Entanglement asymmetry in periodically driven quantum systems

This paper studies the time evolution of the entanglement asymmetry in periodically driven systems. The entanglement asymmetry is a novel quantity that measures the extent to which a symmetry is broken in a portion of a many-body quantum system. As the authors correctly point out, its time evolution has been recently analysed after a global quantum quench in a wide variety of integrable and non-integrable systems, but
there are not works that consider time-dependent periodically driven systems. The present paper fills this gap and explores, combining analytic and numeric methods, the behaviour of the entanglement asymmetry in the periodic driven XY spin chain, in a Rydberg atom chain effectively described by a PXP Hamiltonian with a time dependent periodic magnetic field and in driven CFTs. As in the case of quantum quenches, the authors take an initial state that breaks certain symmetry that is respected by the dynamics. Then they examine whether the symmetry is restored at late times and the occurrence of the quantum Mpemba effect, i.e. the more the symmetry is initially broken, the faster is restored. The authors find a rich phenomenology. For both the XY spin chain and the PXP model, they obtain that, for certain special Floquet frequencies, the symmetry is restored at long times and they observe the quantum Mpemba effect. On the other hand, the entanglement asymmetry exhibits a drastically different behaviour in driven CFTs. Here, instead of decreasing in time, it grows with the number of Floquet cycles, if I correctly understand.

I think that this work presents interesting, novel, and timely results that make it suitable for being published in SciPost. In my opinion, the paper is, in general, well written and complete, except the CFT part. I have several doubts in that section that should be clarified before I can accept it:

i) In Sec. 4.1, the authors seem to mix several setups. They apply the methods from Ref. [21], which studies the resolution of the entanglement entropy in a CFT on the complex plane with respect to a symmetry of the theory. Formula in Eq. (38) seems to assume a cylinder geometry. However, the final result (Eq. (40)) is compared with results from Ref. [24], where the entanglement asymmetry of an interval attached to a boundary that breaks the symmetry is studied. I am confused by the exact setup the authors are considering. This needs to be clarified.

ii) The formulas in Eq. (37) are derived using the Appendix of Ref. [21], where the vertex opertors that implement the conserved charge $e^{-i\alpha Q}$ create a topological defect line. In that case, one can split the flux $\alpha$ among the replicas such that $\sum_{j=1}^n \alpha_j=\alpha$. This mimics the case of asymmetry in Eq. (33). However, the result I obtain for $d_n$ is different from the expression in Eq. (37). I am obtaining $d_n=c(n-1/n)/24+\Delta(\sum_{j=1}^n \alpha_j)/n^2$. What is the crucial point I am missing to get Eq. (37) instead of the $d_n$ I am obtaining?

iii) I think there is a typo in Eq. (38). The last equality should be $=\mathrm{Tr}(\prod_{j=1}^{n-1} \rho_A e^{i\alpha_j Q})$ instead of $=\mathrm{Tr}(\rho_{QA}^n)$, which would be consistent with Eq. (39).

iv) I think Eq. (39) is only valid for Renyi index $n=2$, but it is written $\Delta S_n$.

v) In the first sentence of Sec. 4.1, the authors write "The computation $\Delta S_n$ in equilibrium has been carried out for cylinder geometry in several works [21, 22]". I would like to point out that, in Ref. [21], the entanglement asymmetry is not computed but rather the symmetry-resolved entanglement, which is the opposite situation. I would suggest to cite instead the paper JHEP 05(2024) 059 where the entanglement asymmetry is studied in the ground state of CFTs breaking a symmetry in the bulk and, in particular, in the Ising CFT. Ref. [25] also investigates asymmetry at equilibrium in CFTs in the complex plane.

Apart from the concerns above, I would like to ask the following question:

vi) In the driven XY spin chain and PXP model, does the quantum Mpemba effect always occur when the symmetry is restored? It is not entirely clear to me from the discussion.

Ask for minor revision

The entanglement asymmetry, which is a useful tool to study the symmetry breaking in the subsystem level, has received extensive attention in the community. This work gives an initial study of entanglement asymmetry in time-dependent driven systems, based on both numerical and analytical methods, in both lattice systems and conformal field theories. The results are novel and interesting. I would recommend the publication of this work after the authors address the following questions/remarks:

— Is there any scaling behavior in the time evolution in Fig.2? Since this is essentially a free-fermion calculation, I think the scaling behavior can be obtained if there is.

— The results in driven spin chains and in driven CFTs are qualitatively different. Is there any physics to understand this difference? Apparently, the driving protocols are very different in these two cases. But it is not clear to me what essential factors result in this qualitative difference in the entanglement asymmetry evolution.

— The analytical results of entanglement asymmetry in driven CFTs are very interesting. I have a technical question here. In Fig.4, the parameters are chosen as l=100 and L=1000. It is known that in driven CFTs, whether one includes the energy density peak(s) in the subsystem or not will give different features in the von Neumann entropy evolution. For the entanglement asymmetry studied here, I wonder if this choice (with energy peak(s) included in the subsystem or not) is still important.

— Since the driven CFT can also be analytically studied when the initial state is a thermal state, as has been recently studied in literature, I think it may be interesting to check how the finite temperature affects the entanglement asymmetry in a driven CFT. This may be an interesting future work to study.

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