Dynamical localization and slow thermalization in a class of disorder-free periodically driven one-dimensional interacting systems

+ Very well written.
+ Comprehensive and systematic analysis of a class of systems with space-time periodic drive.
+ Very nice combination of analytical and numerical work.

+ Incomplete analysis of the interacting system and the scaling with system size.

The authors consider a class of periodically driven quantum many-body systems of spinless fermions in one dimension. The Hamiltonian consists of a hopping term and nearest neighbor density-type interactions in the presence of a time periodic on-site potential which is also periodic in space with period $m=2$ and $m=4$. The noninteracting models $V=0$ exhibit the phenomenon of dynamical localization, which at special points in parameter space leads to an effective suppression of the hopping in first order Floquet perturbation theory.

The authors argue that the dynamical localization provides a mechanism for Hilbert space fragmentation since a kinetic constraint emerges. This is reflected in the entanglement properties of the Floquet eigenstates, some of which exhibit an entanglement deficit reminiscent of what is known in quantum many-body scar states in other constrained models.

The paper is very well written and takes a systematic approach to this class of models, analyzing them in detail using a combination of Floquet perturbation theory and exact numerical solutions of the models in the high frequency limit.

My main criticism concerns the analysis of the effect of interactions: It seems that the analysis of the interacting models is essentially limited to a single system size ($L=16$) and there is no systematic analysis of the scaling of the results with $L$. This is particularly important in the interacting case, since the many-body bandwidth grows with $L$ and with growing $L$ and at fixed frequency $\omega$, one should expect that the effect of wrapping quasienergies around the unit circle will become important. In particular, this means that in Figures similar to Fig. 5 I would expect that for increasingly large $L$, the dependence on quasienergy should vanish and it would be extremely interesting to see what happens to the putative scar states in this limit.

It is numerically challenging to access larger system sizes, but perhaps this question can be addressed by lowering the driving frequency and considering several smaller sizes to identify the trend when $L$ increases.

In summary, I find this paper very well written and an interesting systematic analysis of this class of models. For a complete picture, a discussion of the thermodynamic limit in the interacting case is necessary, although the results at finite (fixed) size are of course correct and relevant for small size experiments e.g. in ultracold atomic setups.

If a proper discussion of the scaling with $L$ is provided, this paper is suitable for publication in SciPost Physics Core.

1- The set of references cited for dynamical localization at the end of p. 2 [74-79] seems to be a mistake and probably should be [58-64] (cf. top of the page).
2- Can the authors check if the $\sum_n$ in Eq. (13) needs to exclude $m=n$? Presumably the matrix element of $H_1$ vanishes in this case, but it's perhaps better to be explicit.
3- The figures appear to be rasterized in low resolution. To reach publication quality, the figures should be vectorized. To limit file sizes, in some cases only the panel content should be rasterized (e.g. Fig 4a, 4b etc).
4- In several occasions, the limits $\mu \gg J$ and similar are typeset with a double $>$ sign. It's better to replace this by $\gg$. (p. 6, 8, p. 11, 14, 15, etc.)
5- Fig 2 shows the crossover of the correlation function $\delta C_n$ as a function of the number of cycles. Can the authors show the derived result for the crossover $n_c \approx 1/|\epsilon|^2$ in the figure?
6- The authors show in Fig. 5 the entanglement entropy for exact eigenstates of the Floquet operator and for eigenstates of the 1st order FPT Hamiltonian and say that the quasienergies agree "quite well". It would be interesting to show this comparison directly. One way to do this would be to order both spectra by the phase angle (quasienergy) and then plot $E_{\mathrm{exact}}$ vs. $E_{\mathrm{FPT}}$. A straight line would indicate an exact match, and deviations quantify the agreement.
7- Is the Fig. 5b showing the spectrum of Eq. (36) or Eq. (42)? It would be
helpful to indicate this in the figure caption.
8- Fig 5a) and to a larger extent 19b) show states at zero quasienergy with an excess entanglement entropy compared to the rest of the spectrum. This is reminiscent of what is seen in the PXP model, where the origin is a large degenerate subspace at zero energy and the numerically obtained eigenvectors
are an arbitrary orthonormal basis of this subspace? If so, there is no physical content to these points and it would be good to check if there is such a degeneracy.

1. The paper is well written and all calculations are clearly explained.
2. The results are sound and address the timely topic of quantum non-ergodic behaviour - or failure of thermalization - in Floquet driven quantum many body systems.
3. The presentation to first focus on non-interacting limits and then add interactions on top is very pedagogical.
3. The authors find a wealth of different phenomena, from dynamical single particle localization, to kinetic constrained induced quantum many body scars as well as Hilbert space fragmentation.
4. The mappings to other well studied models, e.g. the SSH chain and the transverse field Ising chain are helpful to appreciate the Floquet engineering aspect.

1. While the paper has many different results I did not find a clear motivation why to study this special set-up and model, e.g. is it experimentally easier to assess than other related proposals?
2. The biggest weakness concerns the novelty of the findings. It seems that each phenomenon - dynamical single particle localization, Floquet driving induced kinetic constraints, Hilbert space fragmentation, slow thermalization from weak (Floquet) integrability , ... have been studied before in other works? What is the new addition which goes beyond any of the existing works?
3. The employed methods are sound but standard and I do not see any methodological advance (which is not a problem if there is new physics or a new experimental proposal, see point 2 above.)

Overall, I found the paper well written addressing timely topics. However, looking at the Acceptance Criteria Expectations I have a hard time to see the breakthrough or groundbreaking discovery. One could argue that a single model can provide a synergetic link via Floquet engineering from dynamical localization of single particle states to many-body kinetic constraints. However, this has been shown in the literature before. The authors need to explain in how far their study goes beyond previous results and what is the novelty.

1. The different panels in Figure 1 all show essentially the same except for a different energy scale. There is no need for three panels but the change in scale can be explained in the caption.
2. Fig.4 c upper panel does show long-lived revivals but there is a clear decay. What is the functional form of the decay and how does it depend on the frequency and parameters?