Wiki. “对称保护拓扑物态” [对称保护拓扑物态]

#数学物理

Topology has established itself as a fundamental principle in condensed matter physics. For gapped ground states of local Hamiltonians, topological invariants can label distinct phases of matter, and these non-local order parameters can be associated with exotic features such as protected edge states or anyonic excitations. While the classification of topological phases has been achieved for non-interacting fermions in arbitrary dimensions, the extension to systems of interacting particles is a matter of ongoing work. For gapped systems in one spatial dimension, however, the general principles have been elucidated. In particular it is known that topological invariants require the presence of an unbroken symmetry in order to be well-defined. These label so-called symmetry protected topological (SPT) phases.

一个有能隙的系统 (gapped system) 在某个对称性下不能连续而无相变地变化为平凡 (乘积) 态, 则称该系统具有对称保护拓扑物态.

性质

没有任意子激发 (anyon excitation).

(对于闭的空间流形) 基态简并度 (GSD) 等于 $1$, “不会有激发态能量越来越低直到和基态一样”.

存在边界态.

例

$1+1$ 维对称保护拓扑物态由群上同调 $H^2(G,U(1))\simeq H^2(BG,U(1)) \simeq H^3(BG,\mathbb{Z})$ 分类. 其中 $H^2(BG,U(1))$ 是连续上同调.