Majorana Zero-modes and Topological Phases of Multi-flavored Jackiw-Rebbi model

TL;DR

This paper classifies topological phases and Majorana zero modes in a 3+1D Jackiw-Rebbi model using K-theory, revealing conditions for zero modes related to symmetries and fermion flavors.

Contribution

It applies K-theory to classify topological phases of the Jackiw-Rebbi model and explicitly constructs Majorana zero modes for different fermion configurations.

Findings

Existence of Majorana zero modes depends on symmetry and fermion flavor number.

Single zero mode found for SU(2) doublet fermions in certain mass parameter regions.

Multiple zero modes can occur with specific mass matrices, consistent with anomaly cancellation.

Abstract

Motivated by the recent Kitaev's K-theory analysis of topological insulators and superconductors, we adopt the same framework to study the topological phase structure of Jackiw-Rebbi model in 3+1 dimensions. According to the K-theory analysis based on the properties of the charge conjugation and time reversal symmetries, we classify the topological phases of the model. In particular, we find that there exist $\mathbf{Z}$ Majorana zero-modes hosted by the hedgehogs/t'Hooft-Polyakov monopoles, if the model has a $T^2=1$ time reversal symmetry. Guided by the K-theory results, we then explicitly show that a single Majorana zero mode solution exists for the SU(2) doublet fermions in some co-dimensional one planes of the mass parameter space. It turns out we can see the existence of none or a single zero mode when the fermion doublet is only two. We then take a step further to consider four-fermion case and find there can be zero, one or two normalizable zero mode in some particular choices of mass matrices. Our results also indicate that a single normalizable Majorana zero mode can be compatible with the cancellation of SU(2) Witten anomaly.