The spin Drude weight of the XXZ chain and generalized hydrodynamics

Submission summary

 As Contributors: Jesko Sirker · Andrew Urichuk Arxiv Link: https://arxiv.org/abs/1808.09033v2 Date accepted: 2018-12-19 Date submitted: 2018-11-26 Submitted by: Sirker, Jesko Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics

Abstract

Based on a generalized free energy we derive exact thermodynamic Bethe ansatz formulas for the expectation value of the spin current, the spin current-charge, charge-charge correlators, and consequently the Drude weight. These formulas agree with recent conjectures within the generalized hydrodynamics formalism. They follow, however, directly from a proper treatment of the operator expression of the spin current. The result for the Drude weight is identical to the one obtained 20 years ago based on the Kohn formula and TBA. We numerically evaluate the Drude weight for anisotropies $\Delta=\cos(\gamma)$ with $\gamma = n\pi/m$, $n\leq m$ integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general $\gamma=\pi n/m$---obtained by analysis of the quantum transfer matrix eigenvalues---agrees with the bound which has been obtained by the construction of quasi-local charges.

Author comments upon resubmission

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 Mazurâ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

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

Submission & Refereeing History

Resubmission 1808.09033v2 (26 November 2018)
Submission 1808.09033v1 (29 August 2018)

Invited Reports on this Submission

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Anonymous Report 1 on 2018-12-11

• Cite as: Anonymous, Report on arXiv:1808.09033v2, delivered 2018-12-11, doi: 10.21468/SciPost.Report.732

Report

I thank the authors for having clarified most aspects. The calculations now seem to be clear and all correct, and I reiterte that the result is interesting.

I agree that the result for the energy current average appeared much before [12,13]. Ref 2 has it explicitly, while Ref 4 might be a more correct earlier reference. Probably the correct earliest reference is Takahashi's paper on the TBA of XXZ, as the formula - at least in thermal states - follows immediately from the free energy. Maybe comment on these papers on page 7 just before sect 3.

I still think there is confusion concerning the results obtained in the context of GHD, in the description of the various methods on pages 3-4. I was probably not as clear as I should have been in my previous report. There are three main references where Drude weights are evaluated following GHD developments: refs 14, 28 and 15. Earliest are refs 14 and 28, doing XXZ (but the derivation is applicable to other models). There the exact solution for nonequilibrium currents in the domain-wall problem (obtained in refs 12,13) is used, along with linear response. This exact solution is based on Euler GHD equations, and so uses the coarse-graining hypothesis, and the derivation of the Drude weight from it is based on a linear response assumption. These indeed fit within what the authors refer to as method 3, and should be mentioned there. Slightly later, ref 15 provided a general formula in the Lieb-Liniger model, derived in a different way (again the derivation is applicable to other models). It does not use coarse graining, or linear response, or equations 1.11 or 1.12 of the present manuscript. It is not based on Euler equations, and no reference is made to the domain-wall problem. It is based on method 2, summing over conserved charges, overlapping with the current, exactly like the method of the present manuscript. The only part of the GHD developments that Ref 15 used, in deriving the Drude weight, is the conjectured expression for currents in arbitrary stationary states (GGEs); these expressions were not known before. Once this formula is known, the required correlation functions on the rhs of 1.7, not just at finite temperatures but in any GGE, follow just by differentiation, and the Drude weight can be evaluated, as done in ref 15. So I think ref 15 should be put in method 2, not method 3.

From this perspective, the main development of the present manuscript - an important development - is the proof of the exact spin current expression in arbitrary stationary states (GGEs). These are then used, much like in ref 15 (but using Mazur's method instead), to derive the Drude weight.

So, the sentence, page 4, starting as "For the Lieb-Liniger model in the linear response regime, in particular, this formalism has been used to obtain formulas for the expectation values of ..." is not accurate. Similarly, in saying the approach presented in the manuscript is complementary to that of GHD, one should bear in mind the above.

I would like the authors to make small adjustments on pages 3-4 to account for the above.

Finally, I still don't see why the authors talk about rays in: "If one assumes that a system which is not in equilibrium is composed of cells which are locally described by the distribution...". The cell-decomposition assumption is already in eq 1.12, and leads to ray-dependent states only in the domain-wall (or similar) problems. I would just take away the discussion of rays - unless the authors want to give more explanations about the domain-wall initial condition problem and the Drude weight calculation presented in refs 14, 28.

Requested changes

1- small changes in pages 3, 4 to account for contributions of refs 14,15,28 as described above.

2- take away discussion of rays on page 4, or described more at length the domain-wall initial value problem for GHD in the context of describing the results of refs 14, 28.

3- maybe add comments on ealier references for energy current in XXZ on page 7 just before sect 3.

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