Mesoscopic fluctuations of off-diagonal matrix elements of the angular   momentum and orbital magnetism of free electrons in a rectangular box
  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 $\cal{N}\gg 1$, our derivation gives $\left< | \hat{L}_{if}| ^{2}\right> \sim \sqrt{\cal{N}}$. 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 $k_{x}=n_{x}\pm 1$, $k_{y}=n_{y}\mp 1$. 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 is much more pronounced for the susceptibility since it is due both to large fluctuations in $\left< | \hat{L}_{if}| ^{2}\right>$ and in level separations $\epsilon_{f}-\epsilon_{i}$ (level bunching). 
M.X. Lou, J. M. A. S. P. Wickramasinghe and R. A. Serota