  Although topological artificial systems, like acoustic/photonic crystals and
cold atoms in optical lattices were initially motivated by simulating
topological phases of electronic systems, they have their own unique features
such as the spinless time-reversal symmetry and tunable $\mathbb{Z}_2$ gauge
fields. Hence, it is fundamentally important to explore new topological phases
based on their unique features. Here, we point out that the $\mathbb{Z}_2$
gauge field leads to two fundamental modifications of the conventional $k\cdot
p$ method: (i) The little co-group must include the translations with
nontrivial algebraic relations; (ii) The algebraic relations of the little
co-group are projectively represented. These give rise to higher-dimensional
irreducible representations and therefore highly degenerate Fermi points.
Breaking the primitive translations can transform the Fermi points to
interesting topological phases. We demonstrate our theory by two models: a
rectangular $\pi$-flux model exhibiting graphene-like semimetal phases, and a
graphite model with interlayer $\pi$ flux that realizes the real second-order
nodal-line semimetal phase with hinge helical modes. Their physical
realizations with a general bright-dark mechanism are discussed. Our finding
opens a new direction to explore novel topological phases unique to artificial
systems and establishes the approach to analyze these phases.
