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Exact large-scale correlations in integrable systems out of equilibrium

by Benjamin Doyon

Submission summary

As Contributors: Benjamin Doyon
Arxiv Link:
Date accepted: 2018-11-14
Date submitted: 2018-11-05
Submitted by: Doyon, Benjamin
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Mathematical Physics


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.

Current status:

Ontology / Topics

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Euler hydrodynamics Fluctuation-dissipation theorem Generalized hydrodynamics (GHD)

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

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