$2+1$ dimensional Floquet systems and lattice fermions: Exact bulk spectral equivalence

1- This paper presents a principled derivation of the static Hamiltonian matching the desired Floquet spectrum. I find the presented rationale in deriving the appropriate static Hamiltonian very thorough and likely helpful to future investigations in this direction.
2- A clear discussion of both the analogies and differences between the constructed static Hamiltonian and Floquet system are presented.

1- As someone not deeply familiar with Floquet theory, the introduction did not contain a sufficiently pedagogical description of the background work in 1+1D for me to follow in a self-contained way. This work was properly cited, but as the introduction does attempt to survey this work, a little further clarity and detail would be very helpful.
2- In the derivation of the two choices of static Hamiltonians, it is quite difficult to follow the full spacetime structure of the resulting operators, due to the introduction of additional staggering partway through the derivation.

This article meets the criteria for publication in SciPost Physics Core. I recommend publication after some minor changes are made following the suggestions described below.

1- Please clarify the meaning of $\epsilon$ in the first paragraph. I assume these are the quasienergies noted later on?
2- Above eq (1.2), I find the phrase "time evolution operator measured at time T" unclear. As I understood Floquet theory, the notation $U_F(T)$ should here refer to time evolution for a time interval of length T. If this is the case, please clarify this text.
3- In eq (2.2) and below, the index $i$ can easily be confused with the imaginary unit. I suggest changing this to more clear notation, e.g. the notation $n$ used in [25].
4- In eq (3.8) and below, I don't follow what the tensor product space is precisely. I understand $H_s(p)$ as acting in the 2-dimensional space of sublattices, but what about $\sin(p_0 T/2)$ later? Please clarify this briefly here.
5- The formatting should be corrected in eq (3.16)
6- As noted above, I suggest summarizing the complete spacetime structure of the Hamiltonians in (3.32) and (3.33) after Fourier transforming back into position space, following eq (3.37) and (3.38).
7- The index structure in (3.28) is unclear, as j appears on the right side but not on the left. Likewise in (3.41). Please clarify the index structure here.
8- Waiting to describe the discrete-time structure of $d(p_0)$ does not in my view help the reader in any significant way. I suggest moving this result from eq (3.39) to near eq (3.7).

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1-The approach is rigorous, methodologically interesting and (mostly) pedagogical.
2-The results and its limitations are clearly stated and can be reproduced.
3-The study succeeds at extending previous research to 2D systems for a specific model.

1-The relevance of the study is not too well supported. It only deals with a specific Floquet model and it does not give general insights into the relation between Floquet and discrete time systems in 2D overall.
2-The information in some parts of the text lacks structure which makes it hard to read.
3-The consistency in notation, units and plot labeling can be improved.

The study extends previous research on a connection between Floquet a and static discrete time lattice systems from 1+1 to 2+1 dimensions. The authors manage to find a static discrete time Hamiltonian that reproduces the bulk spectrum of a specific anomalous Floquet system up to a 4-fold degeneracy and up to a $\pi/T$ quasienergy offset. In addition, this static model reproduces the open boundary conditions spectrum of the Floquet system with its corresponding topological phases in a strip geometry for a particular lattice termination. Unlike in the Floquet system, the edge modes of the static model are 4-fold degenerate and not chiral.

The paper is written in a mostly clear and concise way except for section 3B, which is much longer than the rest and can benefit from further structure and consistency in notation. In this section, the main thread of the explanation of the construction of the static Hamiltonian is intertwined with other considerations like the limitations of the 2D Wilson-Dirac Hamiltonian or the procedure to switch from momentum to position basis. These considerations may be relevant, but they make the reading difficult in the current structure. Also, notation and references to figures and equations is sometimes inconsistent.

The explanation is detailed and reproducible, but it is sometimes inconsistent in notation and contains typos.

Relevant previous research is correctly referenced.

The abstract and conclusion sections are clear.

The introduction section gives adequate context, but it does not provide a reason for the relevance of the specific Floquet model under consideration, which seems arbitrary. The authors get this model from reference 25, but in that publication the model has other parameters that are set to zero here. What does this specific connection between these models teach us about Floquet and static discrete time systems in general? Can this approach be generalized to all Floquet models or at least to a class of them?

Also, the results are contingent on several conditions like ignoring degeneracies and chirality of the edge modes, assuming a specific lattice termination and adding a $\pi/2$ quasienergy offset. The authors should make a stronger case defending the relevance of this specific model given these restrictions.

1- The words "observed" and "measured" are used in the introduction to refer to the fact that the state of a Floquet system at a discrete set of moments in time can be reproduced by a static (continuous) Hamiltonian. These words can mislead non-expert readers to believe that actual measurements are taking place at every period. The phrasing should be modified.
2- It seems that the units of the Floquet period $T$ are $\hbar$ divided by the unit of energy (which is fixed to have $J=1$). These units should be included in the text and in the labels of the figures.
3-It seems like the action approach is used to solve the fermion doubling problem in time, but all that is done in the end is to use the Susskind method to discretize the time derivative. Is the discussion about the action necessary?
4-It seems like when the staggered time lattices are introduced, the distance between $\phi_+$ and $\phi_-$ is $T/2$. This should be stated explicitly.
5-The simbols $k_0$ and $p_0$ seem to represent the same quantity in 3.3, 3.9 and other places. Why are two different symbols used?
6-The $\sin$ in the second line of Eq. 3.4 seems to be a typo.
7-In Eq. 3.14, the simbol $\zeta$ is introduced to refer to the quasienergy of the target Hamiltonian of Eq. 3.13, but in this equation, the simbol $\epsilon_s$ is used for that. Why are two different symbols used for the same thing?
8- Figures 7 and 8 both plot the eigenvalues of the target Hamiltonian for different boundary conditions. However, in the first one the vertical axis is labeled $\zeta$ and in the second one $\sin(T\epsilon')$. Both plots should be labeled the same. Also units should be provided (the units of energy seem to be fixed by $J=1$)
9- In the paragraph after Eq. 3.12, there is a typo "quai-energy".
10- The caption of Fig. 6 has typos.
11- Section 3B contains a lot of information and the aforementioned intermissions distract from the main point of the discussion. This section should be reorganised.
12- Eq. 3.16 does not fit in the width of the text.
13- Does Eq. 3.18 refer to PBC or OBC? If the answer is both, it should be explained how this is possible.
14- At the end of section 3B, the expressions of $\gamma_i$ nad $f_i$ are given, but not $F_i$. In particular it is never said how many flavors $\alpha$ the final model has. This is crucial for the reproducibility of the results.
15- The sentence "We can then demand that $\pm\sqrt{\sum_if_i^2}$ contains all the eigenvalues found in the expression for $\zeta$ in Eq. (3.14)." is confusing, there are only 2 $k$-dependent eigenvalues. Doesn't this just mean that we should demand $\zeta=\pm\sqrt{\sum_if_i^2}$?
16-The discussion in the last paragraph of the left column/first of right column in page 8 seems to imply that any $f_{i\neq1}$ different from zero would yield a lower bound of the PBC eigenvalues that surpasses $1/T$ for $k_+=\pi/a$. If this is not the case, it should be explained why.
17- In the left column of page 8 there seems to be an upper case $K_-$ that should be $k_-$. This also happens in page 9 bottom left.
18- Along section 3B, it is not clear when we are referring to OBC and when to PBC. This should be made more clear, maybe by restructuring the whole section.
19-Equation 3.28 is an equality between a matrix and an object with 1 index, this should be fixed.
20- The sentence "Opening the boundary in $x_-$ is also equivalent to introducing a domain-wall-anti-wall pair in m, as a function of $x_-$." is not clear what this means?
21- In the paragraph after 3.29 there is the expression $H_s-\gamma_1f_1$. Shouldn't this $f_1$ be $F_1$ for consistency with 3.18? In general, the way $f_i$ and $F_i$ are used in the whole text does not look consistent.
22-In Eqs. 3.35 and 3.36, $f_3$ seems to have different dimensionality than $f_1$ and $f_2$. Should they be $f_i$ or $F_i$?
23-It may be helpful to provide plots of some eigenstates (especially the edge states) for facilitating understanding.
24-In the first column of page 11 the authors write "We plot the solutions for $T = 1.5\pi$ which quite clearly reproduces the right panel of Fig. 6 with OBC." This points to a relation between Fig. 6 and some other figure which is not referred to. Do they refer to Fig. 10?

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