for all two-dimensional (2D) symmorphic layer group symmetries [@graph]; these results are also compactly presented in Table \[tab:class\]. Possible extensions of our work in treatment magnetic crystals as well as defected and higher-dimensional systems are discussed in Sec. \[sec:discussion\].

Methodology of Classification {#sec:meth}
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Physical Paradigm {#sec:paradigm}
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The many-body Hilbert space is a complex linear vector space. A symmetry action on the states within the Hilbert space is either a unitary or anti-unitary action. These imply that the Hilbert space forms a real module over the algebra of symmetries, denoted $B^0$. When no crystalline symmetry is present, $B^0$ is generated by some selection of the following non-spatial symmetries [@schnyder2009classification; @ryu2010topological; @kennedy2016bott]: charge conservation $Q$, time reversal $T$, particle-hole symmetry $C$, and spin rotations $S_1,S_2,S_3$. Such a selection corresponds to an AZ symmetry class [@Altland1997Nonstandard; @heinzner2005symmetry; @zirnbauer2010symmetry]; see Table \[tab:per\]. Any translation invariant non-interacting quantum dynamics is described by a Hamiltonian, $\mathcal{H}(\mathbf{k})$, which is quadratic in creation-annihilation(/Majorana) operators of the many-body Fock(/Nambu) space representation. When using this representation, the Hamiltonian may be linearized around the $\Gamma$-point, which is the high-symmetry time-reversal-invariant point, such that [@kitaev2009periodic; @schnyder2009classification; @ryu2010topological; @franz2013topological; @witten2015three; @abramovici2012clifford] $$\label{gammapt}
\mathcal{H}(\mathbf{k})=iM+\bm{\gamma}\cdot\mathbf{k},{{\;\;\:}}\{\gamma_i,\gamma_j\}=2\delta_{ij}.$$ Here, the Dirac gamma matrices $\bm{\gamma}=\gamma_1,\gamma_2,\ldots,\gamma_d$ (in $d$ 