Abstract
Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings, and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is fractionalized and this is related to the presence of fermion zero-energy excitation modes. A zero-energy mode can be regarded as a Majorana fermion mode when the chemical potential vanishes. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable wave function of fermion zero-energy mode does not exist in the core of the half-flux quantum vortex. The index of Dirac operator and the fermion number have additional contributions when a soliton scalar field has a singularity.
Highlights
Topological materials have been attracted much attention in physics
We have investigated fermion zero-energy modes and the index of the Dirac operator in vortex–Dirac fermion systems in (1 + 2) dimensions
Dirac fermions play an important role in many electron systems such as topological insulators, topological superconductors, graphene [32,33,34], and Kondo systems [35,36,37]
Summary
Topological materials have been attracted much attention in physics. New interesting topological properties will emerge in the study of quantum systems from the viewpoint of topology. Dirac fermions sometimes exist on the surface or in the bulk. There exist zero-energy bosonic modes on solitons [9,10,11], and both bosonic and fermionic zero-energy modes will emerge in the presence of solitons These exotic quantum states carry fermionic quantum numbers that can be fractional [12,13,14,15]. If a bilayer system including superconductors and a topological insulator is synthesized, the Dirac fermion on the surface of the topological insulator may cause a zero-energy mode in a vortex. The index of a Dirac operator is closely related to the fermion number and η invariant.
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