Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings by Weak Paramagnetism of the Singlet

David Heson, Shannon Starr, Jacob Thornton

Published 2023-07-24Version 1

For the quantum Heisenberg antiferromagnet with spin-$j$ on a bipartite, balanced graph, the Lieb-Mattis theorem, ``Ordering of energy levels,'' guarantees that the ground state is a spin singlet, and moreover, defining $E^{\textrm{AF}}_{\min}(S)$ to be the minimum eigenvalue of the Hamiltonian in the invariant subspace consisting of all spin $S$ vectors, $\boldsymbol{S}_{\mathrm{tot}}^2 \psi = S(S+1)\psi$, the function $E^{\textrm{AF}}_{\min}(S)$ is monotonically increasing for $0\leq S\leq j|\mathcal{V}|$. For the ferromagnet, the absolute ground state is $E_{\min}^{\textrm{FM}}(j|\mathcal{V}|)$. We say that the graph satisfies ``ferromagnetic ordering of energy levels'' at order $n$, or FOEL-$n$, if two properties hold: (1) $E_{\min}^{\textrm{FM}}(j|\mathcal{V}|)\leq \dots \leq E_{\min}^{\mathrm{FM}}(j|\mathcal{V}|-n)$, and (2) $E_{\min}^{\mathrm{FM}}(j|\mathcal{V}|-n)\leq E_{\min}^{\mathrm{FM}}(j|\mathcal{V}|-m)$ for all $m\geq n$. Caputo, Liggett and Richthammer proved a theorem which generally implies FOEL-$1$ is true. Apparently $E_0^{\mathrm{FM}}(0) <E_0^{\mathrm{FM}}(1)$ for sufficiently long spin rings, $\mathbb{Z}/L\mathbb{Z}$ with even length $L$. So FOEL-$n$ does not hold for $n=jL-1$. We consider $E_0^{\mathrm{FM}}(1)-E_0^{\mathrm{FM}}(0)$ using linear spin-wave analysis and numerical computation. Using the Bethe ansatz, Sutherland already considered the spin ring with $j=1/2$ and notably proved weak paramagnetism. But we also present evidence for $j>1/2$.