Properties of the temporal transfer matrix in integrable Floquet circuits

The Authors show that the influence matrix encoding the reduced nonequilibrium dynamics of the boundary degree of freedom in an integrable circuit is itself an object that originates in the integrability structure. In particular, they show that it can be obtained as the leading Bethe vector of a particular non-diagonalizable transfer matrix. In the free-fermionic circuit they investigate the Jordan block structure of this transfer matrix.

The results themselves are interesting and warrant publication (perhaps, due to the technical nature of this work, SciPost Phys Core would be a better venue). However, I am sorry to say that, in its current form, the manuscript is practically unreadable. Parts if it are simply too cryptic and unexplained. Additional explanations should be provided. Below are some comments to help the Authors identify the most problematic parts.

(1) Above eq. (2.1) it is unclear what is meant by ``The IM takes values on the trajectories of a boundary spin.’’ Firstly, the boundary spin should be specified, since the chain is infinite. Secondly, which spin is $N$? Or is $N$ related to time up to which the evolution has been conisdered? Otherwise, why is $N$ relevant / why do the Authors focus on it? Finally, what is a trajectory of the boundary spin, and what does it mean ``to take value on the trajectory’’? Does it mean that the values of IM are trajectories, or that it acts on a trajectory? Until seeing this presentation I thought I understood what an influence matrix was. I think a nicely explained picture would be far more telling. Before attempting any definitions, I suggest the Authors to explain precisely (but in words), what an influence matrix is. Without this context the first part is unreadable.

(2) Still in eq. (2.1): why are $(U^+)^{-1}$, which act on the left half of the chain, sandwiched between bra-s and ket-s with positive indices, i.e., on the right half of the chain? I made a circuit diagram of what is written and it makes no sense: are the bra-s and ket-s correctly paired? Should the $U$ on the right be $U^+$? Why not simply focus on the picture.

(3) If I am not wrong, $b$ in $\rho_b$ refers to the ``bulk’’. But above eq. (2.3) we then have an operator ${\cal O}_0$, and in eq. (2.3) we have $\rho_0$. What does the index $0$ refer to? Also why is there $2N$ in the argument of ${\cal O}$? In eq. (2.3) the $\rho_{s_1,\bar{s}_1}$ has no index $0$. The element of which $\rho$ is that? I would suggest the Authors to carefully explain all of the notation.

(4) Fig. 1 requires an explanation. In fact, I would suggest the Authors to center the introduction of the IM around that figure, explaining it thoroughly. Also, what are now the time and space directions?

(5) For the unitarity of the R matrix in (3.1) one of the two parameters should be purely real, the other one purely imaginary.

(6) Between eq. (3.4) and eq. (3.5), site $i$ -> site $3$.

(7) After eq. (3.13): why is the singlet state in the kernel (nullspace) of $R_{0,1}(v)R_{0,2}(v-\eta)$? In particular, suppose I apply eq. (3.13) on the singlet state from the left: on the RHS I obtain zero, since this state is destroyed by $R_{1,2}(\eta)$. However, on the LHS I can obtain zero also in the case that $R_{0,1}(v)R_{0,2}(v-\eta)$ preserves the singlet subspace, since it is anyway then destroyed by the left-most $R_{1,2}(\eta)$. Perhaps the Authors meant invariant subspace instead of kernel?

(8) It is not clear why the transfer matrix should be non-diagonalizable from the above: if it is clear, then perhaps an additional sentence will suffice to explain it or remind the reader why it is so.

(9) The opening sentence of section 5 is superfluous: one does not require mentioning infrared degrees of freedom here, as it adds nothing to the discussion and does not clarify how we will identify them etc. Instead, the Authors should specify what kind of objects their ``infrared dof’’ are, i.e., what they are looking for. In particular, from what is written, it is not even clear what a degree of freedom is in this case.

(10) Section 5.1: using ``non-local’’ for a dense matrix that does not seem to be expressed in terms of fermionic operators, is misleading. Instead, the conjugated fermionic creation operator in eq. (5.6) is nonlocal, because the matrix of coefficients $M^+$ is dense. On the other hand, it seems that $H_{l,r}^\pm$ are indeed local operators expressed in terms of fermionic ones—eq. (5.15). But I do not see how the matrix of coefficients $M^+$ can then be compared to $H_{l,r}^\pm$, which is an operator? What does it mean that hese two objects commute?

(11) What is a classic fermionic variable?

(12) After eq. (5.15): while the transfer matrix is noninvertible, could it instead satisfy the so-called inversion identity? I.e., perhaps in the thermodynamic limit $T(v)T(v+\eta)$ approaches identity (up to a prefactor), or something similar? Then, maybe one could construct quasilocal Hamiltonians, as in [https://arxiv.org/abs/1506.05049]?

(13) In which space do canonical Hamiltonians in eq. (5.18) act?

(14) What is the analogy between spin operators in eq. (5.24) and ${\cal H}_{\alpha,d_\alpha,i_{d_\alpha}}$? How does it lead to lemma 5.4? I suggest the Authors to add more explanations.

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The paper studies the Jordan structure of the influence matrix. This is an relevant issue which has not been much explored before. The results obtained are interesting. However the presentation is too technical and poorly explained, making its physical impact difficult to grasp. The explicit analysis is confined to free fermions and the interacting case is only very briefly mentioned. It would be good to further discuss what aspects of the presented results should extend to the interacting case and what not. I think that the paper requires a major revision before being considered for publication.

I list concrete issues that would require revision.

Section 1: There is a brief paragraph on the explanation of the transfer matrix non-diagonalizability in terms of quasi particles, and the spin and masses of the latter. This discussion should be continued and enlarged in the main sections, but on the contrary it is practically absent .

Section 2: 1- The evolution operators in equation (2.1) should be U^+ both for the forward and backward contours. Besides, there is no explanation about the density matrix rho_b and its expression (2.2). What is the physical meaning of the parameter q?

2- In (2.1) and Fig.1 the boundary indices are traced. This corresponds to the infinite temperature limit, which is not mentioned. I guess that the yellow lines in Fig.1 correspond to the insertion of rho_b. If this is so, why there is a light cone structure centered on the joining line between forward and backward contours?

Section 3: 1- The R-matrix (3.1) is only unitary for restricted values of the spectral parameter u. This should be clarified.

2- It should be better explained why the transfer matrix (3.12) is not diagonalizable.

Section 4: 1- Better explain why (4.1) solves the degeneracy of the transfer matrix and why this is necessary.

2- It is not explained why the BAE can be solved explicitly in the limit epsilon->0, and how the important expressions (4.6)-(4.7) are obtained.

3- It is mentioned that (4.6)-(4.7) lead to an exact expression for the IM in terms of multiple integrals. Again this requires further comments.

Section 5: 1- In eq. (5.1) \hat R is introduced without explanation. Clarify the difference with the R-matrix previously introduced.

2- Why (5.7) and (5.8) are referred to respectively as classical and quantum fermionic variables?

3- Are the fermionic creation operators \psi_{alpha,i} supported on the temporal sites corresponding to the corresponding inhomogeneity? And what about those defined in (5.32)?

4- Comment why the IM has an even number of fermionic excitations.

5- In (5.23) \prod Lambda should not multiply the parenthesis?

6- The decomposition (5.25) should be clarified. Do the Jordan blocks on the rhs of (5.25) mix different \alpha sectors? Why the Jordan blocks on (5.22) should further decompose? The analogy with the spin operators of SU(2) in (5.24) is unclear. Why in Subsection 5.3 \alpha is used again to labels Jordan vectors instead of following the notation in (5.25)?

7- What is meant in Fig.6 by first order approximation?

8- Why the IM in (5.33) only involves two out of the four Jordan blocks?

9- Explain how the entanglement properties of the IM relate to its Jordan vector decomposition.

10- Elaborate further on why (5.28) guarantees the existence of a quasi-local gapped Hamiltonian.

Section 6: 1- A discussion on the relation between the exponential number of Jordan blocks and the entanglement barrier should also be included in Section 5.2.

2- It is mentioned that temporal entanglement associated to the IM for the XXZ chain grows logarithmically, while in the paragraph on the XX model it is said that the temporal entanglement exhibits an exponential barrier. These two statements seem contradictory.

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