Protected Fermionic Zero Modes in Periodic Gauge Fields

TL;DR

This paper demonstrates that in two-dimensional systems with doubly-periodic magnetic or pseudo-gauge fields, only two zero-energy fermionic states exist, with implications for graphene and related materials.

Contribution

It extends known zero-mode results to systems with zero net flux and doubly-periodic fields, revealing a strict limit of two zero-energy states per spin flavor.

Findings

Only two Bloch-normalizable zero-energy states exist for such systems.

The results apply to graphene multilayers under doubly-periodic strain fields.

The study explores related models with nonlinearly-dispersing bands and singly-periodic fields.

Abstract

It is well-known that macroscopically-normalizable zero-energy wavefunctions of spin-$\frac{1}{2}$ particles in a two-dimensional inhomogeneous magnetic field are spin-polarized and exactly calculable with degeneracy equaling the number of flux quanta linking the whole system. Extending this argument to massless Dirac fermions subjected to magnetic fields that have \textit{zero} net flux but are doubly periodic in real space, we show that there exist \textit{only two} Bloch-normalizable zero-energy eigenstates, one for each spin flavor. This result is immediately relevant to graphene multilayer systems subjected to doubly-periodic strain fields, which at low energies, enter the Hamiltonian as periodic pseudo-gauge vector potentials. Furthermore, we explore various related settings including nonlinearly-dispersing band structure models and systems with singly-periodic magnetic fields.