  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.
