Entanglement asymmetry as a probe of symmetry breaking

Symmetry and symmetry breaking are two pillars of modern quantum physics. Still, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, this problem is intrinsically bound to the subsystem of interest. Hence, in this work, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. We adapt the quasiparticle picture for entanglement evolution to the analytic determination of the entanglement asymmetry. We find, expectedly, that larger is the subsystem, slower is the restoration, but also the counterintuitive result that more the symmetry is initially broken, faster it is restored, a sort of quantum Mpemba effect, a phenomenon that we show to occur in a large variety of systems.

Editorial Note: This manuscript has been previously reviewed at another journal that is not operating a transparent peer review scheme. This document only contains reviewer comments and rebuttal letters for versions considered at Nature Communications.

REVIEWERS' COMMENTS
Reviewer #2 (Remarks to the Author): The major revision in this submission is the extension to the next-nearest-neighbor XXZ model in Eq. (15). The additional numerical evidences in Fig. 4 show that Mpemba effects exist in non-integrable models, not an edge effect (different boundary conditions) and not related to the groundstate phases (varying Delta). The authors also cite Mermin-Wagner theorem to motivate its generality. By and large, the revision have cleared my concerns. I recommend acceptance.
I have one more technical question that hopefully can improve understanding of Delta S_A in free model.
In Methods, it looks like the entanglement asymmetry calculation of the XY model can be factorized into individual modes, where each mode has a functon(22) similar to Eq. (7) in the warm-up problem. Is the Delta S_A additive for each mode, so that we can integrate the quasi-particle contribution just like the entanglement itself? I'm trying to see the 1/t^3 decay scaling in a simple way.
Reviewer #3 (Remarks to the Author): I would like to thank the authors for the reply and improvement. However, I still believe the significance of the results does not satisfy the criteria for publication in this journal.

Response to Reviewers
Reviewer 2 (Remarks to the Author): The major revision in this submission is the extension to the next-nearest-neighbor XXZ model in Eq. (15). The additional numerical evidences in Fig. 4 show that Mpemba effects exist in non-integrable models, not an edge effect (different boundary conditions) and not related to the groundstate phases (varying Delta). The authors also cite Mermin-Wagner theorem to motivate its generality. By and large, the revision have cleared my concerns. I recommend acceptance.
I have one more technical question that hopefully can improve understanding of ∆S A in free model. In Methods, it looks like the entanglement asymmetry calculation of the XY model can be factorized into individual modes, where each mode has a function (22) similar to Eq. (7) in the warm-up problem. Is the ∆S A additive for each mode, so that we can integrate the quasi-particle contribution just like the entanglement itself ? I'm trying to see the 1/t 3 decay scaling in a simple way.
We thank the referee for the positive comments about the revised manuscript and the recommendation for acceptance. With respect to their question, the factorization of the charged moments Z n (α) in Eqs. (7) and (22) refers to the number n of copies of the projected reduced density matrix ρ A,Q that we take in the Rényi entanglement asymmetry ∆S (n) A . Therefore, the factorization is not related to the modes k of the quasi-particles. To obtain ∆S (n) A , one has to perform the multi-dimensional Fourier transform (4) of the charged moments. Thanks to their factorization in the n copies of ρ A,Q , the Fourier transform can be done analytically at large times and the entanglement asymmetry is given by Eq. (13); observe that, in this regime, ∆S

(n)
A is actually additive in the quasi-particle modes k and we can integrate their contribution just like we do in the entanglement entropy. Analogously to the entanglement entropy, the leading order term for large t, in our case 1/t 3 , is determined by the quasi-particles with the slowest velocity |ϵ ′ (k)|, for free fermions they are those with momentum around k = 0 and π since |ϵ ′ (k)| = | sin(k)|. We are very grateful to the referee for raising this valuable question, and we have added in the final revised version a comment about this after Eq. (14).