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Correlations and commuting transfer matrices in integrable unitary circuits

by Pieter W. Claeys, Jonah Herzog-Arbeitman, Austen Lamacraft

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Submission summary

Authors (as registered SciPost users): Pieter W. Claeys
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-06-22 11:07
Submitted by: Claeys, Pieter W.
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


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:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-8-19 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2106.00640v2, delivered 2021-08-19, doi: 10.21468/SciPost.Report.3416


- computation of dynamical correlation functions in integrable lattice systems is addressed directly in terms fo an appropriate transfer matrix
- the technique has a clear diagrammatic representation in terms of basic objects of integrability


- it is a unfortunate that explicit results on (asymptotic) of correlation functions are still out of reach, even in simple special cases
- it would be good to see some calculations in more explicit form


The paper proposes an explicit expression for 2-point (as well as 4-point, in case of OTOCs) spatio-temporal correlation functions of local observables in terms of a specific inhomogeneous and non-unitary transfer matrix. This approach works particularly nicely for models which are naturally written as integrable brickwork circuits with the elementary two-site gate obeying the braid form of Yang-Baxter equation. Clearly, the approach can be extended to integrable Hamiltonians by means of the Trotter formula.

It is nice to see an alternative, fresh approach to computation of dynamical correlations in integrable systems which circumvents the use of cumbersome form factor expansions. Still, many technical difficulties remain which prevent one from obtaining explicit asymptotic results, say, addressing the anomalous (super-diffusive) KPZ spin transport in trotterised XXX model. These difficulties, mainly related to non-diagonalizability and non-unitarity of the transfer matrix, are clearly discussed and explained. Appropriate Bethe equations for the correlation transfer matrix spectrum are spelled out and related to spin-1 integrable chains.

The paper is clearly written. Several technical derivations are nicely presented using diagrammatic technique. Despite lacking really useful results, I think it will be a valuable addition to the literature and could perhaps stimulate further development of this important problem. Therefore, I recommend it for publication in SciPost.

Requested changes

- For the sake of compatibility with the literature, it is perhaps worth mentioning at some place that what authors denote as $R$-matrix is usually referred to as matrix $\check R = P R$.

- Misleading sentence in the conclusion: KPZ universality scaling of correlation functions is expected only in integrable systems with nonabelian symmetries, e.g. in gapped XXZ model one finds diffusive transport. This has to be stressed correctly.

- Optional: I wander if the formulation of the correlation function transfer matrix would looks simpler if the authors would use the "folded representation" from the beginning (?)

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Author:  Pieter W. Claeys  on 2021-10-08  [id 1823]

(in reply to Report 2 on 2021-08-19)

We would like to thank the Referee for their clear report with a detailed reading of our manuscript and useful comments. As mentioned in our resubmission, we have attempted to address all requested changes and hope this makes our paper acceptable for publication in SciPost Physics Core.

For the authors,
Pieter W. Claeys

Anonymous Report 1 on 2021-8-9 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2106.00640v2, delivered 2021-08-09, doi: 10.21468/SciPost.Report.3355


1) It deals with an interesting and timely problem, namely computation of correlators in unitary circuits
2) Uses advanced techniques in integrable systems
3) Obtains interesting and novel results
4) It is very well written
5) contains an updated list of references.


1) contains some technical parts that could be moved to appendices, making the paper more readable.


This paper proposes an interesting method to compute infinite temperature correlation functions between one-site operators in integrable quantum circuits. The latter are given by a trotterization of the evolution operator of an integrable
Hamiltonian, in this case the spin 1/2 Heisenberg Hamiltonian.
The authors formulate the problem using a non-standard transfer matrix with a two-component structure, that depends on a spectral parameter. They show that transfer matrices, with different spectral parameters, commute thanks to the Yang-Baxter equation satisfied by the R matrix of the XXX model. They use the algebraic Bethe ansatz (ABA) to diagonalize the transfer matrix that,
quite interestingly, turns out to coincide with the Bethe ansatz equation for a spin 1 XXX model. The diagonalization has also some subtleties that lead the authors to consider the inhomogeneous version of the model that enable them to show the completeness of the ABA and other subtleties like the Jordan like structure of the transfer matrix in the homogenous case. In some simple cases the spectrum of the transfer matrix is given analytically showing the nesting property and the Jordan decomposition in the homogeneous case. The paper is very welll written and contains highly interesting results.

Requested changes

1) In the footnote in page 9 it is said that $\tau_m(\lambda)$ is the monodromy matrix and $T_m(\lambda)$ is the transfer matrix. Please correct, the notation is the opposite.

2) The standard RTT equation is formulated in terms of the "universal" R matrix that satisfies the Yang-Baxter eq. in the form

R12(u) R13(u+v) R23(v) = R23(v) R13(u+v) R12(u)

that differs from the one given in eq. (19) that corresponds to "braiding" R matrices. A comment on this issue will be welcome.

3) Section 2.4 is devoted to the inhomogenous model is interesting but a bit technical. I would suggest the authors to move some parts to an appendix to make the manuscript more readable.

4) In page 17 it is explained why the Bethe eq. for the roots is the same as those for the spin 1 - XXX model of Babujian and Taktajan. Is there a relation between this derivation and the one in reference Kulish, Resehtikhin and Sklyanin, Lett. Matt. Phys. 5, 393 (1981) where the R matrices for higher spins are derived using the spin 1/2 R -matrix?

5) I would be helpful for the reader to provide the form of the BAE for a generic spin S, so that the statement that eq.(42) is the case S=1, will be readily understood.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Author:  Pieter W. Claeys  on 2021-10-08  [id 1824]

(in reply to Report 1 on 2021-08-09)

We would like to thank the Referee for their clear report with a detailed reading of our manuscript and useful comments and pointers to the literature. As mentioned in our resubmission, we have attempted to address all requested changes and hope this makes our paper acceptable for publication in SciPost Physics Core.

For the authors,
Pieter W. Claeys

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