物理学报2017，Vol.66Issue(22)：1-10,10.DOI:10.7498/aps.66.220201
时间反演对称性破缺系统中的拓扑零能模
Topological zero-energy modes in time-reversal-symmetry-broken systems
摘要
Abstract
The well-known Su-Schrieffer-Heeger (SSH) model predicts that a chain of sites with alternating coupling constant exhibits two topological distinct phases,and at the truncated edge of the topological nontrivial phase there exists topologically protected edge modes.Such modes are named zero-energy modes as their eigenvalues are located exactly at the midgaps of the corresponding bandstructures.The previous publications have reported a variety of photonic realizations of the SSH model,however,all of these studies have been restricted in the systems of time-reversal-symmetry (TRS),and thus the important question how the breaking of TRS affects the topological edge modes has not been explored.In this work,to the best of our knowledge,we study for the first time the topological zero-energy modes in the systems where the TRS is broken.The system used here is semiconductor microcavities supporting excitonpolariton quasi-particle,in which the interplay between the spin-orbit coupling stemming from the TE-TM energy splitting and the Zeeman effect causes the TRS to break.We first study the topological edge modes occurring at the edge of one-dimensional microcavity array that has alternative coupling strengths between adjacent microcavity,and,by rigorously solving the Schr(o)dinger-like equations (see Eq.(1) or Eq.(2) in the main text),we find that the eigen-energies of topological zero-energy modes are no longer pinned at the midgap position:rather,with the increasing of the spinorbit coupling,they gradually shift from the original midgap position,with the spin-down edge modes moving toward the lower band while the spin-up edge modes moving towards the upper band.Interestingly enough,the mode profiles of these edge modes remain almost unchanged even they are approaching the bulk transmission bands,which is in sharp contrast to the conventional defect modes that have an origin of bifurcation from the Bloch mode of the upper or lower bands.We also study the edge modes in the two-dimensional microcavity square array,and find that the topological zero modes acquire mobility along the truncated edge due to the coupling from the adjacent arrays.Importantly,owing to the breaking of the TRS,a pair of counterpropagating edge modes,of which one has a momentum k and the other has-k,is no longer of energy degeneracy;as a result the scattering between the forward-and backward-propagating modes is greatly suppressed.Thus,we propose the concept of the one-dimensional topological zero-energy modes that are propagating along the two-dimensional lattice edge,with extremely weak backscattering even on the collisions of the topological zero-energy modes with structural defects or disorder.关键词
拓扑绝缘体/拓扑零能模/边缘态/自旋轨道耦合Key words
topological insulators/topological zero-energy mode/edge mode/spin-orbit coupling引用本文复制引用
张卫锋,李春艳,陈险峰,黄长明,叶芳伟..时间反演对称性破缺系统中的拓扑零能模[J].物理学报,2017,66(22):1-10,10.基金项目
Project supported by the National Natural Science Foundation of China (Grant Nos.11104181,61475101) and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20110073120074).国家自然科学基金(批准号:11104181,61475101)和高等学校博士学科点专项科研基金(批准号:20110073120074)资助的课题. （Grant Nos.11104181,61475101）
