TOPOLOGICAL STATES OF MATTER AND NONCOMMUTATIVE GEOMETRY
TOPOLOGICAL STATES OF MATTER AND NONCOMMUTATIVE GEOMETRY
复制标题
物质的拓扑态和非交换几何
DOI：
10.1017/s000497271600037x
复制
发表时间：
2016
影响因子：
0.7
通讯作者：
C. Bourne
中科院分区：
文献类型：
--
作者：
C. Bourne
本文从非交换几何和kk理论的角度研究了物质的拓扑状态。这种物质的拓扑状态的例子包括量子霍尔效应和拓扑绝缘体。对于量子霍尔效应，我们考虑了一个连续模型，并证明了霍尔电导可以用一个无序哈密顿量的费米投影的指数配对来表示，该哈密顿量具有编码样品动量空间几何形状的谱三重。磁场的存在意味着必须使用非交换代数和方法。还考虑了量子霍尔系统的高维类似物，其中索引配对在连续设置中产生“高维陈氏数”。接下来我们考虑一个具有边缘的离散量子霍尔系统。我们证明了集中在边界处的可观测值的拓扑性质可以通过Kasparov积与无边界模型的不变量联系起来。由此，我们用kk理论的语言得到了量子霍尔效应的体边对应。最后，我们考虑拓扑绝缘体，它来自于对凝聚态系统施加（可能是反线性的）对称性，并研究受这些对称性保护的不变量。我们展示了如何将对称数据与真实或复杂kk理论中的类联系起来。最后，我们利用kko理论中的Kasparov积将本体系统和边缘系统连接起来，证明了拓扑绝缘子系统的本体-边缘对应关系。
This thesis examines topological states of matter from the perspective of noncommutative geometry and KK-theory. Examples of such topological states of matter include the quantum Hall effect and topological insulators. For the quantum Hall effect, we consider a continuous model and show that the Hall conductance can be expressed in terms of the index pairing of the Fermi projection of a disordered Hamiltonian with a spectral triple encoding the geometry of the sample’s momentum space. The presence of a magnetic field means that noncommutative algebras and methods must be employed. Higher dimensional analogues of the quantum Hall system are also considered, where the index pairing produces the ‘higher-dimensional Chern numbers’ in the continuous setting. Next we consider a discrete quantum Hall system with an edge. We show that topological properties of observables concentrated at the boundary can be linked to invariants from a boundary-free model via the Kasparov product. Hence we obtain the bulk-edge correspondence of the quantum Hall effect in the language of KK-theory. Finally we consider topological insulators, which come from imposing (possibly anti-linear) symmetries on condensed-matter systems and studying the invariants that are protected by these symmetries. We show how symmetry data can be linked to classes in real or complex KK-theory. Finally we prove the bulk-edge correspondence for topological insulator systems by linking bulk and edge systems using the Kasparov product in KKO-theory.
DOI：
10.1007/s11040-012-9123-9
发表时间：
2013
期刊：
Mathematical Physics, Analysis and Geometry
影响因子：
--
作者：
J. C. Avila;H. Schulz-Baldes;C. Villegas-Blas
通讯作者：
C. Villegas-Blas
与通量管相关的谱流
DOI：
10.1007/s00023-014-0394-5
发表时间：
2016
期刊：
Annales Henri Poincaré
影响因子：
--
作者：
G. De Nittis;H. Schulz-Baldes
通讯作者：
H. Schulz-Baldes
有隙自由费米子系统 $${mathbb{Z}_2}$$Z2 对称基态的 Bott 周期
影响因子：
2.4
作者：
R. Kennedy;M.R. Zirnbauer
通讯作者：
M.R. Zirnbauer