Mesoscopic fluctuations of off-diagonal matrix elements of the angular momentum and orbital magnetism of free electrons in a rectangular box

Abstract

We study, analytically and numerically, mesoscopic fluctuations of the off-diagonal matrix elements of the orbital angular momentum between the nearest energy levels $i=({n}_{x},{n}_{y})$ and $f=({k}_{x},{k}_{y})$ in a rectangular box with incommensurate sides. In the semiclassical regime, where the level number of $i$ is $\mathcal{N}⪢1$, our derivation gives $⟨{\ensuremath{\mid}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{L}}_{if}\ensuremath{\mid}}^{2}⟩\ensuremath{\sim}\sqrt{\mathcal{N}}$. Numerical simulations, using simultaneous ensemble averaging (over the aspect ratios of rectangles) and spectral averaging (over the energy interval), are in excellent agreement with this analytical prediction. Physically, the mean is dominated by the level pairs ${k}_{x}={n}_{x}\ifmmode\pm\else\textpm\fi{}1$ and ${k}_{y}={n}_{y}\ensuremath{\mp}1$. Also in a rectangular box, we investigate the mean orbital susceptibility of a free-electron gas and argue that it reduces, up to a coefficient, to the two-level Van Vleck susceptibility that involves the last occupied (Fermi) level $i$ and the first unoccupied level $f$. This result is confirmed numerically as well, albeit the effect of fluctuations being more pronounced for the susceptibility since it is due to large fluctuations in both $⟨{\ensuremath{\mid}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{L}}_{if}\ensuremath{\mid}}^{2}⟩$ and in level separations ${\ensuremath{\epsilon}}_{f}\ensuremath{-}{\ensuremath{\epsilon}}_{i}$ (level bunching).