Lack of symmetry restoration after a quantum quench: an entanglement asymmetry study

In my opinion, the revised version of the manuscript presents a considerable
improvement over its original version. My previous comments regarding
the conceptual strengths of the manuscript obviously still hold, and
now the technical aspects of the work have been made much clearer
owing to the details and appendices added by the authors. The authors
have largely answered my questions, requests and concerns and I thank
them for their efforts.

In my reading of the new version I still found some minor technical
issues and typos that should be addressed, which I included in the
subsequent list of requested changes. In addition, I begin the list
with a question to the authors on the physics side of things, namely
regarding the identification of an "Mpemba effect" similar to
the one they have observed in their previous work.

I would recommend the manuscript for publication once the remaining
issues are resolved.

1- When reviewing their previous results on the quench from a tilted
ferromagnetic initial state, the authors mention a striking property
to which they refer as a "quantum Mpemba effect", where the rate
at which the initially-broken symmetry is restored increases as the
entanglement asymmetry of the initial state increases. A similar effect
seems to be captured by the right panel of Fig. 3; though the symmetry
is not eventually restored and the asymmetry does not decay to 0,
the rate at which the asymmetry drops towards its saturation value
increases with increasing asymmetry at $t=0$. Could the authors comment
whether this observation is consistent with all the numerical checks
they performed? If this is indeed an effect that applies also for
the N\'eel quench, I believe it would be useful for
the reader if this was emphasized in the text. If so, are the authors
able to provide some analytical insight regarding the effect, for
example through the quasiparticle picture that they present (I am
thinking, for example, on a possible argument where one shows that,
as we vary $\theta$, the contribution of a quasiparticle with momentum
$k$ to the integral defining $B_{n}\left(\alpha,\zeta\right)$ peaks at a different $k=k^{*}\left(\theta\right)$, and that the group
velocity associated with this $k^{*}$ increases with increasing $\theta$)?

2- On page 5, the authors write that, for the initial state they examine,
$\Delta S_{A}$ is monotonic in $\theta$, while it is not so for all $n$ when considering $\Delta S_{A}^{\left(n\right)}$. Could
they mention the source of this claim (numerical checks on a finite
interval, etc.)? Their result for large $\ell$ in Eq. (69) suggests
that $\Delta S_{A}^{\left(n\right)}$ is always monotonic, so I assume
this claim refers to the finite $\ell$ case, but this should be clarified in any case.

3- The definition of $n_{A}$ appears twice: once on page 7 after Eq.
(14), and once on page 23 after Eq. (96). The signs are flipped between
these two definitions, so this should be fixed.

4- On page 10, in Eq. (34), one entry of the matrix says $g_{12}\left(k\right)$
instead of $g_{12}\left(k,\theta\right)$. This typo should be fixed.

5- On page 16, there are typos in Eq. (67): inside the exponent, $l$
should be replaced with $\ell$, and $\beta_{j}$ (in the first sum) should be replaced by $\beta_{j}^{2}$.

6- On page 17, the authors write "one should fix first the value of
$\theta$ in Eq. (56) and then consider the large $\ell$ regime".
However, Eq. (56) is already assuming the large $\ell$ regime, given
that it was produced from the asymptotics of (quasi-)Toeplitz determinants.
I assume that for this purpose $\theta$ should be fixed already at
the stage of calculating the correlation matrices of Eqs. (23) and (34); do the authors agree?

7- On page 20, the authors changed Eq. (83) following my comment on a
discrepancy with Eq. (82), but now the operator that appears in Eq.
(83) is not unitary (in the exponent there is a Hermitian operator
rather than an anti-Hermitian one). There are several modifications
that may solve this issue, and the one the authors choose should match Eq. (82).

8- A small methodological comment regarding the parity argument appearing
on page 23. I find the explanation there a little confusing: in general,
eigenstates of $H_{XY}$ (or $H_{XY}'$) should not necessarily be
eigenstates of $P_{z}$ just because the two operators commute; this
is not an algebraic requirement. The reason why the state should be
an eigenstate of the parity operator $P_{z}$ is fermionic superselection
rules (following the definition of "physical states" for fermions,
e.g. in [Banuls, Cirac, and Wolf, Phys. Rev. A 76, 022311]). I
would suggest changing this short explanation accordingly.

9- On page 24, there is a wrong sign in the expression for $W_{j}$ right
after Eq. (101): the first term in the product should be $I+\Gamma$.

10- On page 25, the time evolution written for $c_{k}\left(t\right)$
before Eq. (115) is incorrect, and should be $c_{k}\left(t\right)=\exp\left(-it\epsilon_{k}\right)c_{k}$.
This might have an effect on some time-dependent expressions written
in the manuscript, but I do not believe this affects the central results
in any way (though it would be good if the authors verified this).