Spin quantum Hall effect in unconventional superconductors

Abstract

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 Ϫy 2ϩ id 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 Ϫy 2ϩ id 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.

ExactS-matrices for supersymmetric sigma models and the Potts model

“…The generators E i then obey the TL algebra (the first relation in each of equations (18) and (19)), where indeed Q = (q 1/2 + q −1/2 ) 2 . The remaining relations of the SO(3) BMW algebra are…”

“…Using only the TL algebra relations (1), one can show (after some effort) that the generators E i and X i defined by (25) satisfy the relations (18), (19) [40]. In this proof, a very useful set of identities is E i = P 2i+2±1 e 2i+2 e 2i+1 e 2i+3 e 2i+2 P 2i+2±1 = P 2i+2±1 e 2i+2 e 2i+1 e 2i+3 e 2i+2 P 2i+2∓1 .…”

Chirald-wave superconductivity in doped graphene

“…This directly gives rise to two co-propagating, i.e. chiral, edge states crossing the bulk energy gap and carrying a spontaneous current [118,28], as well as generates quantized spin and thermal Hall conductances [121]. Moreover, the two chiral states d x 2 −y 2 + id xy and d x 2 −y 2 − id xy are in general degenerate and thus domain walls, with four chiral modes crossing the bulk band gap [118], will likely be present in the superconducting state.…”

“…It was shown in [121] that the spin Hall conductance is quantized in a chiral d-wave state according to σ s xy = 2sgn(∆ dxy ∆ d x 2 −y 2 ) = N for small d xy components. With the quantized value of the spin Hall conductance being a universal property, which cannot change within the same topological phase, the result must valid for any finite size of the d xy component [118].…”

“…The thermal Hall conductance similarly measures the transverse heat current as a function of a temperature gradient. The thermal Hall conductance is given by κ xy = 2π 2 T k 2 B /(3h)sgn(∆ dxy ∆ d x 2 −y 2 ) in a chiral d-wave superconductor and thus κ xy /T is a quantized quantity [121,122].…”