Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings for the Singlet
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| Format: | Preprint |
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2023
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| _version_ | 1866909450113122304 |
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| author | Heson, David Starr, Shannon Thornton, Jacob |
| author_facet | Heson, David Starr, Shannon Thornton, Jacob |
| contents | We demonstrate a violation of the ``ferromagnetic ordering of energy levels'' conjecture (FOEL) for even length spin rings. The FOEL conjecture was a guess made by Nachtergaele, Spitzer and an author for the Heisenberg model on certain graphs: a family of inequalities, the first of which is the statement that the spectral gap of the Heisenberg model equals the gap of the random walk. That first guess was originally a conjecture of Aldous which was later proved by Caputo, Liggett and Richthammer. We claim that for spin rings of even length $L>4$, the lowest spin $S=0$ energy is lower than the lowest spin $S=1$ energy. This violates the $(L/2)$-th inequality in the FOEL conjecture.
Our methodology is largely numerical: we have applied exact diagonalization up to $L=20$. We also rigorously consider the Hamiltonian of the Heisenberg spin ring for even length $L$ projected to the spin $S=0$ sector. We prove that it has a unique ground state. Then, using the single mode approximation the uniqueness explains the energy turn-around. Important insight comes from reconsideration of previous work by Sutherland, using the Bethe ansatz. Especially important is a work of Dhar and Shastry that goes beyond the Bethe ansatz. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_12773 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings for the Singlet Heson, David Starr, Shannon Thornton, Jacob Mathematical Physics Statistical Mechanics 82B10, 81R05, 81R50 We demonstrate a violation of the ``ferromagnetic ordering of energy levels'' conjecture (FOEL) for even length spin rings. The FOEL conjecture was a guess made by Nachtergaele, Spitzer and an author for the Heisenberg model on certain graphs: a family of inequalities, the first of which is the statement that the spectral gap of the Heisenberg model equals the gap of the random walk. That first guess was originally a conjecture of Aldous which was later proved by Caputo, Liggett and Richthammer. We claim that for spin rings of even length $L>4$, the lowest spin $S=0$ energy is lower than the lowest spin $S=1$ energy. This violates the $(L/2)$-th inequality in the FOEL conjecture. Our methodology is largely numerical: we have applied exact diagonalization up to $L=20$. We also rigorously consider the Hamiltonian of the Heisenberg spin ring for even length $L$ projected to the spin $S=0$ sector. We prove that it has a unique ground state. Then, using the single mode approximation the uniqueness explains the energy turn-around. Important insight comes from reconsideration of previous work by Sutherland, using the Bethe ansatz. Especially important is a work of Dhar and Shastry that goes beyond the Bethe ansatz. |
| title | Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings for the Singlet |
| topic | Mathematical Physics Statistical Mechanics 82B10, 81R05, 81R50 |
| url | https://arxiv.org/abs/2307.12773 |