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Diffusion in generalized hydrodynamics and quasiparticle scattering

by Jacopo De Nardis, Denis Bernard, Benjamin Doyon

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Submission summary

Authors (as registered SciPost users): Jacopo De Nardis · Benjamin Doyon
Submission information
Preprint Link:  (pdf)
Date submitted: 2019-03-05 01:00
Submitted by: De Nardis, Jacopo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical


We extend beyond the Euler scales the hydrodynamic theory for quantum and classical integrable models developed in recent years, accounting for diffusive dynamics and local entropy production. We review how the diffusive scale can be reached via a gradient expansion of the expectation values of the conserved fields and how the coefficients of the expansion can be computed via integrated steady-state two-point correlation functions, emphasising that PT-symmetry can fully fix the inherent ambiguity in the definition of conserved fields at the diffusive scale. We develop a form factor expansion to compute such correlation functions and we show that, while the dynamics at the Euler scale is completely determined by the density of single quasiparticle excitations on top of the local steady state, diffusion is due to scattering processes among quasiparticles, which are only present in truly interacting systems. We then show that only two-quasiparticle scattering processes contribute to the diffusive dynamics. Finally we employ the theory to compute the exact spin diffusion constant of a gapped XXZ spin 1/2 chain at finite temperature and half-filling, where we show that spin transport is purely diffusive.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2019-3-24 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1812.00767v3, delivered 2019-03-24, doi: 10.21468/SciPost.Report.888


as before


as before


The authors have responded to all the points raised in my previous report and I am satisfied with the response in most cases.

I have two comments with regard to the amended version of the manuscript:

1) In my view, a distinction between phenomenological results, conjectures of possibly exact formulas (e.g. for form factors) and proofs is still not always clear. The manuscript, however, allows readers to form their own views so I do not want to insist on further changes.

2) I thank the authors for extending the section about the gauge and the diffusion matrix. Clarifying that the Onsager coefficients L_{ij} are gauge invariant is important. However, I am now wondering why choosing any particular gauge (such as the one using PT symmetry) is of relevance at all. If the L_{ij}'s are invariant: Why can I not choose any arbitrary gauge as long as I stay consistent?

Requested changes

i) The authors should further clarify point 2) above.

  • validity: high
  • significance: high
  • originality: good
  • clarity: good
  • formatting: excellent
  • grammar: excellent

Author:  Jacopo De Nardis  on 2019-04-04  [id 483]

(in reply to Report 3 on 2019-03-24)

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.

Anonymous Report 2 on 2019-3-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1812.00767v3, delivered 2019-03-18, doi: 10.21468/SciPost.Report.875


as before


as before


I thank the authors for their in general adequate response and for the
amendments of their manuscript.

I made a few observations.

1) In the new final paragraph of section 1 the following sentence needs
a reformulation:

"That is, we suppose that, at large times, the relevent of degrees of freedom
is reduced to the local mean charge densities..."

2) Thank you for extending appendix for explanations on "gauge covariance".

(In appendix C the word "gauge" is mispelled as "gaude".)

It is good to know that (top of page 11):

"It is possible to show, assuming the validity of the hydrodynamic projection
[72, 101], that the Onsager coefficients Lij are invariant under (2.28)."


"The hydrodynamic approximation of the currents (2.9) is explicitly dependent
on the choice of densities. See Appendix C."

but why do you say

"One must therefore choose a gauge in order to fix the diffusion matrix

Why not "Use any gauge and stick to it"? Maybe only very special gauges allow
for a hydrodynamical approach?

3) I am still having problems with appendix B. I find (B.3) problematic. The
derivation uses a generating functional with time-independent fields
\beta_j(x). Hence on the RHS of (B.3) instead of a two-point correlator with
second time variable identical to 0 an integral over a two-point correlator
with the second time variable as variable of integration should appear.

If (B.3) in the literal form is to be derived, the generating functional
should involve fields like \beta_j(x,tau).

Requested changes

see report

  • validity: high
  • significance: high
  • originality: high
  • clarity: ok
  • formatting: reasonable
  • grammar: excellent

Author:  Jacopo De Nardis  on 2019-04-04  [id 484]

(in reply to Report 2 on 2019-03-18)

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.

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