Haldane Phase

Abstract

Theoretical studies of the \(S=1\) antiferromagnetic spin chains exhibiting the Haldane gap have led to an understanding that the corresponding ground states are not simply disordered but exhibit rich phenomena, which should be called the Haldane phenomena (Sect. 8.1). It was then pointed out that all of the Haldane phenomena can be naturally understood as consequences of spontaneous breakdown of hidden \({\mathbb {Z}_{2}}\times {\mathbb {Z}_{2}}\) symmetry (Sect. 8.2). But it was soon realized that the picture of hidden \({\mathbb {Z}_{2}}\times {\mathbb {Z}_{2}}\) symmetry breaking is far from enough to fully characterize antiferromagnetic spin chains with the Haldane gap. The notion of symmetry protected topological (SPT) phase (Sect. 8.3), which was proposed by Gu and Wen in 2009 and refined by Pollmann, Turner, Berg, and Oshikawa in 2010, finally gave satisfactory understanding of the phenomena. Very recently, in 2018 and 2019, fully rigorous index theorems based on operator algebraic techniques that characterize SPT phases were developed by Ogata. The rigorous index theorems essentially complete the study of the Haldane phase. In this chapter we also discuss closely related idea of topological order, and give an introductory review of Kitaev’s toric code model (Sect. 8.4).