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Density matrices in integrable face models
by Holger Frahm, Daniel Westerfeld
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Submission summary
Authors (as Contributors):  Holger Frahm · Daniel Westerfeld 
Submission information  

Arxiv Link:  https://arxiv.org/abs/2104.06269v2 (pdf) 
Date accepted:  20210902 
Date submitted:  20210716 10:14 
Submitted by:  Frahm, Holger 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Using the properties of the local Boltzmann weights of integrable interactionroundaface (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 solidonsolid type and quantum chains of nonAbelian $\bm{su(2)_3}$ 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 nearestneighbour 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
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 2site correlations are given in terms of 1point functions alone. We have added results on the 3site 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 $D_{N1}$ 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 twopoint 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)
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 3 on 2021815 (Invited Report)
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.
Anonymous Report 2 on 202182 (Invited Report)
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.
Anonymous Report 1 on 2021727 (Invited Report)
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.