Spin conductivity of the XXZ chain in the antiferromagnetic massive regime

1- Fast converging series expansion for a particular set of correlation functions in quantum integrable models

1- The paper is not self-contained, as the derivation of the results (and more general ones) will be presented in a forthcoming publication
2- The state of the art regarding spin transport, Drude weights, etc..., in XXZ chains, is very vaguely referred to.

The authors expand on a previous publication, Ref. [27], which introduced a novel approach to the computation of correlation functions in quantum integrable models, based on a summation of form factors in a transverse channel (associated with the "quantum transfer matrix"). Ref. [27] was concerned with correlation functions of operators acting on one lattice site.
A forthcoming technical publication will generalize the results of [27] to more general operators, and the present work borrows from this in order to present results for the dynamical correlations of spin current (in the antiferromagnetic regime, and at zero temperature), which has been the subject of much attention due to its relation with spin transport.

This paper is not really self-contained, as the central formulae (2.11) and (2.12) are presented without any derivation. However, due to its excellent convergence properties, the series representation (2.11) is a very practical tool which allows the authors to draw some physical conclusions, and hence may deserve publication on its own.

1- The abstract should make clear that the results here are for the ground state, or, said differently, at zero temperature

2-The introduction should describe in more detail the relation between the Drude weight, conductivity and current dynamical correlation functions, paying some attention to discussing which conclusions hold for generic non-zero temperatures, and which hold at $T=0$.

3- The above point also applies to the discussion around Fig. 2 : according to the authors the observations lade on the current autocorrelation indicate a non-ballistic transport. It should be recalled what to expect in the case of ballistic transport (plateau at large times ? any further subtleties at zero temperature ?)

4- On p.6 there seems to be two typos in the beginning of the second paragraph : $N_0$ should be $N_{>0}$, and "large $m$" should be "large $L_c$"

5- eqs. (3.1) and (3.2) are singular at $T=0$, so the authors should clarify how to use these formulae at $T=0$.

6- This paper has the potential to interest readers not too familiar with integrability, but mostly interested in the physical conclusions and transport properties. For those readers, it would be instructive, and make the paper a little more self-contained, if a short recap on the "thermal form factor" approach could be given, and an explanation why it works so well with respect to other methods. One may wonder if the quality of the series convergence has to do with the "transverse channel" used in this method, just like in quantum transfer matrix methods the thermodynamic limit projects onto the leading eigenstate. Is this indeed the case ? If so, I think this would be worth making explicit.

it is a controlled and accurate evaluation of the spin conductivity in the easy axis spin-1/2 Heisenberg chain at T=0

technically very involved

In this work the authors study the spin conductivity of the Heisenberg chain in the massive regime, extending a thermal form factor approach
they have introduced in ref. 27.
It is actually a sequel of works where they developed and applied this approach, ref. 34+35.
It has the advantage of avoiding the introduction of string states
giving a clear picture of the processes involved.
They further corroborate their analysis with numerical light-cone renormalization group (LCRG) simulations.
They presentation is well structured and clear, nevertheless the analysis
is technically fairly involved and based on previous work by the authors.

It should be clarified in the abstract that it is a T=0 calculation

- exact results for the spin conductivity of an interacting system

- "exactness" of the results is not really checked
- some comments on references and other minor points

The paper presents solid results on calculations that some of the authors are pushing since a number of years. I have only minor comments:

1- the "exactness" of the results is not well checked. Figure 1 does not show a comparison with quasi-exact dmrg calculations to their precision (i.e. 10^-6). In principle there could be deviations of order 10^-1 not visibles from the figure.
2- The values of velocities in table 1 do not seem to make much sense, (they should be not larger than 2) or at least they do not agree with other approaches (i.e. TBA). Maybe the author consider a different rescaling of the Hamiltonian or a value of coupling J not equal to 1?
3- The DC conductivity, i.e. sigma(omega->0^+), is very well know to be zero at T=0 in gapped XXZ. This is not really evident from Fig. 4. Comments?
- Comments on the text:

"The theoretical emphasis in recent
years was on a systematic justification of phenomenological approaches and on numerical
work in a setting that often attempted to go beyond the framework of linear response." This sentence is not clear and possibly misleading. The authors here are considering a linear response quantity (conductivity)

" The calculation of the spin conductivity requires an honest calculation of the
dynamical correlation function of two spin-current density operators."
It is not clear what is an honest calculation.

"So far, the most successful attempts to exactly calculate dynamical correlation functions
of the XXZ chain were based on different types of form factor series expansions."
I understand the authors here refer to their attempts. However, it would be preferable to refer also to other "successful attempts" of the past years. I invite the authors to refer to ref [2], in case they are interested only in the most recent review paper, for calculations of spin diffusion constants and Drude weights, at any temperature and magnetic field.

"The Lorentziantype peak around ω = 0 observed in this paper, which seems to decrease with increasing T,
therefore appears to be a genuine finite-temperature effect related to the expected diffusive
behavior. "
This sentece seems to ignore all the theoretical work on the spin diffusion at half filling in gapped XXZ. Again, I invite the authors to consult ref [1] and [2] and references inside. It is indeed well established that at finite T DC conductiviy is finite (and its value is known) , namely sigma(omega) has a Lorentziantype peak. Therefore, this reported conclusion is far from new.

- Address point 1 and 3 by slightly modifying the figures. Comment on point 2. Reformulate the text when needed.