# Exact large-scale correlations in integrable systems out of equilibrium

### Submission summary

 As Contributors: Benjamin Doyon Arxiv Link: https://arxiv.org/abs/1711.04568v4 (pdf) Date accepted: 2018-11-14 Date submitted: 2018-11-05 01:00 Submitted by: Doyon, Benjamin Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

Using the theory of generalized hydrodynamics (GHD), we derive exact Euler-scale dynamical two-point correlation functions of conserved densities and currents in inhomogeneous, non-stationary states of many-body integrable systems with weak space-time variations. This extends previous works to inhomogeneous and non-stationary situations. Using GHD projection operators, we further derive formulae for Euler-scale two-point functions of arbitrary local fields, purely from the data of their homogeneous one-point functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuation-dissipation principle 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 sinh-Gordon model and the Lieb-Liniger 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 Leclair-Mussardo-type infinite form-factor series in integrable quantum field theory, and exact Euler-scale two-point functions of exponential fields in the sinh-Gordon model and of powers of the density field in the Lieb-Liniger model. We also analyze correlations in the partitioning protocol, extract large-time asymptotics, and, in free models, derive all Euler-scale $n$-point functions.

### Ontology / Topics

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Published as SciPost Phys. 5, 054 (2018)

### Author comments upon resubmission

I thank the referees for their careful consideration of the manuscript, and especially for both pointing out that the assumption about monotonicity of the effective velocity in rapidity is too strong; I was clearly too fast in making the assertion. [I think in the free-chain examples that both referees gave, the assumption *is* satisfied, with an appropriate choice of spectral space: one may divide the momenta into two regions, in such a way that within each region the velocity is monotonic, and one can see each regions as corresponding to a different particle type. However I don't think such a construction can be done generically in interacting systems.]

Indeed as pointed out the assumption is not necessary for the solution to the partitioning protocol. In fact, I realised that it was not necessary for any result I have presented - it was just simplifying my life in characterising the solutions to certain equations, but is in fact not strictly needed. Thus I have modified the discussion of this assumption on page 13, making it a remark only, and I have make appropriate modifications throughout in order to account for this: all places where the derivative of the effective velocity appeared through Jacobian I have added absolute values; in sections 5.3 and E.2 I have taken away the requirement of the monotonicity assumption, and I have adjusted the sentence between eq 3.24 and 3.25 on p 19.

However, perhaps the most interesting realisation from thinking about this is that in general, the rapidity derivative of the effective velocity may vanish. In this case, some large-time asymptotics, at certain rays for instance in the partitioning protocol (e.g. near the maximal velocity), may be modified. I think this is a potentially very interesting effect, which I keep for future works. I have added a paragraph about this in the conclusion, and also a short comment in the Remark on page 13.

I have also corrected all typos found by referee 2.

### List of changes

Absolute value for derivative of effective velocity in eqs. 3.36, 4.19, 4.23, 5.12, 5.17, 5.19, 5.21, E.15, E.17, E.20, E.22, E.23, E.24, E.28 and eq above - E.31, E.33

paragraph added in conclusion

discussion adjusted in section 5.3 (p35) and E.2 (p47)

discussion adjusted and remark added p13

adjusted the sentence between eq 3.24 and 3.25 on p 19

### Submission & Refereeing History

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Resubmission 1711.04568v4 on 5 November 2018
Resubmission 1711.04568v3 on 18 August 2018
Submission 1711.04568v2 on 20 February 2018