identifies and compute some quantities that can be accessed within the framework of the so-called ``generalized hydrodynamics'', which is a theory recently developed to deal with inhomogeneities in integrable systems. In particular, the author exhibits analytic expressions for space-time averaged dynamical correlation functions (he calls it ``Eulerian scaling limit for correlation functions'').
The paper is very long and rather technical, but, undoubtedly, the author made an effort to present the results in a simple way. Considering also their generality, the results are extremely interesting, therefore I strongly recommend this paper for publication in Scipost after minor revision, detailed in ``Requested changes''.

1- In the middle of page 4, the author writes ``In integrable quantum spin chains, two-point functions in Gibbs state have been calculated [63,64], but it is unclear how to extend to GGEs''. I think that the situation is less obscure than it is presented. Indeed, also Ref. [83] is a generalization of [63,64] to GGEs; as far as I know, the first papers generalizing [63,64] to GGEs were
[] B. Pozsgay, J. Stat. Mech. (2013) P07003;
[] M. Fagotti and F.H.L. Essler, J. Stat. Mech. (2013) P07012.
2- I think that there is a typo in the definition of $T^T$ just below (2.4).
3- The author presents the theory in a very general way. There are however equations that could be less general than expected, and I wonder whether such a general presentation is really worth. For example, I'm not completely sure that the first equation in (2.6) holds true also in the gapless XXZ model, where the equation could be correct only up to the sign. Can the sign be simplified by redefining the various quantities?
4- I think that I understand the logic behind (2.19) and (2.20), however I'm wondering whether the limiting procedure could have subtleties. In (1.5) the operators are averaged over a a space-time region whose extent scales as $\lambda^\nu$. If, in (2.19), one considers the case $x_n-x_m\sim O(\lambda^\beta)$, the validity of (2.19) could depend on how big $\beta$ is with respect to $\nu$, couldn't it?
5- The variable $y$ in the definition (2.25) is not defined.
6- Two lines below (2.26), ``... are monotonically increasing functions of the velocity.'' Is ''velocity'' a typo? If not, which velocity?
7- Could the author explain the comment above (3.1), that is to say, ``In quantum models, terms coming from nontrivial commutators between local conserved densities are negligible in the Euler scale, contributing only to higher-order derivatives''?
8- I'm not sure to understand the physical meaning of ``observables perturb the state'' at the end of section 5.2.1.

1- Important advances on a very interesting problem;
2-Timely;

- See the report

The paper studies nonequilibrium dynamics of integrable systems in inhomogeneous settings focussing on the determination of dynamical correlation functions. Specifically, the author considers situations treatable using the recently introduced theory of generalised hydrodynamics and proposes new formulae for the dynamical connected two-point functions of generic local observables in the ``Eulerian scaling limit" of large distances and times. In this limit, the state of system can be thought of as a collection of stationary states one at each space-time point. States on a given time slice are uncorrelated, however, non trivial correlations can be observed considering observables at different times. The author combines a generalised fluctuation dissipation theorem with the a "non-linear method of characteristics" and some Thermodynamic Bethe ansatz identities to determine new formulae for the connected two-point functions of charge densities and currents in the scaling limit. The latter are used to determine two point functions of generic operators in the framework of "hydrodynamic projection theory".
Two point-functions of charge densities and currents have a relatively simple expression, and require the solution of a linear integral equation. Instead, two-point functions of generic observables also depend on a function written in terms of an infinite form-factor series. In some special cases, using some known results for one-point functions in homogeneous settings, this function can be written in a simpler way in terms of the solution of an integral equation. The author also outlines a general inductive strategy to determine $N$-point functions and carries it out explicitly in the non-interacting case. Finally, he applies his results to the case of partitioning protocol and finds that the two point function $\langle{{q}_i(\xi t,t)q_j(y,0)}\rangle$, where $t=0$ is the initial time and $y=0$ is the junction, depends on how precisely the two states are connected.

The examples given in the paper are centred on the case of integrable quantum field theories but the treatment is kept at a general level and it is applicable also to classical integrable field theories and, with some caveats (see below), to integrable quantum spin chains.

I think that the paper is very interesting, it provides novel and highly non-trivial results further expanding the generalised hydrodynamics theory, and the derivation is mathematically sound. Therefore, I recommend the publication of this paper in SciPost. Before publication, however, the author should improve some aspects. First of all assumption (i) at page 12 (namely $p'(\theta)>0$ and $(v^{eff})'(\theta)>0$) is not fulfilled in integrable quantum spin chains (in short-ranged quantum spin chains both the momentum and the effective velocities are non monotonic functions of the rapidity, this is the case for the XY and the XXZ models for example). This means that all the results obtained on the basis of this assumption are not immediately applicable to integrable quantum spin chains but need some modification. This point should be clearly stressed and the text modified accordingly. Moreover, even if the author moved most of the technical parts in the appendices, some passages are difficult to read (see the detailed points below).

Another point that the author might want to consider is to add some further numerical checks, complementing those of [16], at least in the case of non-interacting systems. Such checks would in my opinion improve the paper.

1- In the introduction I suggest to move the paragraph

"Here we use a continuous space notation x, and the trace notation Tr. This is for convenience, and the problem is posed in its most general setting, for classical (where the trace means a summation over classical configurations) or quantum models, on a one-dimensional infinite space that can be continuous or discrete."

after Eq. (1.1), as the notation described is introduced there.

2- When discussing the various spectral expansions for GGE two-point functions (or Gibbs two-point functions that can be extended to GGE) I suggest to add also "Essler and Konik, J. Stat. Mech. (2009) P09018" to Refs.[53-56];

3- Can the author explain why the averaging in Eq. (1.5) is not expected to be necessary for one point functions?

4- What do the author mean with "involving TBA strings if the fundamental scattering is nondiagonal" at page 7? TBA strings are interpreted as bound states, they are not related non-diagonal scattering.

5- I suggest to give more explanation on the physical meaning of vector and scalar fields in Sec 2.1.

6- For consistency, in the discussion after Eq. (2.14) I suggest to use $\boldsymbol \theta$ also for the argument of $n_t(x,\boldsymbol \theta)$.

7- I find the discussion after Eq. (2.2) confusing. I suggest of using something on the lines of "the space of pseudolocal charges" instead of "the space spanned by $h_i(\boldsymbol\theta)$" because this gives the wrong impression that the author considers $h_i(\boldsymbol\theta)$ observables in the Hilbert space.

8 - In the discussion above equation 3.2 I suggest adding "of charge densities and currents", as the previous discussion was generic.

9- It seems that there is some confusion in the referencing of equation 3.4. In several cases (e.g. above Eq. 3.5 and in Appendix B1) what is referenced as Eq. 3.4 is the unnumbered equation above.

10- At the top of page 17 I suggest to replace "these ingredients" with " Eq. (3.11)" as that is the only ingredient needed. Moreover, I suggest to include Eq. (3.11) among the main results of the paper stressed in the discussion after Eq. (3.18).

11- I find Sec. 3.3 difficult to read. First of all, when talking about the hydrodynamic projection theory, I suggest to give the main ideas of the theory. For example the author could move there the brief discussion which is now at the beginning of Sec. 3.4. Second, Eqs. (3.28) and (3.29) appear to me as a rewriting of (3.16) - (3.18) and not a derivation of them as stated at the beginning of the sub-section. In my understanding such re-derivation is carried out in the remark. In summary: I suggest the author to reorganise this sub-section to improve readability.

12- I think that Eq. (3.34) should be explained in more detail. In particular it could be helpful for the reader to stress that the particular form of the resolution of the identity used is due to the non-orthogonality of the basis. Moreover, the author could add another step between the second and the third line.