  In 2+1D, topological electromagnetic phases are defined as atomic-scale media
which host photonic monopoles in the bulk band structure and respect bosonic
symmetries. Additionally, they support topologically protected spin-1 edge
states, which are fundamentally different than spin-1/2 and pseudo-spin-1/2
edge states arising in fermionic and pseudo-fermionic systems. The striking
feature of the edge state is that all electric and magnetic field components
vanish at the boundary. This surprising open boundary solution of Maxwell's
equations, dubbed the quantum gyroelectric effect [Phys. Rev. A 98, 023842
(2018)], only occurs in the presence of temporal as well as spatial dispersion
(nonlocality) and is the supersymmetric partner of the topological Dirac edge
state where the spinor wave function completely vanishes at the boundary. In
this paper, we generalize these topological electromagnetic phases beyond the
continuum approximation to the exact lattice field theory of a periodic atomic
crystal. To accomplish this, we put forth the concept of microscopic (nonlocal)
photonic band structure of solids, analogous to the traditional theory of
electronic band structure. Our definition of topological invariants and
topological phases uses optical Bloch modes and can be applied to naturally
occurring crystalline materials. For the photon propagating within a crystal,
our theory shows that besides the Chern invariant $\mathfrak{C}\in\mathbb{Z}$,
there are also symmetry-protected topological (SPT) invariants
$\nu\in\mathbb{Z}_N$ which are related to the cyclic point group $C_N$ of the
crystal $\nu=\mathfrak{C}\mod N$. Due to the rotational symmetries of light
$\mathcal{R}(2\pi)=+1$, these SPT phases are manifestly bosonic and behave very
differently from their fermionic counterparts $\mathcal{R}(2\pi)=-1$
encountered in conventional condensed matter systems.
