  The Hofstadter model exemplifies a large class of physical systems
characterized by particles hopping on a lattice immersed in a gauge field.
Recent advancements on various synthetic platforms have enabled
highly-controllable simulations of such systems with tailored gauge fields
featuring complex spatial textures. These synthetic gauge fields could
introduce synthetic symmetries that do not appear in electronic materials.
Here, in an SU(2) non-Abelian Hofstadter model, we theoretically show the
emergence of multiple nonsymmorphic chiral symmetries, which combine an
internal unitary anti-symmetry with fractional spatial translation. Depending
on the values of the gauge fields, the nonsymmorphic chiral symmetries can
exhibit non-Abelian algebra and protect Kramer quartet states in the bulk band
structure, creating general four-fold degeneracy at all momenta. These
nonsymmorphic chiral symmetries protect double Dirac semimetals at zero energy,
which become gapped into quantum confined insulating phases upon introducing a
boundary. Moreover, the parity of the system size can determine whether the
resulting insulating phase is trivial or topological. Our work indicates a
pathway for creating topology via synthetic symmetries emergent from synthetic
gauge fields.
