Randomness-InducedXYOrdering in a Graphene Quantum Hall Ferromagnet

Valley-polarized quantum Hall states in graphene are described by a Heisenberg O(3) ferromagnet model, with the ordering type controlled by the strength and the sign of the valley anisotropy. A mechanism resulting from electron coupling to the strain-induced gauge field, giving a leading contribution to the anisotropy, is described in terms of an effective random magnetic field aligned with the ferromagnet $z$ axis. We argue that such a random field stabilizes the $XY$ ferromagnet state, which is a coherent equal-weight mixture of the $K$ and $K^{\prime }$ valley states. The implications such as the Berezinskii-Kosterlitz-Thouless ordering transition and topological defects with half-integer charge are discussed.