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.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.

arXiv:2307.12773Journal of Statistical Mechanics Theory and ExperimentJan 1, 20250 citations

Asymptotic Ferromagnetic Ordering of Energy Levels for the Heisenberg Model on Large Boxes

Bruno Nachtergaele, Spitzer, Wolfgang, Shannon L. Starr

We prove a result for the spin- $1/2$ quantum Heisenberg ferromagnet on $d$ -dimensional boxes $\{1,\dots,L\}^d \subset \mathbb{Z}^d$ . For any $n$ , if $L$ is large enough, the Hamiltonian satisfies: among all vectors whose total spin is at most $(L^d/2)-n$ , the minimum energy is attained by a vector whose total spin is exactly $(L^d/2)-n$ .We prove a result for the spin-$1/2$ quantum Heisenberg ferromagnet on $d$-dimensional boxes $\{1,\dots,L\}^d \subset \mathbb{Z}^d$. For any $n$, if $L$ is large enough, the Hamiltonian satisfies: among all vectors whose total spin is at most $(L^d/2)-n$, the minimum energy is attained by a vector whose total spin is exactly $(L^d/2)-n$.

arXiv:1509.00907Sep 3, 20151 citation

Ground States of the Heisenberg Model

Assa Auerbach

Graduate texts in contemporary physicsJan 1, 19942 citations

Inequalities between ground-state energies of Heisenberg models

Jacek Wojtkiewicz, Rafał Skolasiński

arXiv:1410.5322Physica A Statistical Mechanics and its ApplicationsOct 9, 20140 citations

Density-matrix renormalization group study of the extended Kitaev-Heisenberg model

Kazuya Shinjo, Shigetoshi Sota, Takami Tohyama

We study an extended Kitaev-Heisenberg model including additional anisotropic couplings by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy, entanglement entropy, and spin-spin correlation functions, we make a phase diagram of the extended Kitaev-Heisenberg model around the spin-liquid phase. We find a zigzag antiferromagnetic phase, a ferromagnetic phase, a ${120}^{\ensuremath{\circ}}$ antiferromagnetic phase, and two kinds of incommensurate phases around the Kitaev spin-liquid phase. Furthermore, we study the entanglement spectrum of the model, and we find that entanglement levels in the Kitaev spin-liquid phase are degenerate forming pairs, but those in the magnetically ordered phases are nondegenerate. The Schmidt gap defined as the energy difference between the lowest two levels changes at the phase boundary adjacent to the Kitaev spin-liquid phase. However, we find that phase boundaries between magnetically ordered phases do not necessarily agree with the change of the Schmidt gap.We study an extended Kitaev-Heisenberg model including additional anisotropic couplings by using the two-dimensional density-matrix renormalization group method. Calculating the ground-state energy, entanglement entropy, and spin-spin correlation functions, we make a phase diagram of the extended Kitaev-Heisenberg model around the spin-liquid phase. We find a zigzag antiferromagnetic phase, a ferromagnetic phase, a ${120}^{\ensuremath{\circ}}$ antiferromagnetic phase, and two kinds of incommensurate phases around the Kitaev spin-liquid phase. Furthermore, we study the entanglement spectrum of the model, and we find that entanglement levels in the Kitaev spin-liquid phase are degenerate forming pairs, but those in the magnetically ordered phases are nondegenerate. The Schmidt gap defined as the energy difference between the lowest two levels changes at the phase boundary adjacent to the Kitaev spin-liquid phase. However, we find that phase boundaries between magnetically ordered phases do not necessarily agree with the change of the Schmidt gap.

arXiv:1410.4790Physical Review BFeb 4, 201536 citations

We prove a result for the spin-

1/2

quantum Heisenberg ferromagnet on

d

-dimensional boxes

\{1,\dots,L\}^d \subset \mathbb{Z}^d

. For any

n

, if

L

is large enough, the Hamiltonian satisfies: among all vectors whose total spin is at most

(L^d/2)-n

, the minimum energy is attained by a vector whose total spin is exactly

(L^d/2)-n