Non-Abelian SU(2) gauge fields through density wave order and strain in graphene

Spatially varying strain patterns can qualitatively alter the electronic properties of graphene, acting as effective valley-dependent magnetic fields and giving rise to pseudo-Landau-level (PLL) quantization. Here, we show that the strain-induced magnetic field is one component of a non-Abelian SU(2) gauge field within the low-energy theory of graphene and identify the other two components as period-3 charge-density waves. We show that these density waves, if spatially varied, give rise to PLL quantization. We also argue that strain-induced magnetic fields can induce density-wave order in graphene, thus dynamically gapping out the lowest PLL; moreover, the ordering should generically be accompanied by dislocations. We discuss experimental signatures of these effects.