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Density matrices in integrable face models

by Holger Frahm, Daniel Westerfeld

Submission summary

As Contributors: Holger Frahm · Daniel Westerfeld
Arxiv Link: (pdf)
Date accepted: 2021-09-02
Date submitted: 2021-07-16 10:14
Submitted by: Frahm, Holger
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian $\bm{su(2)_3}$ or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be 'factorized', i.e.\ expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.

Published as SciPost Phys. 11, 057 (2021)

Author comments upon resubmission

We thank the referees for their positive feedback and the helpful suggestions.

List of changes

- we have rewritten and extended the introduction significantly to put our work into the proper context of existing literature, in particular on correlation functions for SOS models (Referee 1: References to previous works) and the factorization of correlation functions of the XXZ model (Referee 2).

Referee 1:

"Notations and definitions could sometimes be improved."

- We have implemented the suggestions. To discriminate between indices representing states in the auxiliary space and corner values we have changed the notation for the former in (2.7)

- explicit analytical formulae have been added to (2.12)-(2.16) and (4.1)

"About the solution of the inverse problem"

- The solution of the inverse problem for CSOS models in Refs. [25,66] relies on the formulation of these models as a dynamical vertex model. In our proof of Theorem 1 the existence of a dynamical $R$-matrix is not used. The corresponding formulations in the introduction and Section 3 (p.7) are changed accordingly.

- The proof of Theorem 1 for general L>2 is given in Appendix A.

"About the application of the functional equation"

- We expect no problems with the solution of the functional equations for more general models (at least for finite systems). To study the factorization property in these cases, however, we would need a suitable ansatz which for the models considered in our paper is motivated by results from the XXZ model, i.e. (5.32). (no change)

- Our results for the CSOS model are obtained for the reference state. Due to the simple nature of this state the 2-site correlations are given in terms of 1-point functions alone. We have added results on the 3-site density matrix where the same behaviour is observed.

Referee 2:

"Uniqueness of the solution to the discrete functional equations"

- we have, in fact, observed (although this is not proven) the equivalent of the "asymptotic reduction" in certain topological sectors, see Eq. (5.30). As stated below Eqs (5.29) and (5.47), however, we find that it is sufficient to use the (weaker) relation between $D_N$ and $D_{N-1}$ obtained by taking partial traces for the recursive calculation of the density matrices in a given eigenstate of the transfer matrix (at least for the $r=4$ and $5$ RSOS models considered in the paper). We have clarified the corresponding statement in the conclusion.

- Number of unknown functions in the 'physical part' of the correlation functions:
For the $r=4$ and $r=5$ RSOS we find that two functions, in the topological sectors with quantum dimension a single function is sufficient. Preliminary results for models with $r>5$ indicate that the latter holds there, too. This is formulated as a conjecture in the conclusion.

- We have added a comment on the characterization and calculation of the two-point function $f(\lambda,\mu)$ in the conclusion

- Details on the algorithm for the computation of the structure coefficients have been added on p.18/19.

- A Reference to [R6] has been added in the conclusion.

Referee 3:

- On the terminology "generalized transfer matrices":
We introduce this notation in the context of generic (not necessarily integrable) face models on p. 5,6. Here the concepts of a transfer matrix generating a family of commuting operators or of generators of the dynamical algebra do not exist. The difference between (2.8), (2,9) and (2.10) is only in the boundary conditions in the horizontal direction. That's why we have chosen this name (or "transfer matrices with generalized boundary conditions" in the introduction). (no change)

Reports on this Submission

Anonymous Report 3 on 2021-8-15 (Invited Report)


The main changes requested by the referees have been implemented, and the quality of the paper is now much better, both concerning references to the existing literature and clarity of the presentation and notations. I now recommend the publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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


The authors have implemented the main requirements and further improved the presentation of their results. The manuscript seems now suitable for publication on scipost.

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

Anonymous Report 1 on 2021-7-27 (Invited Report)


The authors have responded very positively to all three referee reports. In my understanding they have managed to considerably improve the quality of the presentation of their important results. The embedding of their achievements into the context of the existing literature is also much better now. I recommend the publication of the manuscript in its present form.

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

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