Spin quantum Hall effect in unconventional superconductors

We study the properties of the “spin quantum Hall fluid”—a spin phase with quantized spin Hall conductance that is potentially realizable in superconducting systems with unconventional pairing symmetry. A simple realization is provided by a $d_{x^2-y^2}{+id}_{\mathrm{xy}}$ superconductor which we argue has a dimensionless spin Hall conductance equal to 2. A theory of the edge states of the $d_{x^2-y^2}{+id}_{\mathrm{xy}}$ superconductor is developed. The properties of the transition to a phase with vanishing spin Hall conductance induced by disorder are considered. We construct a description of this transition in terms of a supersymmetric spin chain, and use it to numerically determine universal properties of the transition. We discuss various possible experimental probes of this quantum Hall physics.