Geometric phase for a dimerized disordered continuum: Topological shot noise

Geometric phase shift associated with an electron propagating through a dimerized-disordered continuum is shown to be 0, or $\pm \pi$ (modulo 2$\pi$), according as the associated circuit traversed in the two-dimensional parameter space excludes, or encircles a certain singularity. This phase-shift is a topological invariant. Its discontinuous dependence on the electron energy and disorder implies a statistical spectral and conductance fluctuation in a corresponding mesoscopic system. Inasmuch as the fluctuation derives from the discreteness of the phase shift, it may aptly be called a topological shot-noise.