# How generalized hydrodynamics time evolution arises from a form factor expansion

### Submission summary

 As Contributors: Axel Cortes Cubero Arxiv Link: https://arxiv.org/abs/2001.03065v2 (pdf) Date submitted: 2020-09-24 13:50 Submitted by: Cortes Cubero, Axel Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

The generalized hydrodynamics (GHD) formalism has become an invaluable tool for the study of spatially inhomogeneous quantum quenches in (1+1)-dimensional integrable models. The main paradigm of the GHD is that at late times local observables can be computed as generalized Gibbs ensemble averages with space-time dependent chemical potentials. It is, however, still unclear how this semiclassical GHD picture emerges out of the full quantum dynamics. We evaluate the quantum time evolution of local observables in spatially inhomogeneous quenches, based on the quench action method, where observables can be expressed in terms of a form factor expansion around a finite-entropy state. We show how the GHD formalism arises as the leading term in the form factor expansion, involving one particle-hole pair on top of the finite-entropy state. From this picture it is completely transparent how to compute quantum corrections to GHD, which arise from the higher terms in the form factor expansion. Our calculations are based on relativistic field theory results, though our arguments are likely generalizable to generic integrable models.

###### Current status:
Editor-in-charge assigned

I thank the referees for their time and their valuable comments on the manuscript. I have addressed the several issues raised by both referees in this version, so I hope it is now suitable for publication. Below is the list of changes, corresponding to the referee requests.

### List of changes

Referee 1:

I added a discussion at the end of Section 5, pointing out that indeed, only part of the TBP formalism is used or tested in this paper, and a number of TBP axioms were not used at all. This is because in the leading term we consider, only zero-momentum particle hole form factors are needed, and these can be computed explicitly, without relying on all of the axioms. If we compute subleading terms, such as those discussed in Section 8, then we would need form factors away from the zero-momentum limit, in which case they need to be computed through the TBP axioms (or some other method).

The definition of connected form factors and their regularization is elaborated more in Section 5, and we added reference to the original papers. In the previous version it was not clearly specified what the finite part meant (specifically the term that remains finite when any of the regulators goes to zero, individually), but it should now be clear.

Referee 2:

Indeed, I agree the definition of “dressed” treated loosely in the previous version. This is now clarified in the discussion of Section 2, defining the two different quantities as “dressed” and “effective”.

I don’t know of a formal mathematical proof that these linear quench actions have only one saddle point. I modified the discussion to express the point that the corresponding saddle point equation can be solved numerically with a straightforward iterative approach. This seems to converge to one solution saddle point, with no indication that there is a problem of many saddle points, so that the existing evidence is only numerical in this sense.

3) A comment has been added pointing out the local density approximation.

4) I removed the use of the word “explicitly” as indeed, the solution is expressed in terms of another equation that needs to be solved. This expression, however, makes the physical content of the evolution equation very clear, in a way that can be directly compared with the results from the form factor approach.

5) I agree this discussion was perhaps unclear, and I now think distracting from the arguments of the paper, so I decided to remove it entirely. Certain aspects of the coarse grained approach that were still needed for the calculation were moved to Section 6.

6) From the subleading terms that were discussed in this paper, there doesn’t appear to be a term which decays as $1/\sqrt{t}$, but the leading correction is instead $1/t$. I added a discussion pointing out that in fact there may still be other corrections with possibly square root decay if we consider a different initial state. We studied an initial state which is already slowly varying and the inhomogeneity is very smooth, such that hydrodynamical scale arguments could be applied since $t=0$. There could be additional terms if the initial state is less smooth, then perhaps only at late times it becomes smooth.

7) I corrected the typos, thanks for pointing them out!

### Submission & Refereeing History

Resubmission 2001.03065v2 on 24 September 2020
Submission 2001.03065v1 on 6 February 2020

## Reports on this Submission

### Strengths

1. Timely subject.
2. Establishing a link between different approaches (GHD vs. form factor methods).
3. Generally applicable framework.
4. Proposal for the leading quantum corrections to the GHD approach.

### Weaknesses

Mostly addressed by the author in reply to the previous referee report.

Only remaining one:

The derivation of leading quantum corrections is unfinished.

However, that can be accepted as a task for future work.

### Report

This paper treats an important subject, and represent a significant and interesting advance in understanding Generalised Hydrodynamics (GHD). Based on these, I do consider that the results merit publication and I recommend the revised paper for publication in Scipost Physics.

### Requested changes

The author satisfactorily replied to requests in the earlier report.

However, the paragraphd added in response to the end of section 5 does hava some minor issues:
- the equation reference 48 appears without parentheses,
- misprint: "the de definition",
- "we will only need", "we will need" - I think it's better to use simple present here and elsewhere, too.

I ask the author to correct these before the paper is published.

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