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Diffusion in generalized hydrodynamics and quasiparticle scattering
by Jacopo De Nardis, Denis Bernard, Benjamin Doyon
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Authors (as Contributors):  Jacopo De Nardis · Benjamin Doyon 
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Arxiv Link:  https://arxiv.org/abs/1812.00767v2 (pdf) 
Date submitted:  20181220 01:00 
Submitted by:  De Nardis, Jacopo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
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 steadystate twopoint correlation functions, emphasising that PTsymmetry 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 twoquasiparticle 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 halffilling, where we show that spin transport is purely diffusive.
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Anonymous Report 2 on 2019213 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1812.00767v2, delivered 20190213, doi: 10.21468/SciPost.Report.825
Strengths
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Weaknesses
See report
Report
In this article the authors review and expand upon recent attempts to
study diffusive corrections to the generalized hydrodynamic (GHD)
theory for integrable models.
The discussed theory and the obtained results are certainly of
interest and worth to be published. I do, however, see some problems
with the manuscript in its current form. My main criticisms are:
1) There is often no clear distinction made between phenomenological
theory, conjectures for exact or asymptotic formulas, and formulas
which are exact and proven.
The hydrodynamic equation (2.5) and the NavierStokes corrections
(2.8) are a phenomenological ansatz which might or might not provide
good approximations of the true dynamics in the appropriate limit. In
this regard one should perhaps mention that even the seemingly
simplest case of noninteracting particles (where according to the
manuscript the hydrodynamic description should be exact in the
appropriate limit without any corrections) does not appear to be
completely settled, see e.g. Ref. [86].
Accepting GHD+corrections as the phenomenological framework, the paper
also makes several claims of a proof for certain formulas (e.g. for
the current (3.18), see also Appendix D) which in my view are instead
rather limited consistency checks.
2) A problematic point when considering corrections to hydrodynamics
is that the corrections terms are not uniquely defined. This point is
discussed in the paper and a solution is proposed choosing a
particular 'gauge' by PT symmetry. It has not at all become clear to
me why this is the proper choice leading, for example, to the same
diffusion constant obtained from a microscopic calculation.
3) I do find some of the notation very cumbersome and also used
inconsistently. In chapter 2, for example the authors define
\bar{q}_i(x,t)=:<q_i(x,t)> and
<j_i(x,t)>=:\bar{j}_i[\bar{q}(.,t)](x,t). Why are there two different
notations for the same quantity? The notation with the 'bars' is then
used in (2.7) and (2.8) but then almost completely disappears for the
rest of the paper.
Further points:
i) Large parts of chapters 14 have already been published
elsewhere. For a regular article the percentage of material which is a
review appears quite high.
ii) Page 4: 'no diffusive dynamics in noninteracting systems'. This is
too general. Diffusive dynamics can, for example, arise in
noninteracting systems in the presence of timedependent potentials,
see e.g. PRL 39, 1424 (77).
iii) Eq. (2.7): F_i does not seem to be defined.
iv) Eq. (3.183.20) are simply stated. Below it says that Appendix D
provides an alternative proof implying that a proof has already been
provided. I also fail to see why App. D is a proof.
v) There are several typos which make it unclear what the authors are
refering to. On pages 21, 22 the authors refer for example to points
(i) and point (ii) but there are no such points.
vi) There has been a long discussion whether or not there is spin
diffusion in the XXZ chain. I find it misleading to state that a
standard diffusion equation for the local magnetization is obtained,
see (6.5). This is simply a consequence of the phenomenological ansatz
made which is purely diffusive in the absence of a ballistic
part. This is not related to the particular model considered and does
not provide any information about that model.
vii) Below (6.7) a figure is not properly referenced.
xiii) The authors only mention that the phenomenological GHD+corrections
theory might possibly fail in the last paragraph of the
conclusions. This point does deserve a broader discussion and should
already be made in the introduction.
ix) I wonder how much formulas like (3.51) help in explaining the final
formulas for the diffusion matrix. From the information given I find
it impossible to judge whether the conjectured form for the two
particlehole form factors is valid. The formulas which are finally
evaluated (6.326.33) are perhaps easier to understand directly as a
conjecture.
Requested changes
see report
Anonymous Report 1 on 2019213 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1812.00767v2, delivered 20190213, doi: 10.21468/SciPost.Report.824
Strengths
1) The manuscript deals with the interesting topic of extending the hydrodynamic theory for integrable systems beyond the Euler scale, namely to account for diffusive dynamics.
2) The manuscript is wellwritten and provides many steps
that are necessary for the treatment of the subject.
Weaknesses
1) Unfortunately, many of the necessary techniques, e.g. TBA and "backflow" for excited states, are quoted from the literature without attempting a derivation from just few principles. This observation would not be a critique if the manuscript was compact, but it comprises more than 60 pages. Many of the comments on equations should be replaced by derivations, even if they were brief.
2) There are many instances where assumptions are used that are not spelled out.
Report
The manuscript deals with the interesting topic of extending the hydrodynamic
theory for integrable systems beyond the Euler scale, namely to account for
diffusive dynamics. The manuscript is wellwritten and provides many steps
that are necessary for the treatment of the subject. Unfortunately, many of
the necessary techniques, e.g. TBA and "backflow" for excited states, are
quoted from the literature without attempting a derivation from just few
principles. This observation would not be a critique if the manuscript was
compact, but it comprises more than 60 pages. Many of the comments on
equations should be replaced by derivations, even if they were brief.
In (2.6) and (2.7) functions \cal F and \cal D are postulated allowing to
express the expectation value of j from the expectation value of q and its
derivative in case of inhomogeneous systems. From (2.7) the relation (B.3)
between the <jq> and <qq> correlations is "derived". I do not trust this
derivation as (B.2) is used with a generating function depending on some
generating field beta(x) which may or may not depend on time. If it does not
depend on time, then a time integral on the r.h.s. of (B.2) should appear. If
the field beta(x) is time dependent, then relation (2.7) must have built in a
dependence on this field and then additional terms appear in (B.3).
The next step in the computation of the diffusion matrix makes use of the
partial differential equation for the <qq> correlator and explicit results for
this correlator.
It is, however, quite disturbing that the results for the diffusion matrix
depend on the "gauge" of the local conserved operator. Adding a divergence
field preserves the continuity equation, but changes the diffusion matrix. The
authors state this honestly by "One must therefore choose a gauge in order to
fix the diffusion matrix". And they do it by choosing a gauge rendering q and
j invariant under PT transformation. They even prove that there is a unique
gauge to this end. But they do not prove why this yields the physical
diffusion matrix. I would find the procedure presented by the authors
convincing if they could prove that the two differing diffusion matrices for
the left hand side and the right hand side of (2.26) would result in
compatible Lcoefficients (by use of (2.14)).
The authors use the current formula (3.18) and comment after (3.20) that "in
Appendix D we also provide an alternative proof for this formula". I do not
see any proof of (3.18) in the main body of the manuscript. So it is not
appropriate to call Appendix D an "alternative" proof. (In addition I have to
note that the "proof" of Appendix D is full of assumptions.)
Section 4 is very technical and hard to digest. Many calculations depend on
regularization schemes like (3.37), but I trust that the authors here apply
(their own) state of the art results for the LiebLiniger gas. Section 5 reaps
nice physical results.
In summary, I like to ask the authors to sharpen their presentation. There are
many instances (see above) where assumptions are used that are not spelled
out.
Requested changes
1) I like to ask the authors to generally sharpen their presentation.
2) Please replace general comments on equations by deductions.
3) Please spell out assumptions and keep them separate from proper proofs.
Author: Jacopo De Nardis on 20190305 [id 459]
(in reply to Report 1 on 20190213)
We thank the referee for his.her insightful comments which led, we believe, to clarifications of important aspects of the discussion that are now in the resubmitted version.
We have spelled out in the main text which are the assumptions underlying our derivation. In the introduction, we have clarified the main hypotheses for the paper: those underlying the hydrodynamic approach, and those used for the form factors. The former is a standard set of hypotheses dating from more than one century. The latter is grounded on many examples, but a general theory is still missing and worth investigating further (see the recent paper [137]).
“In (2.6) and (2.7) functions \cal F and \cal D are postulated allowing to express the expectation value of j from the expectation value of q and its derivative in case of inhomogeneous systems.”
it maybe worth clarifying that we actually compute the Onsager matrix, which code for the microscopic diffusive spreading of the charges correlation functions. The diffusion coefficient \mathfrak{D} is then reconstructed using the hydrodynamics assumptions. The validity of the hydrodynamic assumptions have been proved in mathematically rigorous way only for a very restricted set of models (which however include the classical hard rod model, see ref.[80]).
“From (2.7) the relation (B.3) between the <jq> and <qq> correlations is "derived". I do not trust this derivation as (B.2) is used with a generating function depending on some generating field beta(x) which may or may not depend on time. If it does not depend on time, then a time integral on the r.h.s. of (B.2) should appear. If the field beta(x) is time dependent, then relation (2.7) must have built in a dependence on this field and then additional terms appear in (B.3).
The next step in the computation of the diffusion matrix makes use of the partial differential equation for the <qq> correlator and explicit results for this correlator.”
Thank you for pointing out the lack of clarity here. The question of the referee concerns the dynamical variables used in hydro. We had discussed aspects of this in section 2, however we have tried to further clarify and added more explanations: sentences around the new equation 2.4, and explanations in the appendix B. The main point is that the dynamical variables of hydro are the conserved densities evaluated on a given time slice. The choice of reference time slice is arbitrary (similarly to, for instance, classical field theory). These dynamical variables evolve in time according to the hydrodynamic equations. Therefore, local changes of initial conditions of this dynamical system  local changes of $\bar{\mathfrak{q}}_i(x,0)$  produce changes of averages evaluated at any positive time, because such averages are evaluated by time evolving from a perturbed initial condition. By statistical mechanics principle, a change of initial condition at some point $x$ is equivalent to the insertion of a local observable at $x$ and $t=0$. There is no time integration, as only the initial condition is changed; the time evolution is unchanged. The effect is of course nevertheless nontrivial on observables evaluated at all later times, and this is described by a correlation, as on the right hand side of B.3. We assume there is a change that exactly inserts local conserved densities; the associated variables are the local effective temperatures $\beta_i(x)$.
“It is, however, quite disturbing that the results for the diffusion matrix depend on the "gauge" of the local conserved operator. Adding a divergence field preserves the continuity equation, but changes the diffusion matrix. The authors state this honestly by "One must therefore choose a gauge in order to fix the diffusion matrix". And they do it by choosing a gauge rendering q and j invariant under PT transformation. They even prove that there is a unique gauge to this end. But they do not prove why this yields the physical diffusion matrix. I would find the procedure presented by the authors convincing if they could prove that the two differing diffusion matrices for the left hand side and the right hand side of (2.26) would result in compatible Lcoefficients (by use of (2.14)).”
Thank you again for this good question. There was already a comment in particular about the independence of the L matrix upon choice of gauge, but it was not clear and not well stated. We have clarified the discussion in the main text, see the first paragraph of section 2.4, and we have added further details in appendix C, sub appendix C.1. The Drude weight $D_{ij}$ and the Onsager coefficients ${\cal L}_{ij}$ are indeed gauge invariant (and the latter code for the microscopic diffusive spreading of the correlation function). The coefficients ${\cal D}_{i}^{~j}$ is on the hand not invariant, but covariant, under gauge transformation, as it depends on the specific gauge for local conserved charges.
“The authors use the current formula (3.18) and comment after (3.20) that in Appendix D we also provide an alternative proof for this formula. I do not see any proof of (3.18) in the main body of the manuscript. So it is not appropriate to call Appendix D an "alternative" proof. (In addition I have to note that the "proof" of Appendix D is full of assumptions.)”
We are sorry about this confusion. What we meant was that there already are proofs of this formula in the literature, we have added references to these (Refs 49,105, 106). As for the proof in appendix D, too many steps were implicit. The main points are as follows. (1) Given the existence of a form factor expansion, the results from thermodynamic Bethe ansatz give us the oneparticle form factors of densities and currents at equal particle and hole rapidities. Indeed, TBA gives eq 3.39, while the form factor expansion gives 3.40. Comparing, we get the density form factors 3.37. The continuity relation then gives the result for the currents. (2) We can then evaluate $B_{ij}$ using form factor expansion. At the same time, the average currents in homogeneous stationary states are from general principle linear functionals of the $h_i(\theta)$, thus expressible as integrals over $h_i(\theta)$ times some statedependent function of rapidity. Thus the formfactor expression for $B_{ij}$ gives a firstorder differential equation for this statedependent function. The effective velocity times particle density is a solution to this equation. We assume uniqueness of the solution. This is the single nontrivial assumption.
"Many calculations depend on regularization schemes like (3.37), but I trust that the authors here apply (their own) state of the art results for the LiebLiniger gas"
We are sorry for the confusion but we stated that the final result for the diffusion matrix does not depend on the details of the regularisation scheme.
We stressed now below eq 3.30 that the only requirement is that the form factor expansion eq 3.30 exist with regularised integrations on the real axis and below eq 4.15 that eventually the final result does not depend on the regularisation.
Author: Jacopo De Nardis on 20190305 [id 458]
(in reply to Report 2 on 20190213)1) We believe that we have spelled out in the introduction which are the assumptions underlying our derivation, especially now in the resubmitted version. Please notice that ref.[86], which deals with free fermionic theories, is not in contradiction with our results as this reference pointed the existence of higher order corrections to Euler hydrodynamics that start at third order in derivative expansion for free theories (while the diffusion processes we are dealing here are at the second order and are absent in free theories)
2) See answer to referee 1. We have added a detailed discussion on gauge invariance and gauge covariance in Appendix C. We also improved clarity in the main text. Please do not confuse the microscopic diffusion, defined via the diffusive spreading of correlation functions, with the diffusion matrix.
3) Thank you for pointing this possibly confusing notation. We have optimised and uniformised our notation.
i) Even if some part of those chapters is partially spread out in the literature, we believe it is useful for the scientific community to have a consistent presentation as we have proposed. Chapters 14 contain a large quantity of unpublished material.
ii) By this statement we mean that the diffusion matrix is zero in free systems. This was recently put forward in ref.[69] and it is confirmed by our analysis. The example mentioned by referee 2 deals with open systems (which of course may show diffusive behaviours) while we are here dealing with closed systems. We actually wrote explicitly in the introduction that « confirming the general intuition that there is no diffusive dynamics in noninteracting systems [69,86] (except, potentially, with external disorder [87–90]) ».
iii) Thank you. We corrected it.
iv) We have added references to the previous proof, and give more detailed steps in the Appendix D. But since both referees are not happy with calling this a proof we changed the title of the appendix to “derivation” instead of proof.
v) Thank you for pointing out these misprints.
vi) It is true that the simple diffusion equation comes directly from the hydrodynamic assumption with the extra information that $v_\infty=0$ (by symmetry) in this particular case. However, we do provide a computation of the diffusion constant as a function of all external parameters (temperature, etc.) in the XXZ chain from the microscopic model and checked it numerically. This prediction contains new information: We show that in the limit of half filling (states with zero total magnetisation) the spin ballistic current (or alternatively, the spin Drude weight) is zero and the diffusion constant is not. We analytically and numerically evaluate the latter, and we compare with stateoftheart DRMG simulations in that specific model.
vii) Thank you. We corrected it.
viii) Actually we already mentioned in Section 2 and 6 the possibility of superdiffusion, which will manifest itself through divergences in the Onsager matrix (for example in the Heisenberg XXX chain at finite temperature). We pointed this again in the introduction as this is an interesting point to investigate further. This is a subject of current research.
ix) Formulas 6.34 for the spins diffusion constant at halffilling (zero magnetic field) is a direct consequences of our main result, formula 4.22 for the Onsager matrix and its derivation from the latter is the main scope of section 6. Formula 4.22 is based on a conjecture on the poles of the form factors. We believe that this is very different from stating that 6.34 is a conjectured equation.