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Lack of symmetry restoration after a quantum quench: an entanglement asymmetry study
by Filiberto Ares, Sara Murciano, Eric Vernier, Pasquale Calabrese
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Authors (as registered SciPost users):  Sara Murciano 
Submission information  

Preprint Link:  https://arxiv.org/abs/2302.03330v3 (pdf) 
Date submitted:  20230606 03:13 
Submitted by:  Murciano, Sara 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider the quantum quench in the XX spin chain starting from a tilted N\'eel state which explicitly breaks the $U(1)$ symmetry of the postquench Hamiltonian. Very surprisingly, the $U(1)$ symmetry is not restored at large time because of the activation of a nonAbelian set of charges which all break it. The breaking of the symmetry can be effectively and quantitatively characterised by the recently introduced entanglement asymmetry. By a combination of exact calculations and quasiparticle picture arguments, we are able to exactly describe the behaviour of the asymmetry at any time after the quench. Furthermore we show that the stationary behaviour is completely captured by a nonAbelian generalised Gibbs ensemble. While our computations have been performed for a noninteracting spin chain, we expect similar results to hold for the integrable interacting case as well because of the presence of nonAbelian charges also in that case.
Current status:
Author comments upon resubmission
quench: an entanglement asymmetry study".
We would like to thank the editors and the reviewers for their work.
We believe that we have addressed the comments of the Reviewer 2 in the new version of the manuscript. Their comment also helped us in improving the final revised version of the manuscript.
List of changes
 We modified all typos/misleading/wrong expressions.
 We have added the result of the entanglement asymmetry for the cat ferromagnetic state.
 We have included several comments, i.e. after Eq. 37, before Eqs. 46 and 53, at the end of Sec. 4 and 5, a paragraph about the multivariable saddlepoint approximation after Eq. 64, after Eq. 92.
 We have added 4 appendices (AD) and modified the Appendix (E).
Submission & Refereeing History
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Report
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.
Requested changes
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 initiallybroken 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 antiHermitian 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 timedependent 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).
Author: Sara Murciano on 20230623 [id 3756]
(in reply to Report 1 on 20230619)1) We thank the referee for their suggestion. Comparing the asymptotic expressions (69) and (74) for $\Delta S_A^{(n)}$ at $t=0$ and $t=\infty$, which we respectively plot in the left and right panels of Fig. 4, we observe that there are values of $\theta$ for which $\Delta S^{(n)}(t=0)<\Delta S^{(n)}(t\to \infty)$. This implies that the asymmetry does not always drop towards its saturation value after the quench, but it can also increase. Therefore, a phenomenon similar to the quantum Mpemba effect found in the quench from a tilted ferromagnetic state cannot be observed in general when we initiate the system in the tilted Néel state and we quench to the XX spin chain. We would like to mention that we are currently working to provide some analytical explanation of this effect.
2) Our claim refers to Fig. 2 of Ref. 5, where we report the result of the entanglement asymmetry of the tilted ferromagnetic state for finite subsystem size $\ell=10$. In that case, despite the expression in the limit $\ell \to \infty$ is monotonic in $\theta$ for any $n$, the same does not hold as $n\to \infty$ for finite $\ell$. We have repeated the same exercise for the tilted N\'eel state, and we have found that the entanglement asymmetry is monotonic in $\theta$ when $n\to \infty$ even for finite subsystem size. We do not report the computation here since we are mainly interested in the result at $\ell \to \infty$. We thank the referee for the question and we have erased that sentence from the main text.
3)4)5) We thank the referee for pointing out these typos, which we have fixed.
6) We agree with the observation done by the referee and we have modified the main text accordingly.
7) We thank the referee for pointing out the inconsistency between Eqs. (82) and (83). This was due to a sign problem in Eq. (83), which we have now fixed. The operator of Eq. (83) is now unitary, and implements the Bogoliubov transformation (82).
8) We agree that the eigenstates of $H_{XY}$ and $H'_{XY}$ are not necessarily eigenstates of $ P_z$, we have rephrased the explanation according to the suggestion of the referee mentioning Ref. [77].
9)10) We thank the referee for pointing out these typos, which we have fixed.