Effect of magnetic flux on the critical behavior of a system with long-range hopping

We study the effect of magnetic flux in the spectral properties of a one-dimensional disordered wire with long-range hopping. It is shown that for all values of flux and disorder this model presents typical features of a metal-insulator transition. In the weak disorder regime a smooth transition between orthogonal and unitary symmetry is observed as the flux strength increases. By contrast, in the strong disorder regime the spectral correlations are almost flux independent. It is also conjectured that the two level correlation function is given by the dynamical density-density correlations of the Calogero-Sutherland model at finite temperature. Finally we describe the classical dynamics of the model and its relevance to quantum chaos.

Figure 1

Level spacing distribution $P\left(s\right)$ for $\lambda =2$ and different fluxes $\alpha$. A transition between the Gaussian orthogonal $\left(\mathrm{GOE}\right)$ and Gaussian unitary $\left(\mathrm{GUE}\right)$ ensemble is observed as the flux is increased, even the asymptotic behavior of $P\left(s\right)$ is modified by the flux.

Figure 2

$P\left(s\right)$ for $\lambda =0.44$ and different fluxes. The flux effect is observed in the $s⪡1$ limit only. For $s⪢1$ all curves become indistinguishable (see inset).

Figure 3

Number variance ${\Sigma }^2\left(L\right)$ for weak disorder $\lambda =2$ and different flux values. Lines are the analytical prediction (3) and symbols are the results obtained from the numerical diagonalization of (1) for $N=8000$.

Figure 4

Number variance ${\Sigma }^2\left(L\right)$ for a disorder $\left(\lambda =0.44\right)$ mimicking a true 3D Anderson transition. Lines represent the analytical prediction (3) $\left(h=0.267\right)$ for no flux (segment) and maximum flux (solid). Symbols correspond to the numerical diagonalization of (1) for $N=8000$. The dependence on the flux observed for small $L\sim 1$ (see inset) is also reproduced by the analytical result (3).