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The Floquet Baxterisation
by Yuan Miao, Vladimir Gritsev, Denis V. Kurlov
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Authors (as registered SciPost users):  Yuan Miao 
Submission information  

Preprint Link:  scipost_202208_00012v1 (pdf) 
Date submitted:  20220805 14:39 
Submitted by:  Miao, Yuan 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
Quantum integrability has proven to be a useful tool to study quantum manybody systems out of equilibrium. In this paper we construct a generic framework for integrable quantum circuits through the procedure of Floquet Baxterisation. The integrability is guaranteed by establishing a connection between Floquet evolution operators and inhomogeneous transfer matrices obtained from the YangBaxter relations. This allows us to construct integrable Floquet evolution operators with arbitrary depths and various boundary conditions. Furthermore, we focus on the example related to the staggered 6vertex model. In the scaling limit we establish a connection of this Floquet protocol with a nonrational conformal field theory. Employing the properties of the underlying affine TemperleyLieb algebraic structure, we demonstrate the dynamical antiunitary symmetry breaking in the easyplane regime. We also give an overview of integrabilityrelated quantum circuits, highlighting future research directions.
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Report
In this paper, the authors introduce a wide class of integrable quantum circuits built from inhomogeneous monodromy matrices obeying the YangBaxter 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 6vertex model. Lastly, the authors discuss the spontaneous breaking of the antiunitary symmetry of the Floquet Hamiltonian in the easyplane 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 settheoretic YangBaxter equation, TemperleyLieb 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 nonexpert readers with only a little background on the topics. For example, the connection between the first nontrivial charge and the blackhole sigma model related to certain noncompact 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 antiunitary symmetry. This is however only discussed at the end of the paper in Section 5. 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. It is however not easy to discern new results from the compilation of alreadyknown results. I believe some improvements and clarifications will be valuable.
Other (technical) remarks to be addressed:
 what does "FloquetBaxterisation" signify? Is this a precise mathematical notion or some "poetic" name? I am only aware of the "Baxterisation" procedure, referring to promoting solutions to the constant YangBaxter equation with a complex spectral parameter. While Baxterisation is achieved at the level of Rmatrices, 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 lightcone discretizations. I see little reason to coin new names.
 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 sigmamodel." Besides that, I cannot make sense of "a 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))?
 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 Hamiltonain H_F to coincide with the spectrum of a noncompact CFT?
 claiming that Ref. [17] computes multipoint 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.
 in eq. (3.13), the notation for the listed classes is not explained in the text
 it is unclear what "the inhomogeneous transfer matrix with one additional spectral parameter" is referring to. Any object on a twofold tensor space should possess two spectral parameters.
 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.
 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?
 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.
 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.
 In Section 5.2, or preferably even earlier when first introducing the notion of "FloquetBaxterisation", it would deserve to acknowledge some of the older works with staggered inhomogeneous rowtorow transfer matrices, particularly in the context of lightcone discretizations of integrable QFTs (sineGordon, sigma model etc.), e.g. by Destri and De Vega and Faddeev, Volkov and Reshetikhin.
 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 complexconjugate pairs representing bound states, whereas the second category is single roots with a finite imaginary part?
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").
Requested changes
I would recommend a revision. I do not think it would be unreasonable to restructure the paper and only retain the necessary background information, or even attempt rewriting it from the ground up. I would suggest better stressing the new contributions (which should be briefly announced and discussed in the introductory sections) and focusing more on the key results and specifically on the symmetry breaking. Detours should be avoided as much as possible.
Report
This manuscript presents the construction and the study of integrable lattice models describing 1d quantum systems governed by Floquet Hamiltonians. The timedependent 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 Rmatrix satisfying the YangBaxter 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 Rmatrix, without assuming the regularity and difference properties. Then, the sixvertex model with a staggering of order two is studied in detail. It corresponds to a stroboscopic XXZ Hamiltonian, which is Hermitian in the easyaxis regime Delta>1, and nonHermitian but PTsymmetric in the easyplane regime Delta<1. The finitesize spectrum of the Floquet Hamiltonian is derived through the Bethe Ansatz Equation. In the easyplane regime, a phase transition is observed between two subregimes, which correspond to different eigenvalues of the ground state under an antiunitary 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.
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 sixvertex model in Sections 56 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.
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 sixvertex model is related in the scaling limit to the "blackhole" sigma model CFT. This is based on the results of [4243], namely that the scaling limit of the staggered sixvertex model shares part of its operator spectrum and operator density with the "blackhole" sigma model CFT. However, in [4546], 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.
2. There are sign mismatches in (5.5) and (5.6).
3. References [42] and [48] are identical. I suggest to cite also this study of the spectrum of the staggered sixvertex model through NLIEs :
C. Candu and Y. Ikhlef, NonLinear Integral Equations for the SL(2,R)/U(1) black hole sigma model
J. Phys. A: Math. Theor. 46, 415401 (2013)
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.