Symmetry-Protected Topological Phases
Symmetry-Protected Topological Phases
复制标题
DOI：
10.1007/978-1-4939-9084-9_10
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
B. Zeng;Xie Chen;D. Zhou;X. Wen
中科院分区：
文献类型：
--
作者：
B. Zeng;Xie Chen;D. Zhou;X. Wen
短程纠缠态都可以通过局域酉变换相互连接，因此属于同一相。然而，如果需要一定的对称性，它们就会分成不同的阶段。首先，对称性可以在基态下自发破缺，导致对称性破缺相。即使基态保持对称，也可能存在不同的对称保护拓扑（SPT）相，其非平凡性质反映在其对称保护简并或无间隙边缘态中。在本章中，我们将详细讨论这些阶段。使用矩阵积状态形式，我们对一维玻色子/自旋系统中的 SPT 相进行了完全分类。通过乔丹-维格纳变换将一维费米子系统映射到自旋系统，我们也获得了费米子 SPT 相的分类。在二维中，张量积表示无法牢固地建立完整的分类。但我们提出了一种具有对称性的 SPT 相位的完全可解的结构，它可以推广到任何内部对称性和任何维度。
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 withsymmetry, which can be generalized to any internal symmetry and in any dimension.