Galois Solvability of Finite-Size Bethe Solutions in the Heisenberg Chain
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909020296577024 |
|---|---|
| author | Bellwood, Oliver R. Munro, William J. |
| author_facet | Bellwood, Oliver R. Munro, William J. |
| contents | The spin-1/2 Heisenberg antiferromagnetic chain is the canonical example of an integrable quantum many-body model. Despite its exact solvability, explicit finite-size solutions are typically only accessible via numerical evaluation of the Bethe ansatz equations. Here, we analyse the algebraic structure of the exact, symbolic ground states for chains up to ten sites using the coordinate Bethe ansatz. We show that both the ground state wavefunction and the Bethe-roots rapidly develop algebraic complexity with respect to system size, but at different rates. The Bethe-roots appear to become Galois unsolvable for chains of eight or more sites, whereas the ground state wavefunction coefficients and energy appear to become unsolvable for ten or more sites. This demonstrates a lack of explicit analytic tractability in a quantum integrable model due to algebraic complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05589 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Galois Solvability of Finite-Size Bethe Solutions in the Heisenberg Chain Bellwood, Oliver R. Munro, William J. Strongly Correlated Electrons Mathematical Physics The spin-1/2 Heisenberg antiferromagnetic chain is the canonical example of an integrable quantum many-body model. Despite its exact solvability, explicit finite-size solutions are typically only accessible via numerical evaluation of the Bethe ansatz equations. Here, we analyse the algebraic structure of the exact, symbolic ground states for chains up to ten sites using the coordinate Bethe ansatz. We show that both the ground state wavefunction and the Bethe-roots rapidly develop algebraic complexity with respect to system size, but at different rates. The Bethe-roots appear to become Galois unsolvable for chains of eight or more sites, whereas the ground state wavefunction coefficients and energy appear to become unsolvable for ten or more sites. This demonstrates a lack of explicit analytic tractability in a quantum integrable model due to algebraic complexity. |
| title | Galois Solvability of Finite-Size Bethe Solutions in the Heisenberg Chain |
| topic | Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2605.05589 |