[论文解读] Topological phases of time-reversal invariant superconductors and electronic systems with sublattice (chiral) symmetry, in three spatial dimensions

Andreas P. Schnyder, Shinsei Ryu+2 more|arXiv (Cornell University)|Mar 19, 2008

Topological Materials and Phenomena被引用 1

一句话总结

该论文在三维时间反演不变绝缘体和超导体的10个对称性类中识别出五个拓扑非平庸相，其中包含四个超越已知Z₂绝缘体的新拓扑绝缘体。这些相在动量空间中由整数绕数表征，具有对安德森局域化免疫的鲁棒、无能隙的狄拉克或马约拉纳表面态，并且在考虑规范场和能隙涨落时表现出非平凡的拓扑简并基态。

ABSTRACT

We systematically study topological phases of insulators and superconductors (SCs) in 3D. We find that there exist 3D topologically non-trivial insulators or SCs in 5 out of 10 symmetry classes introduced by Altland and Zirnbauer within the context of random matrix theory. One of these is the recently introduced Z_2 topological insulator in the symplectic symmetry class. We show there exist precisely 4 more topological insulators. For these systems, all of which are time-reversal (TR) invariant in 3D, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. 3 of the above 5 topologically non-trivial phases can be realized as TR invariant SCs, and in these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a 2D surface, they support a number (which may be an arbitrary non-vanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations that preserve the characteristic discrete symmetries. In particular, these surface modes completely evade Anderson localization. These topological phases can be thought of as 3D analogues of well known paired topological phases in 2D such as the chiral p-wave SC. In the corresponding topologically non-trivial and topologically trivial 3D phases, the wavefunctions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the SC phases with non-vanishing winding number possess non-trivial topological ground state degeneracies.

研究动机与目标

• 系统分类三维时间反演不变系统在全部10个Altland-Zirnbauer对称性类中的拓扑相。
• 在典型类中识别出超越已知Z₂绝缘体的新拓扑绝缘体。
• 通过动量空间中的整数绕数表征超导相的拓扑子空间。
• 研究表面态在保持对称性的微扰下的鲁棒性。
• 研究电磁U(1)规范场和能隙涨落在拓扑超导相中对拓扑基态简并的影响。

提出的方法

• 基于时间反演和子晶格（轴对称）对称性的离散对称性，对绝缘相和超导相的基态进行分类。
• 使用动量空间绕数作为拓扑不变量，以区分时间反演不变超导体中的相。
• 分析三维拓扑绝缘体中的表面态，表明其对无序和保持对称性的微扰具有鲁棒性。
• 应用随机矩阵理论以识别10个对称性类并确定拓扑不变量。
• 引入U(1)规范场和能隙函数涨落，以研究拓扑基态简并。
• 对希尔伯特空间进行拓扑子空间划分，不同拓扑子空间由量子相变分隔。

实验结果

研究问题

• RQ1 在三维系统中，10个Altland-Zirnbauer对称性类中哪些在时间反演不变条件下具有拓扑非平庸的绝缘体或超导体？
• RQ2 这些系统中的拓扑子空间如何表征，何种拓扑不变量可将其区分？
• RQ3 三维时间反演不变拓扑绝缘体和超导体中的表面态具有何种性质，其鲁棒性如何？
• RQ4 电磁U(1)规范场和能隙涨落在拓扑超导相中如何影响基态简并？
• RQ5 子晶格（轴对称）对称性在稳定这些拓扑相中起何种作用？

主要发现

• 在三维时间反演不变系统中存在五个拓扑相，包括四个超越已知Z₂绝缘体的新拓扑绝缘体。
• 拓扑子空间由量子相变分隔，绝缘基态的空间据此被相应划分。
• 五个非平庸相中的三个可实现为时间反演不变超导体，其特征为动量空间中的整数绕数。
• 这些三维拓扑超导体具有数量确定的无能隙狄拉克或马约拉纳表面态，对任意保持对称性的微扰具有鲁棒性。
• 由于拓扑保护，表面态保持无能隙且完全避免安德森局域化。
• 当引入U(1)规范场和能隙涨落时，具有非零绕数的超导相表现出非平凡的拓扑基态简并。