Neutrino billiards: time-reversal symmetry-breaking without magnetic fields

Abstract A Dirac hamiltonian describing massless spin-half particles (‘neutrinos’) moving in the plane r = (x, y) under the action of a 4-scalar (not electric) potential V(r) is, in position representation, H^=−ihcσ^⋅∇+V(r)σ^z,, where σ̂ = (σ̂x, σ̂y) and σ̂z are the Pauli matrices; Ĥ acts on two-component column spinor wavefunctions ψ(r) = (ψ1, ψ2) and has eigen­values ћckn. Ĥ does not possess time-reversal symmetry (T). If V(r) describes a hard wall bounding a finite domain D (‘billiards’), this is equivalent to a novel boundary condition for ψ2/ψ1. T-breaking is interpreted semiclassically as a difference of π between the phases accumulated by waves travelling in opposite senses round closed geo­desics in D with odd numbers of reflections. The semiclassical (large-k) asymptotics of the eigenvalue counting function (spectral staircase) N(k) are shown to have the ‘Weyl’ leading term Ak2/4π, where A is the area of D, but zero perimeter correction. The Dirac equation is transformed to an integral equation round the boundary of D, and forms the basis of a numerical method for computing the kn. When D is the unit disc, geodesics are integrable and the eigenvalues, which satisfy Jl(kn) = Jl+1(kn), are (locally) Poisson-distributed. When D is an ‘Africa’ shape (cubic conformal map of the unit disc), the eigenvalues are (locally) distributed according to the statistics of the gaussian unitary ensemble of random-matrix theory, as predicted on the basis of T-breaking and lack of geometric symmetry.

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