Non-Abelian nonsymmorphic chiral symmetries

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 $\mathrm{SU}\left(2\right)$ non-Abelian Hofstadter model, we theoretically show the emergence of multiple nonsymmorphic chiral symmetries, which combine an internal unitary antisymmetry with fractional spatial translation. Depending on the values of the gauge fields, the nonsymmorphic chiral symmetries can exhibit non-Abelian algebra and protect Kramers quartet states in the bulk band structure, creating general fourfold 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.

Figure 1

Non-Abelian Hofstadter model (a) and its inhomogeneous real-space magnetic fields (b)–(d). Zero, one, and two nonsymmorphic chiral symmetries involving a half-lattice translation appear in (b), (c), and (d), respectively, as evident from the patterns of the magnetic fields. Four magnetic unit cells (divided by bold black lines) are shown. The real space magnetic fields are characterized by the gauge-invariant, real-space Wilson loop $W=\mathrm{Tr}exp(i\oint \mathit{A}\mathrm{d}\mathbf{l})$ for each unit plaquette.

Figure 2

Nonsymmorphic chiral symmetries and wave functions of Kramers quartets. Nonsymmorphic chiral symmetries [(a) and (b); see Eq. (3)] appear as the product between half-translations and site-dependent phase factors [represented by light and dark shadings for $\pm i$ and $\pm 1$ in (a) and (b), respectively]. When $q_x=4\mathbb{Z}$ and $q_y=4\mathbb{Z}+2$, two Kramers doublets (c) and (d) can be constructed by occupying each bipartite sublattices respectively with ${\alpha }^{\pm }$ [red and blue dots label the $\pm$ superscript; see Eq. (5)] or vice versa. The non-Abelian group ${\mathbb{D}}_8$ protects the degeneracy between the two Kramers doublets and thus enables a Kramers quartet, i.e., a fourfold degeneracy at all momenta and all energies. Here $H(1/4,1/6)$ is illustrated as an example.

Figure 3

Double Dirac points on a real projective plane protected by chiral symmetries. (a) Bulk bands and the four subspaces (b)–(e) of $H(1/4,1/4)$ in the MBZ. All subspaces are aperiodic in the original MBZ, which becomes a real projective plane due to the nonsymmorphic symmetries along both $k_x$ and $k_y$ directions. The double Dirac point at $\mathrm{\Gamma }$ of $H$ is “shared” by its four subspaces at $\mathrm{\Gamma }, 2X, 2Y$, and $2M$.

Figure 4

Trivial and topological quantum-confined phases dictated by the system parity. (a) Projected bulk bands of $H(1/4,1/6)$ near half-filling. (b)–(g) Quantum-confined band gaps (shaded gray) and edge states (red and blue, for chiral and antichiral subspaces, respectively) appear upon introducing a boundary that breaks a nonsymmorphic chiral symmetry $S_y$, which protects the bulk double Dirac points at $\mathrm{\Gamma }$ and $X$. The quantum-confined insulating phase can be trivial [${\mathbb{Z}}_2$ even; (b)–(d)] and topological [${\mathbb{Z}}_2$ odd; (e)–(g)] in both the full manifold and its chiral and antichiral subspaces when the open $y$ direction of the cylinder geometry contains an even or odd number of unit cells. Shaded yellow and green indicates half MBZ for the full manifold and the two subspaces.