Algebraic Structure of Finite-Size Bethe Solutions in the Heisenberg Chain
Abstract
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 twelve sites using the Bethe ansatz. We find that the algebraic degree of the ground state magnon phases and Bethe roots scale with respect to system size at a different rate to the ground-state energy and wavefunction coefficients. The ground-state Bethe roots and energy are found to become Galois unsolvable for ten or more sites, thus demonstrating the lack of a symbolic analytical solution in a quantum integrable model due to algebraic complexity.
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BibTeX
@misc{bellwood2026algebraic,
title = {{Algebraic Structure of Finite-Size Bethe Solutions in the Heisenberg Chain}},
author = {Oliver R. Bellwood and William J. Munro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.05589},
eprint = {2605.05589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.05589},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraic Structure of Finite-Size Bethe Solutions in the Heisenberg Chain
AU - Oliver R. Bellwood
AU - William J. Munro
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2605.05589
UR - https://arxiv.org/abs/2605.05589
AB - 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 twelve sites using the Bethe ansatz. We find that the algebraic degree of the ground state magnon phases and Bethe roots scale with respect to system size at a different rate to the ground-state energy and wavefunction coefficients. The ground-state Bethe roots and energy are found to become Galois unsolvable for ten or more sites, thus demonstrating the lack of a symbolic analytical solution in a quantum integrable model due to algebraic complexity.
ER -