Symmetry-Protected Topological phases and Duality

Symmetry-Protected Topological (SPT) phases are, as the name suggests, topological phases without any conventional local order parameter, but distinct from the trivial phase only in the presence of a certain symmetry. The concept was first proposed by Gu and Wen in 2009 as a generalization of topological insulators discovered earlier. However, the prototypical example of the SPT phases, the Haldane gap phase in odd-integer spin chains, was discovered much earlier in the 1980s. Thanks to the pivotal construction of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state, numerous "topological" properties of the Haldane gap phase were identified by the 1990s. In particular, a non-local transformation introduced by Kennedy and Tasaki maps the Haldane gap phase and a conventional Spontaneous Symmetry Breaking (SSB) phase. The duality picture could have naturally led to the concept of the SPT phases. In this talk, I will review the concept of SPT phases from the duality point of view, and its historical developments. I will also discuss the recent resurgence of the duality approach, with applications including a systematic construction of SPT phases including a novel variety of “gapless SPT phases”.