## SciPost Submission Page

# Diffusion in generalized hydrodynamics and quasiparticle scattering

### by Jacopo De Nardis, Denis Bernard, Benjamin Doyon

### Submission summary

As Contributors: | Jacopo De Nardis |

Arxiv Link: | https://arxiv.org/abs/1812.00767v3 |

Date submitted: | 2019-03-05 |

Submitted by: | De Nardis, Jacopo |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Condensed Matter Physics - Theory |

### Abstract

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:

### Submission & Refereeing History

- Report 3 submitted on 2019-03-24 19:46 by
*Anonymous* - Report 2 submitted on 2019-03-18 19:40 by
*Anonymous*

## Reports on this Submission

Show/hide Reports view### Anonymous Report 3 on 2019-3-24 Invited Report

### Strengths

as before

### Weaknesses

as before

### Report

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.

### Anonymous Report 2 on 2019-3-18 Invited Report

### Strengths

as before

### Weaknesses

as before

### Report

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)."

and

"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

itself."

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