# Correlations and commuting transfer matrices in integrable unitary circuits

### Submission summary

 As Contributors: Pieter W. Claeys Arxiv Link: https://arxiv.org/abs/2106.00640v3 (pdf) Date submitted: 2021-10-08 10:56 Submitted by: Claeys, Pieter W. Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Mathematical Physics Quantum Physics Approach: Theoretical

### Abstract

We consider a unitary circuit where the underlying gates are chosen to be R-matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.

###### Current status:
Editor-in-charge assigned

We would like to thank both referees for their careful reading of our manuscript, their useful suggestions, and their positive assessment. In the revised version we have addressed all requested changes and made some minor revisions which we hope makes our paper acceptable for publication in SciPost Physics Core.

For the authors,
Pieter W. Claeys

### List of changes

The main changes in the manuscript can be found below.

- The notation of transfer matrix and monodromy matrix has been made consistent with the literature throughout the manuscript.
- We have made clear that we are working with the $\check{R}$-matrix rather than the $R$-matrix, and explicitly mentioned the connection between the two when introducing the braiding relation in Eq. (19).
- The connection with the fusion procedure of Kulish, Resehtikhin and Sklyanin, Lett. Matt. Phys. 5, 393 (1981) has been explicitly mentioned at the end of Section 2.4.
- The BAE for arbitrary spin s have been given in Eq. (43), making it easy to identify our obtained BAE [Eq. (42)] as those for the integrable spin-1 chain.
- In the conclusion we now explicitly mention the importance of nonabelian symmetries for KPZ.
- In Table 1 the dimensions of the Jordan blocks associated with the eigenvalue $\lambda^2/(1+\lambda^2)$ have been corrected: rather than three 6x6 blocks, this should have been three 5x5 and three 1x1 blocks. At the end of Section 2.5 we also provide a reference and short discussion of the work of the added Ref. [59], which allowed us to spot this mistake and presents a useful way of constructing the matrix elements of the transfer matrix in the generalized eigenbasis, but unfortunately does not allow for a direct calculation of the dimension of the Jordan blocks in general.
- Minor typographical changes have been made throughout.

We have chosen to keep Section 2.4 on the inhomogeneous model as is. We agree that the section is technical, but we also believe that the results in there are important for establishing the completeness of the Bethe ansatz in the inhomogeneous case, as well as further clarifying in which ways the presented two-component transfer matrices differ from the one-component transfer matrices usually studied. Parts of this section had also already been moved to Appendix before submission. To accommodate this remark we now mention at the beginning of Section 2.4 that readers only interested in the homogeneous model can skip forward to Section 2.5 for a discussion of the eigenvalues and dimensions of the Jordan blocks.

The suggestion to present all calculations in the folded representation is a useful one, but when writing the manuscript we made the decision to keep all calculations as explicit as possible. While the folded representation would definitely simplify some expressions, we thought it might also obscure some calculations and make the connection with some of the established literature on integrability less clear, which is why we have chosen to present these results in this way.

### Submission & Refereeing History

Resubmission 2106.00640v3 on 8 October 2021
Submission 2106.00640v2 on 22 June 2021

## Reports on this Submission

### Report

I believe the authors have successfully addressed all remarks and improved the presentation. I am happy to recommend the publication of the mansucript in SciPost Physics in the present form.

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