Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings for the Singlet

TL;DR

The paper presents a counterexample to the ferromagnetic ordering of energy levels (FOEL) on even-length spin rings, showing $E^{\mathrm{FM}}_{\min}(0;L) < E^{\mathrm{FM}}_{\min}(1;L)$ for $L\ge 6$ via exact diagonalization up to $L=20$. It provides a rigorous spin-singlet ground-state result in the Hulthén bracket basis, uses Perron–Frobenius theory to justify ground-state properties, and employs the single mode approximation to explain the observed energy turn-around, indicating a positive energy cost for SMA excitations. The work connects to Dhar–Shastry and Sutherland's Bethe-ansatz analyses, showing that Bloch-wall pictures are not sufficient and that a multi-vector perturbation yields the lowering of the $S=0$ ground-state energy. The findings motivate extensions to higher spins and dimensions and suggest computational and analytical avenues (e.g., DMRG, parallelization) to map FOEL violations beyond the spin-1/2 ring and to probe the presence of hidden symmetries in integrable Heisenberg models.

Abstract

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.

Figures (5)

• Figure 1: We have printed the values of the lowest energy of $H^{\mathrm{FM}}(T_L)$ for even values of $L$.
• Figure 2: Numerical calculation of $Z_L(\ell) = \|\widehat{S}^+_{\ell} \Theta_L\|^2$ plotted against $\ell$ for $L=20$. The range for $\ell$ is $-9$ to $10$, as plotted.
• Figure 3: The difference $E^{\mathrm{FM}}_{\min}(1;L) - E^{\mathrm{FM}}_{\min}(0;L)$, calculated numerically. The quantity $\varepsilon_{\mathrm{SMA}}(1/2,L)$ is equal to the SMA: $(8/3) \sin^2(\pi/L) (\frac{1}{4}L^2 - E^{\mathrm{FM}}_{\min}(0;L))/\|\widehat{S}^+_{1}\Psi^{\mathrm{FM}}_{\min}(0;L)\|^2$. It is approximately $4$ times too large.
• Figure 4: The black dots are the actual energies $E^{\mathrm{FM}}_{\min}(S;L)$ for $L=20$. The circles (connected by grey lines) are the Bloch wall approximations calculated by Shastry and Dhar to be $\widetilde{E}^{\mathrm{Bloch}}_{20}(S)$. The approximation is fairly close, with deviations primarily near $S=0$.
• Figure 5: Two pictures of the full spectrum of the spin-$1/2$ spin ring for $L=18$ up to a energy cut-offs: $E=2$ for the left plot and $E=4$ for the right plot. In the left plot, the cases of 2-fold degeneracy are indicated with gray line segments with labels $2$. The cases of 4-fold are indicated with longer line segments with labels 4. The non-degenerate energy levels are indicated with black bullet points. The curve is Sutherland's formula.

Theorems & Definitions (6)

• Conjecture 2.1
• Proposition 2.2: Nachtergaele, Spitzer, S NSSfoel
• Lemma 2.3: Perron-Frobenius result
• Corollary 2.4
• Corollary 2.5
• Remark 3.1