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Exact largescale correlations in integrable systems out of equilibrium
by Benjamin Doyon
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Authors (as registered SciPost users):  Benjamin Doyon 
Submission information  

Preprint Link:  http://arxiv.org/abs/1711.04568v2 (pdf) 
Date submitted:  20180220 01:00 
Submitted by:  Doyon, Benjamin 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Using the theory of generalized hydrodynamics (GHD), we derive exact Eulerscale dynamical twopoint correlation functions of conserved densities and currents in inhomogeneous, nonstationary states of manybody integrable systems with weak spacetime variations. This extends previous works to inhomogeneous and nonstationary situations. Using GHD projection operators, we further derive formulae for Eulerscale twopoint functions of arbitrary local fields, purely from the data of their homogeneous onepoint functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuationdissipation theorem along with the exact solution by characteristics of GHD, and gives a recursive procedure able to generate $n$point correlation functions. Owing to the universality of GHD, the results are expected to apply to quantum and classical integrable field theory such as the sinhGordon model and the LiebLiniger model, spin chains such as the XXZ and Hubbard models, and solvable classical gases such as the hard rod gas and soliton gases. In particular, we find LeclairMussardotype infinite formfactor series in integrable quantum field theory, and exact Eulerscale twopoint functions of exponential fields in the sinhGordon model and of powers of the density field in the LiebLiniger model. We also analyze correlations in the partitioning protocol, extract largetime asymptotics, and, in free models, derive all Eulerscale $n$point functions.
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Reports on this Submission
Anonymous Report 2 on 2018522 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1711.04568v2, delivered 20180522, doi: 10.21468/SciPost.Report.463
Strengths
1 New analytic findings for spacetime averaged dynamical correlation functions in integrable systems out of equilibrium
2 The results sound general
3 There are explicit examples
Weaknesses
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
Report
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 socalled ``generalized hydrodynamics'', which is a theory recently developed to deal with inhomogeneities in integrable systems. In particular, the author exhibits analytic expressions for spacetime 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''.
Requested changes
1 In the middle of page 4, the author writes ``In integrable quantum spin chains, twopoint 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 spacetime region whose extent scales as $\lambda^\nu$. If, in (2.19), one considers the case $x_nx_m\sim O(\lambda^\beta)$, 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 higherorder derivatives''?
8 I'm not sure to understand the physical meaning of ``observables perturb the state'' at the end of section 5.2.1.
Anonymous Report 1 on 2018516 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1711.04568v2, delivered 20180516, doi: 10.21468/SciPost.Report.448
Strengths
1 Important advances on a very interesting problem;
2Timely;
Weaknesses
 See the report
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 twopoint 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 spacetime 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 "nonlinear method of characteristics" and some Thermodynamic Bethe ansatz identities to determine new formulae for the connected twopoint 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 pointfunctions of charge densities and currents have a relatively simple expression, and require the solution of a linear integral equation. Instead, twopoint functions of generic observables also depend on a function written in terms of an infinite formfactor series. In some special cases, using some known results for onepoint 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 noninteracting 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 nontrivial 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 shortranged 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 noninteracting systems. Such checks would in my opinion improve the paper.
Requested changes
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 onedimensional 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 twopoint functions (or Gibbs twopoint functions that can be extended to GGE) I suggest to add also "Essler and Konik, J. Stat. Mech. (2009) P09018" to Refs.[5356];
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 nondiagonal 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 subsection. In my understanding such rederivation is carried out in the remark. In summary: I suggest the author to reorganise this subsection 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 nonorthogonality of the basis. Moreover, the author could add another step between the second and the third line.