Energy spectrum of the graphene (with some modification) under a magnetic field.
Y. Hatsugai, T. Fukui, and H. Aoki, Phys. Rev. B 74, 205414 (2006)

Non abelian gauge structures appear naturally in graphene as a generic geometrical phase of quantum system.

$$\psi=(|\psi_1\rangle,\cdots,|\psi_M\rangle)=\psi_g g,\ \psi^\dagger\psi=E_M,\ g\in U(M)$$ $$A=\psi^\dagger d\psi=g^{-1}A_gg +g^{-1}dg $$ $$F=dA+A^2=d\psi^\dagger d\psi+(\psi^\dagger d\psi)^2=g^{-1}F_g g$$ $$P(dP)^2 P=\psi F\psi^\dagger=\psi_gF_g\psi_g^\dagger,\ P = \psi\psi^\dagger $$ $$i \gamma=\int_C \text{Tr}\, A=\int_C \text{Tr}\, A_g=i\gamma_g,\ \text{mod}\, 2\pi$$ $$\text{Ch}^1={\frac 1 {2\pi i}}\int_S \text{Tr}\, F={\frac 1 {2\pi i}}\int_S \text{Tr}\, F_g=\text{Ch}^1_g $$

Y. Hatsugai, J. Phys. Soc. Jpn. 73, 2604 (2004)
“Explicit Gauge Fixing for Degenerate Multiplets: A Generic Setup for Topological Orders”
Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006)
“Quantized Berry Phases as a Local Order Parameter of a Quantum Liquid”
Y. Hatsugai, New J. Phys. 12 065004 (2010)
“Symmetry-protected \(\mathbb{Z}_2\)-quantization and quaternionic Berry connection with Kramers degeneracy”