Topological quantization of ensemble averages

Abstract

We define the current of a quantum observable and, under well defined conditions, we connect its ensemble average to the index of a Fredholm operator. The present work builds on a formalism developed by Kellendonk and Schulz-Baldes \cite{Kellendonk:2004p597} to study the quantization of edge currents for continuous magnetic Schroedinger operators. The generalization given here may be a useful tool to scientists looking for novel manifestations of the topological quantization. As a new application, we show that the differential conductance of atomic wires is given by the index of a certain operator. We also comment on how the formalism can be used to probe the existence of edge states.

16 pages, 2 figures

References in corpus (11)

• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Spin Hall effect
• A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators
• Universality of transport properties in equilibrium, Goldstone theorem and chiral anomaly
• Simultaneous quantization of edge and bulk Hall conductivity
• Equality of bulk and edge Hall conductance revisited
• Equality of the bulk and edge Hall conductances in a mobility gap
• Norm estimates of complex symmetric operators applied to quantum systems
• Topological quantization of boundary forces and the integrated density of states

Cited by in corpus (7)

• Disordered Topological Insulators: A Non-Commutative Geometry Perspective
• Non-Commutative Tools for Topological Insulators
• Quantum transport in disordered systems under magnetic fields: A study based on operator algebras
• The edge spectrum of Chern insulators with rough boundaries
• Topologically Protected Extended States in Disordered Quantum Spin-Hall Systems without Time-Reversal Symmetry
• A Non-Commutative Formula for the Isotropic Magneto-Electric Response
• An edge index for the Quantum Spin-Hall effect