The Affleck-Kennedy-Lieb-Tasaki (AKLT) model (Q2164172)

According to Haldane's conjecture, there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. In the paper the author reviews the Affleck-Lieb-Schultz-Mattis (ALSM) theorem about gapless behavior for half-integer spin [\textit{E. H. Lieb} et al., Ann. Phys. 16, 407--466 (1961; Zbl 0129.46401); \textit{B. Affleck} and \textit{E. H. Lieb}, ``A proof of part of Haldane's conjecture on spin chains'', Lett. Math. Phys. 12, 57--69 (1986; \url{doi:10.1007/BF00400304})] and present results on the Affleck-Kennedy-Lieb Tasaki (AKLT) model [\textit{I. Affleck} et al., ``Rigorous results on valence-bond ground states in antiferromagnets'', Phys. Rev. Lett. 59, No. 7, 799--802 (1987; \url{doi:10.1103/PhysRevLett.59.799}); ``Valence bond ground states in isotropic quantum antiferromagnets'', Comm. Math. Phys. 115, No. 3, 477--528 (1988; \url{doi:10.1007/BF01218021})] where gapped behavior for integer spin was demonstrated. The combination of the ALSM theorem and the AKLT model provided powerful evidence that the Haldan conjecture was correct. For the entire collection see [Zbl 1491.46002].