SciPost Submission Page
Tensor-Network Analysis of Root Patterns in the XXX Model with Open Boundaries
by Zhouzheng Ji, Pei Sun, Xiaotian Xu, Yi Qiao, Junpeng Cao and Wen-Li Yang
Submission summary
| Authors (as registered SciPost users): | Yi Qiao |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202512_00064v1 (pdf) |
| Date submitted: | Dec. 31, 2025, 8:51 a.m. |
| Submitted by: | Yi Qiao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
I used Kimi-K2 and ChatGPT-5 in December 2025 solely for language polishing. No scientific content was generated or altered.
Abstract
The string hypothesis for Bethe roots represents a cornerstone in the study of quantum integrable systems, providing access to physical quantities such as the ground-state energy and the finite-temperature free energy. While the $t-W$ scheme and the inhomogeneous $T-Q$ relation have enabled significant methodological advances for systems with broken $U(1)$ symmetry, the underlying physics induced by symmetry breaking remains largely unexplored, due to the previously unknown distributions of the transfer-matrix roots. In this paper, we propose a new approach to determining the patterns of zero roots and Bethe roots for the $\Lambda-\theta$ and inhomogeneous Bethe ansatz equations using tensor-network algorithms. As an explicit example, we consider the isotropic Heisenberg spin chain with non-diagonal boundary conditions. The exact structures of both zero roots and Bethe roots are obtained in the ground state for large system sizes, up to ($N\simeq 60$ and $100$). We find that even in the absence of $U(1)$ symmetry, the Bethe and zero roots still exhibit a highly structured pattern. The zero roots organize into bulk strings, boundary strings, and additional roots, forming two dominant lines with boundary-string attachments. Correspondingly, the Bethe roots can be classified into four distinct types: regular roots, line roots, arc roots, and paired-line roots. These structures are associated with a real-axis line, a vertical line, characteristic arcs in the complex plane, and boundary-induced conjugate pairs. Comparative analysis reveals that the $t-W$ scheme generates significantly simpler root topologies than those obtained via off-diagonal Bethe Ansatz. The developed framework not only resolves the root configuration problem in \(U(1)\) symmetry-broken systems, but also provides a transferable approach for studying ground states, excitations, and finite-temperature properties in quantum integrable models.
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- 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