The spin Drude weight of the XXZ chain and generalized hydrodynamics

First report:

However, the paper has many flaws, especially in some of its (too strong) claims and in the description of what has been done before. It is essential that these flaws be corrected before the paper can be published.

General reply: The approach presented in our manuscript is complementary and completely independent from the GHD approach. The Bethe-Boltzmann (or Euler equation if this is the term the referee prefers) central to the GHD approach are not used at all. Wherever appropriate, we compare with the GHD and cite accordingly.

The first main logical flaw is in the claim that the Drude weight derivation in section 3 is new and more complete / rigorous than the derivation presented in [15]. This is incorrect. In [15], the Drude weight was derived exactly from the projection formula (essentially eq 1.7, but more written without assumption of orthonormality) along with a completeness assumption, which is indeed the formula / assumption that the authors use (they use Mazur’s minimisation argument instead of the direct calculation of the projection, this is nice and interesting, but this evaluates the same projection formula under the same assumptions). The only conjecture used in the derivation [15], and not used in the present manuscript, was the exact current formula.

Reply: In section 3 we use indeed Mazur's (in)equality as outlined in Mazur's original paper and so does Ref. [15]. The Mazur paper is cited as is Ref. [15]. We do not claim that we have invented the Mazur (in)equality nor do the authors of Ref. [15]. We cannot see any 'main logical flaw' in what we present.

The authors here prove it for the spin current in section 2, which is a nice result. But other than that, no other conjecture is used in [15] that is not used here, in particular there is no need for coarse graining and ray-dependent local stationary states. The last sentence just before section 3.1, on page 8, is incorrect and should be deleted, and it should be stressed that the derivation presented in section 3 is equivalent to that presented in [15].

Reply: It is our understanding that the GHD approach is based on a coarse graining and that one cannot extract short-distance or short-time behavior from it. If the referee is aware of any results to the contrary we would be interested to learn about them. Once more, both the derivation of the Drude weight formula in [15] and in our manuscript use Mazur's result which is cited appropriately. The derivation in [15] is based on a conjectured form of <J_0 Q_n> while we derive this correlator from first principles.

Secondly, the equations 1.11 - 1.13 and the explanations given for generalised hydro are very sloppy. First, the notations Qn and Jn seem to be used for the total, space-integrated (site-summed) charges and currents (this should be made more clear in the paper). However, of course, the quantities that satisfy the continuity relation 1.11 are the densities, not the space-integrated quantities. This equation should be corrected.

Reply: See list of changes.

Second, the form 1.12, that the continuity relations are supposedly taking in GHD, is not correct. The proposal does not say that the local stationary states must depend on the ray xi = x/t. The local stationary states depend in general on both x and t. Please read the papers [12,13] carefully. Indeed many later works have used the full x,t-dependence of the local stationary states. The special dependence on x/t only comes when the initial state is scale invariant, which is the case for domain wall initial condition for instance.

Reply: See list of changes.

Third, I should also say that the terminology Bethe-Boltzmann - no widely used - is a bit misleading. The (corrected) equation 1.12 is a Euler equation, not a (collisionless) Boltzmann equation (which is just a Liouville equation). That is, it is not valid at small densities like a collisionless Boltzmann equation is; it is instead valid at large wavelengths like a Euler equation is.

Reply: See list of changes

Fourth, in eq. 1.13, The form of the quantities of charge carried by the quasiparticles seem to be very special: here the authors assume a particular choice of the charges Q_m with eigenvalues that are powers of q_l(theta). But this seems an inconsistent notation: the choice is not the one where orthogonality holds, and orthogonality is used in 1.7. This should be clarified.

Reply: These are not powers but rather upper indices. A short note has been added.

Finally, other small things should be corrected. In section 4, two numerical schemes are used to evaluate the Drude weight formula. The first is based on a linear response protocol. It should be pointed out that the equivalence of the Drude weight formula as presented eq 3.6 was already proven to be equivalent to the linear response scheme in [15; sect 5.1].

Reply: See list of changes.

Sentence 'This formula agrees with the conjectured general current formula used in GHD and appearing in Ref. [15]' page 6: in fact the general formula for current averages first appeared in ref [12,13]. It was conjectured in XXZ in ref [12] with very strong numerical checks, and it was proven using crossing symmetry in relativistic QFT in ref [13].

Sentence 'The GHD conjectures are thus proved both for J0 and for J1 = JE' page 7: For J1 = JE the conjecture was already proven - as it is indeed a trivial application of TBA (it is immediate from the formulae of [12,13] for instance). I don't think the author should claim they are the first to provide the proof.

Reply: It seems that the formula for JE was first given explicitly in Ref. [2] which is cited. If one wants to say that it immediately follows in TBA anyhow, then one should correctly say that it follows from expressions given in [4] which was published much earlier than [12,13]. Furthermore, we insist that it is important to clearly distinguish between a conjecture (if the numerical data would be in contradiction then it would obviously no longer be a valid conjecture) and a proof from first principles. We are discussing lattice models here, not relativistic QFT's.

Formula 3.1, first equality is only valid for Qn orthogonal basis, and second equality is incorrect because the particular choice of Qn (with powers of ql) is not orthogonal. (The derivation given afterwards is correct however, as it uses Mazur's argument.)

Reply: There are no powers here. The superscript n and the subscript l are just indices. Some comments have been added, see list of changes.

Conclusion, page 12: note that the Lieb-Linger proposal given in [15] was already conjectured there to be applicable in the multiparticle case, and it was mentioned to agree with the 20-years-old results (see the introduction of [15]).

Reply: See list of changes.

Please define more precisely what Ekin is in eq 1.5.

Reply: See list of changes.

Second report:

1- Maybe ref [28] can be quoted already in Sect. 2 (after Eq. (2.10)) where a similarity of formulation to that of GHD is mentioned for the first time.

Reply: We checked the referee's suggestion and found that [28] would be misplaced. The reference is quoted already in Sect. 2 (after Eq. (2.10)).

2- A remark on what exact charges Q_n are used in formula Eq. (2.16) and below, expressing the spin Drude weight in terms of Mazur bound. They can't be the simple local ones introduced in Eq. (1.2), as, as has been argued, these have vanishing overlap with the spin current at vanishing magnetic field? Are Q_n the quasi-local charges, but then they have not been introduced in the text? Perhaps more explanation on this point is needed to avoid confusion (like mine).

Reply: This is indeed an interesting point and we thank the referee for this comment. From a practical perspective, one could take the point of view that it does not matter what the additional charges Q_1, Q_2, ... are as long as they make the set of charges complete. If this is the case, then the Mazur argument can be applied and the formula (3.6) follows in which all other charge densities have dropped out. The essential point to have a finite Drude weight is that the correlator <J_0 Q_n> is known and is nonzero. We therefore believe that a proper first principle derivation of this correlator as presented in our manuscript is an important step in making the result for the Drude weight more rigorous.

3- Using symbols for both temperature (T) and inverse temperature (\beta) is somehow confusing. Perhaps one may only use \beta?

Reply: See list of changes.

4- Is the finite-temperature correction to formula (3.21) really O(\beta^3)? I would find this surprising.. If so, the authors can perhaps comment on where it comes from?

Reply: This is a typo. See list of changes.

List of changes:

-Changed all temperatures `T' to inverse temperatures `\beta' (also the
references to temperature in certain places)

-Eq. 1.5 E_kin to H_kin.

-1.11-1.12 changed to lower-case q and j and added sentence above 1.12.

-1.12: Footnote regarding Euler versus Bethe-Boltzmann Eq. added.

-Sentence around 1.13 to clarify the notation (ie. draw attention to the
superscript).

-Below Eq. 2.18 clarified that this is not the first derivation of the energy
current.

-Below Eq. 3.4 added a paragraph clarifying what is meant by
completeness and what the role of the additional conserved charges is.

-added footnote on page 7: Taking <J0 J0> as shorthand for lim <J0(0)J0(t)>

-Below Eq. 3.8 we now point out that the derivation relies on completeness and
TBA, so is not to be understood as completely independent

-Corrected typo Eq. 3.21 (ie. O(\beta^2) is the next surviving term as 3.21 is
written)

-In section 4 (1st paragraph) point out the equivalence of linear response and
GHD. Citation to Ref. [15] added.

-Added description under figure 2 to clarify what power law was observed.

-Mention conjectured multi-particle formula in Ref. [15] in the conclusions