The effect of disorder on the axial anomaly in a lattice model in 1 + 1 dimensions

Abstract: This work investigates the effect of a disorder potential, that represents impurities/dislocations in the lattice, on the time-dependent signature of the axial anomaly in a one-band tight-binding model in one spatial dimension. The tight-binding model is solved numerically and the simulation results are compared with a semiclassical expression for the time-dependent axial density, n5(t), due to Nielsen and Ninomiya [1]. An estimate of the scattering time, which enters into the semiclassical expression, is computed using the Fermi golden rule (FG) approximation. The inclusion of disorder in the lattice model, in the form of a random disorder potential with Gaussian correlations, alters the time evolution of the axial density, from being oscillating in the clean case, to a curve that saturates at a non-zero value in the presence of sufficiently strong disorder. The semiclassical result is in accordance with the numerical solution in most of the parameter regimes considered. The FG approximation appears to be more accurate in the moderate to strong disorder regime than in the weak regime (in contrast with the validity condition of the FG approximation). A possible extension of the model in 1 + 1 dimensions to a model in 3 + 1 dimensions is briefly discussed.