The Floquet Baxterisation

Reply to Anonymous Report 1 on 2022-10-28 (Invited Report)

Referee: This manuscript presents the construction and the study of integrable lattice models describing 1d quantum systems governed by Floquet Hamiltonians. The time-dependent Hamiltonians considered are periodic and piecewise constant with respect to time. The associated lattice models correspond to quantum circuits. They are constructed from an elementary R-matrix satisfying the Yang-Baxter equation, on a lattice with staggered spectral parameters. This construction, and the commutation relation with the staggered transfer matrix, are carried out for a generic R-matrix, without assuming the regularity and difference properties. Then, the six-vertex model with a staggering of order two is studied in detail. It corresponds to a stroboscopic XXZ Hamiltonian, which is Hermitian in the easy-axis regime $|\Delta|>1$, and non-Hermitian but PT-symmetric in the easy-plane regime $|\Delta|<1$. The finite-size spectrum of the Floquet Hamiltonian is derived through the Bethe Ansatz Equation. In the easy-plane regime, a phase transition is observed between two subregimes, which correspond to different eigenvalues of the ground state under an anti-unitary operator.

Before giving my comments, I want to indicate that, not being an expert in quantum circuits, I am not able to deliver a judgement on the interest of this manuscript from the perspective of quantum circuits, nor in the bibliographic review on quantum circuits given in the manuscript.

After submitting this manuscript for a review, we became aware of an experiment from the Google team {1}. This paper is the first experimental demonstration of integrable Floquet protocol related to the six-vertex model in a field. This is a part of our story.

{1} A. Morvan, T. I. Andersen, X. Mi, C. Neill, A. Petukhov, K. Kechedzhi, D. A. Abanin, A. Michailidis, R. Acharya, F. Arute, et al., Nature 612, 240 (2022)

Referee: The general construction given Section 4 is quite elementary from the point of view of integrable models. The previous section gives convincing arguments that this simple construction deserves attention. The specific example of the six-vertex model in Sections 5-6 also involves simple calculations (i.e. writing the Bethe Ansatz equations and solving them numerically for small system sizes, and examining the relation between the spectral parameter alpha and the period T), but it is treated with care and clarity.

I appreciate the fact that in seemingly unphysical regimes of the six vertex model, e.g. |Delta|>1 and pure imaginary spectral parameter alpha, are given a nice physical interpretation, which is the quantum circuit.

Answer: We highly appreciate this positive feedback of our introduction and an example.

Referee: Here is a list of issues I have identified in the manuscript, which deserve to be corrected:

1. In various places in the manuscript (in particular, in the abstract and in Section 3), it is asserted that the staggered six-vertex model is related in the scaling limit to the black-hole" sigma model CFT. This is based on the results of [42-43], namely that the scaling limit of the staggered six-vertex model shares part of its operator spectrum and operator density with theblack-hole" sigma model CFT. However, in [45-46], with a more careful of the spectrum, using the IM/ODE correspondence, it was shown that the six vertex model actually corresponds to a different CFT.

Answer: The correspondence of the $Z_{2}$ staggered spin lattice model in the continuum limit with the $sl(2, R)/u(1)$ Euclidian black hole CFT which is a non-linear sigma model with a non-compact target space first introduced in the string-theory literature was first made by Ikhlef et al. This observation was supported by a number of evidences, analytical and numerical, including the match of the density of states, various numerical and analytical studies. Finally, a very recent series of works by Bazhanov et al. on the one hand confirms the coincidence of the partition function of the Euclidian black hole CFT with {\it one half} of the partition function arising in the scaling limit of the lattice model with periodic boundary conditions, but on the other hand refines the original identification by proposing that a part of the Hilbert space of the lattice model should coincide with the pseudo-Hilbert space of the non-linear black hole sigma model with Lorentzian signature. We believe that the phase transition found in our paper is related to the compact-non-compact transition in the corresponding spectrum of this non-linear sigma model. We updated the text correspondingly.

Referee: 2. There are sign mismatches in (5.5) and (5.6).

Answer: We are very grateful to the Referee for spotting this misprint. We have corrected Eq. (5.6).

Referee: 3. References [42] and [48] are identical. I suggest to cite also this study of the spectrum of the staggered six-vertex model through NLIEs : C. Candu and Y. Ikhlef, Non-Linear Integral Equations for the SL(2,R)/U(1) black hole sigma model J. Phys. A: Math. Theor. 46, 415401 (2013)

Answer: Thank you for this observation and suggestion! We corrected and updated our manuscript accordingly.

Referee: Overall, the manuscript is well written, and presents some interesting original material. Therefore, I recommend its publication in SciPost Physics, provided the above points are addressed.

Answer: We are grateful to the Referee for his/her good opinion about our paper. He hope we completely implemented all these valuable comments above.

Reply to Anonymous Report 2 on 2022-10-28 (Invited Report)

Referee:

In this paper, the authors introduce a wide class of integrable quantum circuits built from inhomogeneous monodromy matrices obeying the Yang-Baxter equation. After providing a general overview, the authors outline the construction in the case of periodic and open boundary conditions. Subsequently, they detail the algebraic constructions associated with the inhomogeneous 6-vertex model. Lastly, the authors discuss the spontaneous breaking of the anti-unitary symmetry of the Floquet Hamiltonian in the easy-plane regime of the model.

In my impression, the presentation is somewhat eclectic. The manuscript walks the reader through a diverse range of interrelated topics, including the set-theoretic Yang-Baxter equation, Temperley-Lieb algebras, and various types of solutions with different boundary conditions. Certain connections to CFTs are highlighted as well. If I am not mistaken, this part is largely a summary of the existing literature, including recently published results by the authors themselves. I wonder how much value added value can these chapters be to non-expert readers with only a little background on the topics. For example, the connection between the first nontrivial charge and the black-hole sigma model related to certain non-compact CFTs could certainly be explained in more detail, or otherwise (assuming it is of marginal relevance) completely dropped. I also struggled with decoding the meaning of dynamical Floquet criticality and in what way is it relevant for the subsequent discussion. In my perspective, the most interesting, presumably new, result of this manuscript concerns the breaking of anti-unitary symmetry. This is however only discussed at the end of the paper in Section 5.

Answer: Indeed, in the introduction we aimed to provide an overview of existing, and potentially relevant candidates for the Floquet integrable models in the broad sense. We do not think that the paper suffers from it - we think that it serves as a basis for readers who might not be familiar with the field, or interested in the recent developments motivated by the recent experimental progress. Dynamical Floquet criticality is one of the subjects of this paper - we found it in a particular class of models in Sec. 6 and we think it is related to the compact--versus--non-compact nature of the underlying non-linear sigma model CFT Hilbert space structure discussed in the overview section. This could be explained by a spontaneous breaking of the anti-unitary symmetry. We extended this part in the current version.

Referee: My impression is that most of the content from Sections 2 to 4 does not go further than the standard application of the algebraic Bethe Ansatz with inhomogeneous transfer matrices, dating back to works by Faddeev and coworkers.

Answer: We agree with the Referee on this point. These sections serve as an introduction for the readers who might not be familiar to these techniques.

Referee: It is however not easy to discern new results from the compilation of already-known results. I believe some improvements and clarifications will be valuable.

Answer: Our results are new in the sense that they establish a formalism for $\mathbf{dynamics}$ of a quantum many-body system rather than equilibrium statistical-mechanical models or quantum spin chains. Moreover, the previous works are mainly focused on the correspondence between quantum circuits and staggered transfer matrices. As we proved in our paper, this analogue can be easily generalised into the correspondence between quantum circuits and inhomogeneous transfer matrices with any period $n\in \mathbb{Z}_{>0}$, which has not studied systematically before.

Referee: Other (technical) remarks to be addressed: - what does Floquet-Baxterisation" signify? Is this a precise mathematical notion or somepoetic" name? I am only aware of the "Baxterisation" procedure, referring to promoting solutions to the constant Yang-Baxter equation with a complex spectral parameter. While Baxterisation is achieved at the level of R-matrices, Floquet pertains to inhomogeneities monodromy operators with staggered inhomogeneities. This type of construction with commuting staggered transfer matrices has indeed been around for quite some time, at least from the work by Destri and De Vega on light-cone discretizations. I see little reason to coin new names.

Answer:
The reason why we name the procedure as Floquet Baxterisation'' is as follows. First,Baxterisation'' is coined by Vaughan Jones in 1989 to describe a procedure that assign a spectral parameter to the transfer matrices of certain 2D classical statistical mechanical models, such that the transfer matrices are in involution. What we are doing here is in analogue to what Jones did. Namely, we start with an operator $\mathbf{U}_{\rm F}$ and construct the mutually commuting operator $\mathbf{T} (u)$, where $\mathbf{U}_{\rm F}$ is a special case of $\mathbf{T} (u)$. Even though we use different argument compared to Jones', the philosophy is essentially the same. Secondly, the time evolution operator $\mathbf{U}_{\rm F}$ of the quantum circuit describes a Floquet time evolution (discrete-time and periodic) of 1D quantum spin chains. Together we believe that it is justified to use the term ``Floquet Baxterisation'' to describe our procedure to elucidate the integrability of the quantum circuits.

Referee: - I failed to decipher the meaning of when written in the representation of the XXZ Hamiltonian density, coincides exactly with the Hamiltonian of the lattice limit of the SL(2,R)/U (1) black hole sigma-model." Besides that, I cannot make sense ofa lattice limit" (of a QFT). I wonder whether these findings are the novel contribution of this work or perhaps quote some previous studies? What does it mean ``it appears" (before eq. (3.4))?

Answer: Here we refer to the original papers on spin chains which, in the continuum limit, are believed to be related to the SL(2,R)/U(1) nonlinear sigma model CFT. The Hamiltonian density of these spin chains coincide with the first nontrivial conserved charge of our Floquet protocol. We hope that this remarkable (and unexpected to us) fact explains why we have used the wording ``it appears". This is a novel contribution of our work.

Referee: - I do not understand the statement about a novel dynamical Floquet criticality. I hope the authors can elaborate on it in the revised version. Another question related to the aforementioned CFT spectrum (cf. end of page 6): what does it mean for a discrete spectrum of Floquet Hamiltonian $H_F$ to coincide with the spectrum of a non-compact CFT?

Answer: We refer to the Floquet criticality as a drastic reconstruction of the Floquet Hamiltonian spectrum. The spectrum of that non-compact (and non-rational) CFT has essentially two parts, a discrete one and a continuous one. In a series of papers, especially in arXiv:2305.03620 by Kotousov and Lukyanov, a possibility of spontaneous discrete symmetry breaking is not discarded. Concerning the second question: Our integrable Floquet Hamiltonian commutes with the charges defined in our (3.4), (3.5) which is a discrete version of the aforementioned CFT. This implies that the space of states of both coincide. This is what we meant by that statement.

Referee: - claiming that Ref. [17] computes multi-point correlators in (the integrable Trotterisation of) the XXX model seems quite a stretch. While in the quoted paper the authors cast the correlators in terms of a particular transfer matrix, they give no procedure for evaluating them at finite or asymptotically large times.

Answer: We agree with the Referee on this point. We modified the statement in our paper.

Referee: - in eq. (3.13), the notation for the listed classes is not explained in the text

Answer: We agree that the notations are not very transparent. We changed that in the modified version. While A-B-C-D... is clear (just an alphabetical order) the subscript is not obvious. ``P'' means simply permutation. We would like to thank the Referee for pointing out these unfortunate notations.

Referee: - it is unclear what ``the inhomogeneous transfer matrix with one additional spectral parameter" is referring to. Any object on a two-fold tensor space should possess two spectral parameters.

Answer: We thank the referee for pointing this out. What we meant there was the inhomogeneous transfer matrix $\mathbf{T} (u, { u_j })$ contains one additional spectral parameter $u$ compared to the time evolution operator $\mathbf{U}_{\rm F} ({ u_j })$. We have removed the statement in the main text now.

Referee: - In Section 3.4, $u_j$ represent operators. On the other hand, in Section 2 and in 4.1., the same notation is used to represent complex scalars pertaining to inhomogeneities of commuting transfer matrices.

Answer: We would like to thank the Referee for pointing out this unfortunate mismatch in our notations. We have fixed the notation by using the bold font in Sec. 3.4 to signify the fact that they are operators in that section.

Referee: - I do not understand the statement regarding the connection between parameters u and T appearing just before eq. (5.6). Perhaps it was meant $\alpha$?

Answer: We would like to thank the Referee for this remark: this unfortunate mistake is now corrected, as well as eq. (5.6).

Referee: - On page 18, the authors mention a conjecture from an earlier paper, reference [26], but it is unclear what precisely the conjecture is about. Moreover, after eq. (5.7), it is mentioned that the conjecture (again, without further details) has been only partially proven. Overall, I find Section 5.1. form rather obscure in its current form.

Answer: We agree with the Referee that the formulation was not clear. We hope that we clarified it in the updated version.

Referee: - The comments made after eq. (5.19) are not particularly clear, or at least some additional references (about algebraic constructions of the Q operator) would probably be beneficial.

Answer: In our investigation, we have used the Q operator constructed in {2} to study the spectra of the Floquet evolution operator. In {2}, the Q operator is constructed in the homogeneous limit, which can be easily generalised to the inhomogeneous case in this paper. Therefore, we would like to make a short comment on this generalisation in this paper.

{2} Y. Miao, J. Lamers, and V. Pasquier, SciPost Phys. 11, 67 (2021)

Referee: - In Section 5.2, or preferably even earlier when first introducing the notion of ``Floquet-Baxterisation", it would deserve to acknowledge some of the older works with staggered inhomogeneous row-to-row transfer matrices, particularly in the context of light-cone discretizations of integrable QFTs (sine-Gordon, sigma model etc.), e.g. by Destri and De Vega and Faddeev, Volkov and Reshetikhin.

Answer: We thank our Referee for pointing out these earlier works which we forgot to mention. This omission is fixed in the updated version.

Referee: - At the end of page 24, I do not understand the third category (last bullet) of Bethe roots. Presumably, it is meant that roots organize into complex-conjugate pairs representing bound states, whereas the second category is single roots with a finite imaginary part?

Answer:
We would like to thank the referee for the question. What the referee wrote is essentially correct. The third category refers to bound states with complex conjugate pairs of Bethe roots and the second category refers to single Bethe root with imaginary part being $\pi/2$ (also known as the Bethe root with odd parity in the literature). We have added a few words in the list to explain this.

Referee: Other comments: - a number of equations are missing a comma at the end - on pages 7 and 9: for consistency reasons, the spelling should be Trotterised". - below eq. (4.2),baxterised" should be capitalized - in eq. (4.16), there should be some space after the comma . in eq. (4.21), there is a missing comma after the ellipsis - in Fig. 9, there is a typo (``noted").

Answer: We are grateful to the Referee for a very careful reading of our manuscript. We made effort to fix (hopefully) most of these typos.

1-- We have rewritten and reorganised the introduction and the first three sections to improve the presentation of the content.

2-- We have added a more detailed discussion between the staggered 6-vertex model and its field theory limit, and pointed out a possible connection to the dynamical anti-unitary symmetry breaking in Section 6.

3-- We have changed the typos pointed out in the referee reports. See author comments for the detailed changes.

4-- We have added several relevant references.