Two soluble models of an antiferromagnetic chain

Elliott Lieb, Theodore Schultz, Daniel Mattis

Research output: Contribution to journalArticlepeer-review

3390 Scopus citations


Two genuinely quantum mechanical models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the elementary excitations and the free energy are found. A general formalism for calculating the instantaneous correlation between any two spins is developed and applied to the investigation of short- and long-range order. Both models show nonvanishing long-range order in the ground state for a range of values of a certain parameter λ which is analogous to an anisotropy parameter in the Heisenberg model. A detailed comparison with the Heisenberg model suggests that the latter has no long-range order in the isotropic case but finite long-range order for any finite amount of anisotropy. The unreliability of variational methods for determining long-range order is emphasized. It is also shown that for spin 1 2 systems having rather general isotropic Heisenberg interactions favoring an antiferromagnetic ordering, the ground state is nondegenerate and there is no energy gap above the ground state in the energy spectrum of the total system.

Original languageEnglish (US)
Pages (from-to)407-466
Number of pages60
JournalAnnals of Physics
Issue number3
StatePublished - Dec 1961

All Science Journal Classification (ASJC) codes

  • General Physics and Astronomy


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