Classification of topological insulators and superconductors in three spatial dimensions
Classification of topological insulators and superconductors in three spatial dimensions
复制标题
DOI：
10.1103/physrevb.78.195125
复制
发表时间：
2008-11-01
影响因子：
3.7
通讯作者：
Ludwig, Andreas W. W.
中科院分区：
文献类型：
--
作者：
Schnyder, Andreas P.;Ryu, Shinsei;Ludwig, Andreas W. W.
我们系统地研究三个空间维度上的绝缘体和超导体（或超流体）的拓扑相。我们发现十多年前 Altland 和 Zirnbauer 在随机矩阵理论背景下的开创性工作中引入的十分之五的对称类中存在三维 (3D) 拓扑非平凡绝缘体或超导体。其中之一是最近引入的辛（或自旋轨道）对称类中的 Z(2) 拓扑绝缘体。我们证明了恰好存在另外四个拓扑绝缘体。对于这些系统，所有这些系统在三个维度上都是时间反转不变的，满足某些离散对称性的绝缘基态空间被划分为由量子相变分隔的拓扑扇区。上述五个拓扑非平凡相中的三个可以实现为时间反转不变超导体。在这些中，不同的拓扑扇区的特征是动量空间中定义的整数缠绕数。当这种 3D 拓扑绝缘体被二维表面终止时，它们支持多个狄拉克费米子（自旋旋转对称性完全破缺时的马约拉纳费米子）表面模式（对于单线态配对来说可能是任意非零偶数），这些表面模式在哈密顿量的任意扰动下保持无间隙，从而保留了特征离散对称性，包括无序性。特别是，这些表面模式完全避开了随机杂质的安德森局域化。这些拓扑相可以被认为是二维空间维度中众所周知的成对拓扑相的三维类似物，例如无自旋手性 (p(x)+/- ip(y)) 波超导体（或摩尔-里德普法夫态）。在相应的拓扑非平凡（类似于“弱配对”）和拓扑平凡（类似于“强配对”）3D 相中，波函数表现出明显不同的行为。当电磁 U(1) 规范场和间隙函数的波动包含在动力学中时，具有非零绕组数的超导相具有非平凡的拓扑基态简并性。
We systematically study topological phases of insulators and superconductors (or superfluids) in three spatial dimensions. We find that there exist three-dimensional (3D) topologically nontrivial insulators or superconductors in five out of ten symmetry classes introduced in seminal work by Altland and Zirnbauer within the context of random matrix theory, more than a decade ago. One of these is the recently introduced Z(2) topological insulator in the symplectic (or spin-orbit) symmetry class. We show that there exist precisely four more topological insulators. For these systems, all of which are time-reversal invariant in three dimensions, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. Three of the above five topologically nontrivial phases can be realized as time-reversal invariant superconductors. 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 two-dimensional surface, they support a number (which may be an arbitrary nonvanishing 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 of the Hamiltonian that preserve the characteristic discrete symmetries, including disorder. In particular, these surface modes completely evade Anderson localization from random impurities. These topological phases can be thought of as three-dimensional analogs of well-known paired topological phases in two spatial dimensions such as the spinless chiral (p(x)+/- ip(y))-wave superconductor (or Moore-Read Pfaffian state). In the corresponding topologically nontrivial (analogous to "weak pairing") and topologically trivial (analogous to "strong pairing") 3D phases, the wave functions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the superconducting phases with nonvanishing winding number possess nontrivial topological ground-state degeneracies.