Symmetry-Protected Topological Phases

Abstract

Short-range entangled states can all be connected to each other through local unitary transformations and hence belong to the same phase. However, if certain symmetry is required, they break into different phases. First of all, the symmetry can be spontaneously broken in the ground state leading to symmetry-breaking phases. Even when the ground state remains symmetric, there can be different symmetry-protected topological (SPT) phases, whose nontrivial nature is reflected in their symmetry-protected degenerate or gapless edge states. In this chapter, we discuss these phases in detail. Using the matrix product state formalism, we completely classify SPT phases in 1D boson/spin systems. By mapping 1D fermion systems to spin systems through Jordan–Wigner transformation, we obtain a classification for fermionic SPT phases as well. In 2D, the tensor product representation falls short of firmly establishing a complete classification. But we present an exactly solvable construction of SPT phase with \\(\\mathbb {Z}_2\\) symmetry, which can be generalized to any internal symmetry and in any dimension.