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 ^ = − i h c σ ^ ⋅ ∇ + V ( r ) σ ^ z , , where σ̂ = (σ̂ x , σ̂ y ) and σ̂ z are the Pauli matrices; Ĥ acts on two-component column spinor wavefunctions ψ ( r ) = ( ψ 1 , ψ 2 ) and has eigen­values ћck n . 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 ψ 2 / ψ 1 . 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 Ak 2 /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 k n . When D is the unit disc, geodesics are integrable and the eigenvalues, which satisfy J l ( k n ) = J l +1 ( k n ), 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.