SO(5) symmetry in the quantum Hall effect in graphene
Electrons in graphene have four flavors associated with low-energy spin and valley degrees of freedom. The fractional quantum Hall effect in graphene is dominated by long-range Coulomb interactions which are invariant under rotations in spin-valley space. This SU(4) symmetry is spontaneously broken at most filling factors, and also weakly broken by atomic scale valley-dependent and valley-exchange interactions with coupling constants $g_{z}$ and $g_{\perp}$. In this paper we demonstrate that when $g_{z}=-g_{\perp}$ an exact SO(5) symmetry survives which unifies the N\'eel spin order parameter of the antiferromagnetic state and the $XY$ valley order parameter of the Kekul\'e distortion state into a single five-component order parameter. The proximity of the highly insulating quantum Hall state observed in graphene at $\nu=0$ to an ideal SO(5) symmetric quantum Hall state remains an open experimental question. We illustrate the physics associated with this SO(5) symmetry by studying the multiplet structure and collective dynamics of filling factor $\nu=0$ quantum Hall states based on exact-diagonalization and low-energy effective theory approaches. This allows to illustrate how manifestations of the SO(5) symmetry would survive even when it is weakly broken.