SciPost Submission Page
Locally quasi-stationary states in noninteracting spin chains
by Maurizio Fagotti
This is not the latest submitted version.
This Submission thread is now published as
|Authors (as registered SciPost users):||Maurizio Fagotti|
|Preprint Link:||https://arxiv.org/abs/1910.01046v2 (pdf)|
|Date submitted:||2019-10-09 02:00|
|Submitted by:||Fagotti, Maurizio|
|Submitted to:||SciPost Physics|
Locally quasi-stationary states (LQSS) were introduced as inhomogeneous generalisations of stationary states in integrable systems. Roughly speaking, LQSSs look like stationary states, but only locally. Despite their key role in hydrodynamic descriptions, an unambiguous definition of LQSSs was not given. By solving the dynamics in inhomogeneous noninteracting spin chains, we identify the set of LQSSs as a subspace that is invariant under time evolution, and we explicitly construct the latter in a generalised XY model. As a by-product, we exhibit an exact generalised hydrodynamic theory (including "quantum corrections").
Submission & Refereeing History
You are currently on this page
Reports on this Submission
- Cite as: Anonymous, Report on arXiv:1910.01046v2, delivered 2020-01-16, doi: 10.21468/SciPost.Report.1454
1- the paper addresses a very timely subject as the systematic computation of corrections to GHD
2- the developed framework is completely general for noninteracting theories and elegant
3- generalizations to the interacting case can be envisaged, although this can be hard in practice
1- the paper is very technical and to maintain its degree of generality, it is based on a notation which is not known to everybody
2- many references are made to previous papers by the same author, which makes the current manuscript less self-contained
3- there is no quantitative numerical verification of the higher corrections for GHD. This could be useful for Sec. 4 for instance.
This paper considers the problem of developing a complete theory for the inhomogeneous dynamics of 1d noninteracting fermionic systems. At the lowest order in the inhomogeneity, GHD provides the full answer but several efforts have been recently made to characterize higher order corrections. This paper takes a step back which is important and useful in general: to provide unambiguous definitions of the quantities which GHD (and its higher order corrections) tries to describe (e.g. root densities, etc). At least, in the noninteracting case, this work provides an elegant approach, which by making use of the available gauge freedom, identifies a whole invariant subspace of conserved charges, currents, currents of currents, etc.
This results in Eq. (44) and (45) which compactly represent the main result of the paper.
This is for sure a relevant and interesting paper that deserves publication. The main issue I can see is that the paper is rather technical, makes use of a particular notation which is not easy to grasp, and requires knowledge of more than one paper from the same author. So it is not easy to read even for someone in the same field.
I understand that for most of its content, technicalities are practically unavoidable. However, I think it would be useful if the introduction of the paper was expanded in order to give a schematic intuition of its content and main results. For instance, a brief recap of the problems encountered in  which this new manuscript is solving, would certainly help.
1- In Eq. (4) the definition of G is not given.
2- On page 4, can the author comment on why is it obvious that Eq. (5) is quasilocal for C[Q] = 0 for noninteracting systems?
3-Page 7: the notion of "physical part" is not entirely clear. Can the author provide an example for a simple case (e.g. the Ising spin chain)
4- I had some troubles understanding the definition of "global" / "local" root density and auxiliary complex field. The quantities introduced in Eq. 23 are global but do depend on the space index x. Some additional comments would be useful and could help understanding the Localisation procedure performed in page 14.
Report 1 by Olalla Castro-Alvaredo on 2019-11-14 (Invited Report)
- Cite as: Olalla Castro-Alvaredo, Report on arXiv:1910.01046v2, delivered 2019-11-14, doi: 10.21468/SciPost.Report.1316
1) It addresses several problems of current interest in this area of research.
2) It proposes a novel solution in the form of a new Mathematical framework to treat a variety of models and out-of-equilibrium situations.
3) It is very well written.
4) The solution proposed in particular at the level of (generalized) GHD equations is very elegant and natural. In particular the use of Moyal products is a very interesting feature that might suggest a way to further generalizations.
1) The paper is very technical so not particularly easy to read but as clear as possible given the nature of the results.
2) The techniques presented only apply to free theories.
In my view this paper contains (at least) two important results: the introduction of an unambiguous definition of the charge densities and associated currents, by introducing precisely defined locally quasi-conserved operators, and a hydrodynamic description of the dynamics of a large class of non-interacting theories by means of (what could be called!) a generalized GHD.
I find the latter result most impressive. The proposed generalization captures all higher order (quantum) corrections to the known GHD equations within an extremely elegant, and seemingly very natural, mathematical framework. “Standard” products are replaced by carefully defined Moyal products, a solution that is very satisfying given that this replacement is supposed to generate all quantum corrections to GHD.
The paper is very technical and therefore not very easy to read but I find that it is well written, and as clear as possible given the nature of the results.
Among the many papers currently being produced in relation to different applications/generalizations of hydrodynamics and find this an impressive and original contribution. Beyond specific models and quenches, it proposes a new mathematical framework to treat a large variety of problems of current interest and an elegant solution to a family of such problems within that framework.
Of course, the results are not fully model-independent (as they only apply to free theories) and the big question is how this might be generalized to interacting systems, as acknowledged in the paper itself. I am sure there will be subsequent work on this. For the moment I think this is a very good contribution to the field and should certainly be published in SciPost.
I have only small comments to make as I found the paper very good overall.
1) I have a small comment/question regarding the statement at the top of page 3 “incorrect definition of space dependent root densities”.
I think I understand what is meant by this, specially given Section 3, however reference  is also cited at this point and I do not think this statement can apply to the QFT case. Certainly “root densities” do not play a role there (particle densities do), but perhaps this “incorrect definition” translates into some other issue in the QFT framework. Could the author clarify this a little bit? Or remove the reference at that stage, if the statement does not apply?
2) In general, I find the paragraph containing this sentence and the one before it, in the previous page, a little bit too negative about GHD. First, GHD was formulated to describe the Euler scale and at that scale none of the problems mentioned here arise, so there is no “incorrect definition of the roots densities” in that original set-up, they are perfectly well defined within that framework. Second, the treatment of diffusive phenomena is referred to as if it were a completely separate development from GHD. In my view it is rather an extension of GHD beyond Euler scale. It is not the complete extension that is presented in this paper, but it is a further refinement of GHD that now includes diffusive phenomena. And within the framework of diffusion, particularly the works [78,79], the “ambiguity” problem raised in this paper is dealt with by fixing a particular gauge, the PT-invariant gauge. Within that gauge, as far as I understand, their results are fully well defined. I think this deserves a mention in this introductory section and maybe in section 3 too, in particular when the issue of gauge fixing is dealt with.
3) I also found the last sentence before section 1.2 a bit obscure. Would it be a possible to say something a bit more concrete about these possible “unfortunate conventions” and how they might generate “cumbersome” large time corrections? Are there existing examples of this in the literature that he could cite?