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On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint
by G. Niccoli, V. Terras
|Authors (as registered SciPost users):||Giuliano Niccoli · Véronique Terras|
|Preprint Link:||https://arxiv.org/abs/2208.10097v1 (pdf)|
|Date submitted:||2023-01-19 18:05|
|Submitted by:||Terras, Véronique|
|Submitted to:||SciPost Physics|
This paper is a continuation of , in which a set of matrix elements of local operators was computed for the XXZ spin-1/2 open chain with a particular case of unparallel boundary fields. Here, we extend these results to the more general case in which both fields are non-longitudinal and related by one constraint, allowing for a partial description of the spectrum by usual Bethe equations. More precisely, the complete spectrum and eigenstates can be characterized within the Separation of Variables (SoV) framework. One uses here the fact that, under the constraint, a part of this SoV spectrum can be described via solutions of a usual, homogeneous, TQ-equation, with corresponding transfer matrix eigenstates coinciding with generalized Bethe states. We explain how to generically compute the action of a basis of local operators on such kind of states, and this under the most general boundary condition on the last site of the chain. As a result, we can compute the matrix elements of some of these basis elements in any eigenstate described by the homogenous TQ-equation. Assuming, following a conjecture of Nepomechie and Ravanini, that the ground state itself can be described in this framework, we obtain multiple integral representations for these matrix elements in the half-infinite chain limit, generalizing those previously obtained in the case of longitudinal boundary fields and in the case of the special boundary conditions considered in .
Submission & Refereeing History
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Reports on this Submission
- New result about correlation functions for the open XXZ chain (with constraint)
- Explicit boundary bulk decomposition formula for the gauged boundary Bethe state.
- involve only the usual TQ equation without the additional inhomogeneous term.
The authors produce a new step torward the general integrable boundary correlation function. To overcome the lack of technique to study thermodynamical limit of the TQ equation with an additional inhomogeneous term, they impose one constraint. They perform the calculation of correlation functions for specific "local" operators and study the thermodynamic limit. The paper is well writed and deserve to be publish.
1. Relevance - deals with a problem and set-up which has attracted attention in the literature.
2. The main result in principle has numerous applications.
3. Extensive literature review.
4. Careful and precise definitions of all quantities used.
1. At times the paper is excessively technical.
2. It is not clear how feasible it is to implement the main result in practice. For example for spin chains of some large but finite size, in a numerical set-up.
The authors consider the problem of computing correlation functions in integrable spin chains with open boundary conditions in the separation of variables (SoV) approach.
The paper is another in a series by the authors. In previous works the authors computed correlation functions of local operators in the set-up where one of the boundary fields was longitudinal and the other was non-longitudinal. In the current article the authors consider a more general set-up where both boundaries are non-longitudinal but, rather than being completely generic, are related by a certain constraint. This constraint allows part of the spectrum of the model to be described by a homogeneous Baxter TQ equation and Bethe Ansatz equations. The authors refer to a conjecture in the literature that these equations also apply to the ground state at half filling. They then develop tools for computing correlation functions of bulk local operators which, under the assumption of the aforementioned conjecture, allows them to compute these correlation functions in the thermodynamic limit at zero temperature, which constitutes the main result of the article.
The main technical advancement is the “bulk-boundary decomposition” in the present set-up, which was initially developed by the authors and their collaborators in earlier work. Aided by this decomposition, bulk local operators are embedded into the SoV framework, and their finite-volume matrix elements are computed. Then, the thermodynamic limit is taken.
The result constitutes a major advancements in the computation of correlation functions in integrable systems with open boundary conditions. Owing to the relevance of such models in physical applications, the result is of particular importance. I recommend the article for publication.
I should however follow up on the point I listed in the "Weakness" category above. While I understand the authors desire for generality and precision, at times this is to the detriment of the reader. Indeed, propositions are proved in full generality in one go, while it might be more advantageous for the reader to see some simple set-ups considered first in order to get a feel for the results. This is also a common feature in other works of the authors. For example, Theorem 4.1 is an important result, but practically impossible to get a feel for what it is really saying in its current form, without spending time to work out all definitions and notations used, of which there are many. For the average reader who is not already an expert in the exact methods and notations and simply wants to have an overview of the results, it is unreasonable to expect them to do so. For example, presenting the results of Theorem 4.1 in the case m=1, M=2 before presenting the most general case would already help quite a bit.
At this point I do not think it feasible to implement such changes, as doing so would likely require a substantial amount of rewriting. However, I would ask that the authors consider such an approach in their future publications.
1. The paper would benefit from a careful proof reading, while paying attention to English sentence structure, as sometimes it reads in an unnatural way to the point of somewhat obscuring the scientific content, which may in particular cause trouble for non-native speakers of English. For example "In  was also introduced for the study of these open spin chains a full algebraic formalism" -> "In  a complete algebraic formalism for the study of these open spin chains was introduced". "Reflexion" -> Reflection".
1. New results for the correlation functions of open spin chains
2. Use of boundary-bulk decomposition for the gauged double-row monodromy matrix
3. Multiple integral representations in the thermodynamic limit
1. Complicated form for the local operators due to the gauge transformation
2. Relation between the constraint and selection of the ground state in the thermodynamic limit should be clarified
The paper ``On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint'' is a logical continuation of the work started by the same authors in their previous article. They study an extremely important case of the open spin chains with non-parallel boundary fields: the case with a boundary constraint relating the parameters on the sides of the lattice (compatible with usual homogeneous Baxter equation). The authors remind the construction of the eigenstates using the SOV approach to obtain finally the eigenstates in a form similar to the usual algebraic Bethe Ansatz. This representation in turn permits them to apply the boundary-bulk decomposition for the gauged operators and compute elements of the reduced density matrix as multiple sums and finally as multiple integrals in the thermodynamic limit. This result is new, important and extremely interesting for the study of open spin chains, for these reasons I recommend this paper for publication in SciPost Physics.
1. The quantum inverse problem is solved in terms of gauge transformed local operators. The authors prove that these operators form a basis in the corresponding matrix space and thus give all the elements of the reduced density matrix. It would be very helpful to have some illustrations (for one or maybe two sites) of the local operators obtained by this procedure.
2. The authors claim that the expression in thermodynamic limit works when the constraint fixes the number of Bethe roots at $N/2-k$ with $k$ remaining finite in the thermodynamic limit.The ground state density computed from the integral equation fixes the number of roots at $N/2$ and boundary complex roots can change it by one. Is there a description for a ground state with $N/2-k$ roots with finite $k$? Should it include some holes? Will it influence the corresponding correlation functions or it leads only to finite size corrections? I think that authors should add some comments to clarify these questions.
3. There are some typos in the text for example on page 8 ``up an overall factor'' should be replaced by ``up to an overall factor''; in the first line of the Conclusion ``which non-longitudinal'' should be replaced by ``with non-longitudinal'' etc., an additional proofreading can be useful.