  We construct and solve a two-dimensional, chirally symmetric model of Dirac
cones subjected to a quasiperiodic modulation. In real space, this is realized
with a quasiperiodic hopping term. This hopping model, as we show, at the Dirac
node energy has a rich phase diagram with a semimetal-to-metal phase transition
at intermediate amplitude of the quasiperiodic modulation, and a transition to
a phase with a diverging density of states and sub-diffusive transport when the
quasiperiodic hopping is strongest. We further demonstrate that the
semimetal-to-metal phase transition can be characterized by the multifractal
structure of eigenstates in momentum space and can be considered as a unique
"unfreezing" transition. This unfreezing transition in momentum space generates
flat bands with a dramatically renormalized bandwidth in the metallic phase
similar to the phenomena of the band structure of twisted bilayer graphene at
the magic angle. We characterize the nature of this transition numerically as
well as analytically in terms of the formation of a band of topological zero
modes. For pure quasiperiodic hopping, we provide strong numerical evidence
that the low-energy density of states develops a divergence and the eigenstates
exhibit Chalker (quantum-critical) scaling despite the model not being random.
At particular commensurate limits the model realizes higher-order topological
insulating phases. We discuss how these systems can be realized in experiments
on ultracold atoms and metamaterials.
