Quantum mechanical stability of fermion-soliton systems

TL;DR

This paper investigates the stability of fermion-soliton systems with zero-energy modes, showing that non-integer fermionic charge can stabilize topological objects and establishing superselection rules for their decay.

Contribution

It introduces a framework for understanding the stability of fermion-soliton systems with zero modes, including the concept of a Majorana pond and superselection rules.

Findings

Zero-energy fermion modes can stabilize topological objects.

Half-integer fermion number states cannot decay in isolation.

Superselection rules prevent decay of objects with half-integer fermion number.

Abstract

Topological objects resulting from symmetry breakdown may be either stable or metastable depending on the pattern of symmetry breaking. However, if they acquire zero-energy modes of fermions, and in the process acquire non-integer fermionic charge, the metastable configurations also get stabilized. In the case of Dirac fermions the spectrum of the number operator shifts by 1/2. In the case of majorana fermions it becomes useful to assign negative values of fermion number to a finite number of states occupying the zero-energy level, constituting a \textit{majorana pond}. We determine the parities of these states and prove a superselection rule. Thus decay of objects with half-integer fermion number is not possible in isolation or by scattering with ordinary particles. The result has important bearing on cosmology as well as condensed matter physics.