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Enriched classification of parafermionic gapped phases with time-reversal symmetry
Physical Review B ( IF 3.7 ) Pub Date : 2018-03-21 00:00:00 , DOI: 10.1103/physrevb.97.125135
Wen-Tao Xu , Guang-Ming Zhang

Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with time-reversal (TR) symmetry satisfying T2=1. Without extra symmetry, it has been found that Zp parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labeled by the divisors n of p. In the presence of TR symmetry, however, the enriched classification is characterized by three indices n, κ, and μ, where κZ2 denotes the linear or projective TR actions on the edges, and μZ2 indicates the commutation relations between the TR and (fractionalized) charge operator. For the Zr-symmetric parafermionic ground states, where r=p for trivial or topological phases, and r=p/n for SSB phases, each original gapped phase with odd r is divided into two phases, while each phase with even r is further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of these resulting topological phases, we can obtain the topologically protected degeneracies of their entanglement spectra.

中文翻译:

具有时间逆向对称性的铁粉下间隙相的富集分类

基于最近建立的相差铁离子矩阵乘积状态,我们研究了满足时间逆向(TR)对称性的相差铁离子的一维带隙相的分类 Ť2个=1个。没有额外的对称性,已经发现žp 伪铁离子间隙相可分为拓扑相,自发对称断裂(SSB)相和琐碎相,它们由除数唯一地标记 ñp。但是,在存在TR对称性的情况下,富集分类的特征在于三个指标ñκ, 和 μ, 在哪里 κž2个 表示边缘上的线性或投影TR作用,并且 μž2个表示TR和(分式)电荷算子之间的换向关系。为了ž[R不对称超铁离子基态,其中 [R=p 对于琐碎或拓扑阶段,以及 [R=p/ñ 对于SSB阶段,每个原始间隙阶段均具有奇数 [R 分为两个阶段,而每个阶段都具有偶数 [R进一步分为四个阶段。具有TR对称性的空位二聚铁离子相包括对称保护的拓扑相,对称富集的拓扑相和SSB并存的对称保护的拓扑相。通过分析这些结果拓扑相的密度降低矩阵的结构和对称性,我们可以获得它们的纠缠谱的拓扑保护简并性。
更新日期:2018-03-21
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