Extended Data Fig. 7: The manifolds \(\boldsymbol{\mathcal{M}}\) in the three-dimensional d-space, and the corresponding band structures, with different values of A0.
From: Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions
When dx ≠ ±dy, that is, the vector gauge potentials are non-Abelian, the manifold \({\mathcal{M}}\) has a geometry of a double-covered rhombus centered at d0. The manifold \({\mathcal{M}}\) is a closed surface: it is composed of two pieces of rhombus-shaped sheets overlapping with each other and “glued” together on the four edges. (a) When d0 = 0, the origin of d-space resides within \({\mathcal{M}}\) and therefore Dirac points exist in the band structure, regardless of the values of Ax and Ay. In this plot we have Ax = (π/2)σz, Ay = (π/2)σx, A0 = 0. (b) When d0 ≠ 0, it is possible that \({\mathcal{M}}\) is translated such that the origin of d-space resides outside \({\mathcal{M}}\), and hence the band structure is gapped. In this plot we have Ax = (π/2)σz, Ay = (π/2)σx, A0 = 0.9σy. In this figure the origins of the d-spaces are highlighted by the black dots.
