Ferromagnetic Ordering of Energy Levels

Abstract

We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the Lieb-Mattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin-1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the Temperley-Lieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.

22 pages, 3 figures, corrected version to appear in J. Stat. Phys

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• Ground state properties of a fully frustrated quantum spin system
• On the flux phase conjecture at half-filling: an improved proof
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• Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite, and related graphs
• Ordering of Energy Levels in Heisenberg Models and Applications
• Ordering of Energy Levels for Extended SU(N) Hubbard Chain
• Ferromagnetic Ordering of Energy Levels for $U_q({\mathfrak{sl}}_2)$ Symmetric Spin Chains
• Lowest-energy states in parity-transformation eigenspaces of SO(N) spin chain
• Counterexamples to Ferromagnetic Ordering of Energy Levels
• Transport of interface states in the Heisenberg chain