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Fusion approach for quantum integrable system associated with the gl(1|1) Lie superalgebra
by Xiaotian Xu, Wuxiao Wen, Tao Yang, Xin Zhang, Junpeng Cao
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Submission summary
| Authors (as registered SciPost users): | Xiaotian Xu |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202510_00026v1 (pdf) |
| Date submitted: | Oct. 16, 2025, 9:28 a.m. |
| Submitted by: | Xiaotian Xu |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this work we obtain the exact solution of quantum integrable system associated with the Lie superalgebra gl(1|1), both for periodic and for generic open boundary conditions. By means of the fusion technique we derive a closed set of operator identities among the fused transfer matrices. These identities allow us to determine the complete energy spectrum and the corresponding Bethe ansatz equations of the model. Our approach furnishes a systematic framework for studying the spectra of quantum integrable models based on Lie superalgebras, in particular when the $U(1)$ symmetry is broken.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
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Strengths
1- exact solution of the gl(1|1) superspin chain for various boundary conditions based on operator identities derived from the fusion hierarchy
Weaknesses
1- the results are not sufficiently discussed in the context of previous work on this problem 2- the results of the present approach are still limited to eigenvalues of the transfer matrix, the dependence of eigenstates on the off-diagonal elements of the reflection matrices appears to be still out of reach.
Report
For periodic BC the model is a free fermion model and the Bethe equations (44) describe the quantisation of single particle momenta as found by elementary methods.
Generic open boundary conditions allowing for the transformation of bosons into fermions and vice versa are described by reflection matrices with Grassmann valued off-diagonal elements.
The Bethe eqs. (68) derived in the present manuscript coincide with the ones derived in [23] for diagonal and 'quasi-diagonal' BCs (essentially one diagonal and one triangular reflection matrix) using the graded algebraic Bethe ansatz on a reference state constructed from a fermionic coherent state.
For generic off-diagonal BC only the diagonal elements of the reflection matrix enter in the transfer matrix eigenvalues and Bethe eqs. The latter appear to coincide with the ones proposed (and verified for small systems) in Ref. [23] based on a single TQ-relation (i.e. the first of the relations (67)).
In summary, the authors have rederived the Bethe equations describing the spectrum of an integrable superspin chain based on gl(1|1)-symmetric R-matrices. Their analysis is based on operator identities following from the fusion hierarchy of transfer matrices and complements the construction used in Ref. [23] where a single TQ-relation for generic BCs has been 'guessed' from the one for diagonal or quasi-diagonal ones.
Requested changes
1- The authors should add a discussion of their results in the context of those from Ref. [23] 2- If possible they should also extend their remarks at the end of Section 3 concerning the construction of a reference state and/or the application of SoV to construct eigenstates of the model.
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Ask for minor revision
Author: Xiaotian Xu on 2026-01-19 [id 6245]
(in reply to Report 1 on 2025-12-12)Reply to Referee 1
Q1. In summary, the authors have rederived the Bethe equations describing the spectrum of an integrable superspin chain based on gl(1|1)-symmetric R-matrices. Their analysis is based on operator identities following from the fusion hierarchy of transfer matrices and complements the construction used in Ref. [23] where a single TQ-relation for generic BCs has been 'guessed' from the one for diagonal or quasi-diagonal ones.
R1. We agree with the referee's comment. In Ref. [23] (J. Phys. A: Math. Theor. 43, 045207 (2010)), the authors consider the gl(1|1) super spin chains under three types of boundary conditions: diagonal, “quasi-diagonal” and generic off-diagonal. For the first two cases, analytical results are obtained via the algebraic Bethe ansatz (ABA) method. For the model with generic off‑diagonal boundaries, a suitable “reference” state is absent, so the ABA method cannot be employed. Consequently, in Ref. [23], the result corresponding to the third case is a hypothesis supported only by
numerical verification for small system sizes, and its proof was left as an open problem. In Section 3 of our paper, we consider the generic off-diagonal boundary and derive the exact spectrum of the transfer matrix analytically, thereby providing a complete proof of the earlier hypothesis.
The overlap between our manuscript and Ref. [23] (J. Phys. A: Math. Theor. 43, 045207 (2010)) is indeed notable. Although we acknowledge that the model studied (i.e., the gl(1|1) super spin chains) and some of the results in these two papers are consistent, the techniques employed are quite different. In Ref. [23], the standard algebraic Bethe ansatz method is employed to diagonalize the transfer matrix. This approach relies crucially on a suitable “vacuum” state for constructing eigenstates and then getting the spectrum of the transfer matrix (and also the Hamiltonian). In contrast, we employ the fusion technique to obtain the exact eigenvalue spectrum of the transfer matrix. This method depends only on functional relations regarding the transfer matrix and eliminates the need for a “vacuum” state.
In our work, we focus specifically on the quantum integrable system associated with the gl(1|1) Lie superalgebra. However, a key contribution of this paper is thought to be the generalization of the fusion approach from quantum integrable models based on Lie algebras to those based on Lie superalgebras. As mentioned in the conclusion, we have already found this approach to be applicable to other graded systems as well.
Q2. The authors should add a discussion of their results in the context of those from Ref. [23] - If possible they should also extend their remarks at the end of Section 3 concerning the construction of a reference state and/or the application of SoV to construct eigenstates of the model.
R2. In response to the referee's comment, we have added a new section (Section 4) detailing the construction of Bethe states for the gl(1|1) superspin chain with generic open boundaries. We think this addition can address the referee's concern and substantially strengthen our paper.
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gl11-1.19.pdf