Domain wall problem in the quantum XXZ chain and semiclassical behavior close to the isotropic point

- Timely topic and interesting results
- Careful and exhaustive analysis

- Interpretation of entropy growth may be too naive

The authors study the time evolution of a domain wall
in the XXZ chain and propose an effective description
of the dynamics via the anisotropic Landau-Lifshitz
(LL) equation. The results of DMRG calculations on the
XXZ are compared to the simulations of the LL dynamics
for various observables. If the LL anisotropy parameter
is fixed in an appropriate way, the results show a
quite good qualitative agreement. On the top of that
it is found that, in the isotropic Heisenberg limit,
the agreement becomes even quantitative, suggesting
that the quantum dynamics of the domain wall becomes
effectively classical.

The manuscript presents a very detailed and exhaustive
analysis of the problem at hand, with a particular
focus on the $\Delta=1$ case. In particular, the authors
identify two different scaling regimes of the dynamics,
namely a diffusive one (with logarithmic corrections)
as well as a ballistic tail regime. The numerical results
are corroborated by perturbative analytical calculations
with a good agreement.

In my opinion this is a well written manuscript that
treats a problem of current interest, and thus well
deserves to be published. I have only one point of
criticism related to the calculation of the entropy.

The main issue:

I have serious doubts about the correctness of the
heuristic argument presented by the authors to interpret
the entanglement entropy growth. Indeed, they essentially
invoke only the U(1) symmetry, which is present for
arbitrary values of $\Delta$. However, the authors
themselves make a footnote that the argument does not
apply away from $\Delta=1$. But why?
In particular, the same argumentation would give
$R(t) \sim t$ and thus $S(t) \sim \ln t$ for $\Delta < 1$
which is not the correct prefactor.

I believe, that this argument is just too naive to be able
to interpret the entanglement. In fact, having a look at
Fig. 11, one even observes that the ansatz log(t)/2 gives
a rather poor description of the numerical data.


The smaller corrections:

(1) In Fig. 4 the labels (1) and (2) should be shifted
in order not to overlap with the x axis. There is also
an extra parenthesis in the y axis label of (2).

(2) In Fig. 7, there is a typo in the legend,
$\Omega^x$ should read $\Omega^z$.

(3) In Fig. 8 is $I(t)$ the local (at $r=0$) or the global
current plotted? How comes that it is nonzero at $t=0$?

(4) Before Eq. (39): “filament function” $\psi(r,t)$
I guess this should read $u(r,t)$.

(5) Typo later —> latter (appears many times in text)

1- Semiclassical description of the out of equilibrium Heisenberg chain close to the isotropic point in terms of the classical Landau Lifshitz equation.
2- State of the art numerics on a difficult problem.

1- Some overlap with Reference [27].

This paper presents a study of the emerging classical behavior following a inhomogeneous quench from a domain wall state in the (integrable) spin-1/2 anisotropic Heisenberg chain. The particular setup has been studied in the past, and can be seen as a quantum version of the famous Riemann problem in classical hydrodynamics.

The aim is to better understand the vicinity of the isotropic point, for which it is very difficult to use standard quantum integrability tools such as the Thermodynamic Bethe Ansatz. The main claim is a quantitative description in terms of the integrable Landau Lifschitz (LL) integrable PDE, at least for local observables. This claim is checked numerically in considerable detail. A nice and simple heuristic argument, based on energy conservation, is also provided to justify the emergence of the classical LL equation.

The paper is well written and interesting. It also raises a number of interesting questions regarding the connection between out of equilibrium quantum dynamics and integrable PDEs. I recommend publication, provided the following minor issues are addressed.

1) A tanh-sinh profile similar to the initial data for the LL can also be implemented in the spin chain. Did the authors look whether the results presented here remain valid for such initial states?

2) Regarding the corrections to the linear behavior discussed at the end of page 10. It is not clear to me, because of this, how the agreement between LL and XXZ can be seen as quantitative in a precise sense. Also, it appears that the agreement is not perfect at the isotropic point either. While I understand that the match between the various models is qualitatively remarkable, it is not clear how this may be turned into exact statements, such as the one for the density profile stemming from generalized hydrodynamics.

3) Regarding the discussion of the entanglement entropy in the isotropic case. Does the $\log(R(t))$ argument still underestimates the entanglement entropy upon inclusion of multiplicative logarithm corrections to R(t)?

4) Page 13, last paragraph. 'exponential decay in $r/t$...This exponential decay is however cut for $r/t=1$'. I do not understand this sentence. Wouldn't it be more correct to say that there is no exponential decay in the quantum case, since $e^{-1}=cst$?

5) Page 21, caption to figure 11. '$t=185$' contradicts the claim '$t=210$' at the end of page 19. Also, inconsistent notations $\tau$ vs $dt$.

6) Section 7.3. It is not clear to me how the LL and XXZ chain calculations should match, since the assumption of ferromagnetic correlations breaks down at very large distances. Usually when dealing with hydrodynamics descriptions of quantum systems, it is necessary to 're-quantize' the classical solution in order to access long-range correlations. This is what is done in Luttinger liquid theory, for instance. Could the authors comment on that?

7) Section 7.6, last paragraph. Perhaps it would be better to state more clearly that the divergences are precisely the logarithmic divergences that make up for the logarithmically enhanced diffusion discussed before.

Typos:

Page 2, 'form' -> 'from'.
Page 4, line 12. 'and in characterize'.
Page 4, the last sentence of the introduction starts with 'And'.
Page 10, last line: '$r/r$' ->'$r/t$'.
Page 18, second paragraph in 7.1. 'As can be seen in in' -> 'As can be seen in'
Page 26, line before equation (39). 'the following the filament' ->'the following filament'. Also, the filament function is '$\psi(x,t)$' not $u(x,t)$, so the sentence reads awkwardly.
Page 26, two equations are not numbered.

See report.

1- Direct confirmation of the quantum-classical correspondence of spin transport in the anisotropic Heisenberg model.
2- Careful and detailed analysis with high-quality numerical data.

1- Lack of integrability.

This is an extensive study of the domain-wall dynamics in the Heisenberg quantum spin-1/2 chain with anisotropic interaction. There has been a number of works devoted to this problem in the past. The most recent study, ref. [27], already addressed the problem in the context of the classical Landau-Lifshitz field theory using integrability and other analytical tools, provided theoretical explanations for three distinct dynamical regimes of spin transport which strongly hint on a remarkable quantum-classical correspondence.

In the present work, the authors carry out a thorough quantitative analysis on this problem which further vindicates the conjectured quantum-classical correspondence. Specifically, they demonstrate that, on sufficiently large time-scales (which crucially depend on the anisotropy parameter), the asymptotic regions effectively behave as large ferromagnetic reservoirs which motivates a semi-classical description. By conjecturing that the intermediate (that is interface) region is simply governed by the classical Landau-Lifshitz equation of motion, they go on to convincingly demonstrate that the effective classical description becomes even quantitatively exact provided the interaction anisotropy simultaneously scales zero. Extensive numerical (tDMRG) simulations are supplemented by theoretical arguments based on various approximate models. In addition, the authors briefly address energy transport through the spreading of a localized energy-density profile, where they exploit the known mapping to the focusing NLS equation and solve it approximately using a variational Ansatz. They conclude that the variational approach is not sufficient to capture also the logarithmic deviations of the diffusive spreading.

In my opinion this is a well-rounded and nicely written paper, with a good mixture of analytic approaches and high-quality numerical simulations. Given however that model is exactly solvable, it is a pity that it does not offer any in-depth theoretical understanding using techniques of classical and quantum integrability. But in spite of that, I definitely do recommend this paper for publication after a small revision.

Comments:

There is a somewhat awkward language at certain places which I would like to address first:

i) At the beginning of Sec. 4, it is said " We give below an alternative -- and somewhat direct -- derivation of this solution."

So far I can see, the presented derivation is *exactly* the same as that presented previously in ref. [27]. Well, this is just the simplest Riemann-type hydrodynamic approximation where one neglects the dispersive term and solves the 'Riemann flow' subjected to suitable boundary conditions. This derivation is nothing deep really, merely a short exercise. I do not think it is legitimate to proclaim this derivation as "alternative" or "more direct"...

ii) At the beginning of Sec. 6, we can read " As for the LL problem, the z magnetization problem was observed to freeze at long times [27, 28], and to be described by a soliton solution of the easy-axis LL equation."

One should notice here that ref. [27] obtained the exact spectral data for the domain-wall initial profile. In the easy-axis regime, the spectrum involves a static kink configuration (in general there are additional radiative modes and/or breathers). The absence of spin transport is thus attributed exclusively to the stability of a localized kink, which an exact (and rigorous) statement. Saying "was observed to freeze" may read as an understatement.

Other remarks and questions:

1) I think it would be beneficial to clarify a bit better how much can one strictly infer from the energy conservation argument. Just by knowing that asymptotic regions will remain ferromagnetic does not automatically ensure that the interface region centred at the origin will be free of genuine quantum effects (and therefore effectively governed by the classical ferromagnetic field theory), does it? In my reading, the argument here relies on the empirical evidence (based on e.g. [26] or other simulations) that the magnetization gradient vanishes as $t \to \infty$ on the characteristic length R(t). Naively, one would then expect the semi-classical description should work outside of the dynamical interface [-R(t),R(t)]. But then what guarantees you that long-wavelength semi-classical theory w.r.t. ferromagnetic vacuum remains applicable in the vicinity of the origin (even at $\Delta = 1$ to begin with)?

2) In order to observe the quantum-classical correspondence at the quantitative level one has to probe the dynamics in the weakly anisotropic regime. Then I am curious whether one could also detect classical breather modes (which "live" in the strongly anisotropic regime of the Landau-Lifshitz magnet) in the non-equilibrium dynamics of the quantum spin chain?

3) If the authors could give some quantitative estimates on the "quantum corrections" (when moving away from the $\Delta \to 1^\pm$), that would be very nice.

Other remarks:

- The presentation of the left plot from Fig. 9 can be probably be improved. In particular, it is hard to distinguish the blue and yellow curves by naked eye. Moreover, I have a question here: how does one determine the fitting window? Is there a way to estimate the relevant transient time-scale after which the asymptotic law become effectively valid I can imagine that by some ad-hoc manipulations (e.g. sliding the window to larger times), the green fit for the diffusive curve will presumably look much better. Perhaps some clarifications along these line can be helpful.

Minor remarks:

- Bad spelling: "insure" should be "ensure" (end of page 23).
- Just a suggestion which may slightly improve readability: one could try use explicit symbols to the terms of the sort $1/4 - S \cdot S$ and $1/4 - S^{z} \cdot S^{z}$, e.g. using $H_{iso}$ and $V$.