Anomalous Magnetotransport in Disordered Structures: Classical Edge-State Percolation

By event-driven molecular dynamics simulations we investigate magnetotransport in a two-dimensional model with randomly distributed scatterers close to the field-induced localization transition. This transition is generated by percolating skipping orbits along the edges of obstacle clusters. The dynamic exponents differ significantly from those of the conventional transport problem on percolating systems, thus establishing a new dynamic universality class. This difference is tentatively attributed to a weak-link scenario, which emerges naturally due to barely overlapping edge trajectories. We make predictions for the frequency-dependent conductivity and discuss implications for active colloidal circle swimmers in a hetegogeneous environment.

Figure 1

Typical trajectories for classical magnetotransport. Left: conductive phase well above the magnetic transition ($R/\sigma =2.0$, $n^{*}=0.3$); right: almost at the transition ($R/\sigma =0.9$, $n^{*}=0.1$), see the red dots in Fig. 4.

Figure 2

Mean-square displacements $\delta r^2(t)$ for $R/\sigma =0.9$. The density $n^{*}$ decreases from top to bottom. The dashed line indicates the critical asymptote $\propto t^{2/z}$ with $z=3.44$. Inset: local exponent $\gamma (t)≔d\log [\delta r^2(t)]/d\log (t)$ as a function of time at the magnetic transition, $n^{*}=n_m^{*}(R)$, for three cyclotron radii: $R/\sigma =0.5$ (cyan circles), 0.9 (blue triangles), and 2.0 (red squares). The two horizontal lines mark the subdiffusion exponent for the universality classes of lattice percolation ($2/z_{\mathrm{lat}}$) and of the magnetic localization transition ($2/z$), respectively.

Figure 3

(a) Rectification plot of the squared localization length $(\ell /\sigma {)}^2|ϵ{|}^{2\nu -\beta }$ vs $|ϵ|=|n^{*}-n_m^{*}(R)|/n_m^{*}(R)$ in the insulating phase for different cyclotron radii $R$. Error bars combine statistical errors and uncertainties in reading off ${\ell }^2$ from the long-time asymptotes of $\delta r^2(t)$. Thin solid lines indicate weighted averages of the data points and shaded bars the uncertainty of the respective mean value; only data points covered by the bars were taken into account. (b) Rectification plot of the diffusion constant $(D/v\sigma ){ϵ}^{-\mu }$ vs $ϵ$ in the conducting phase. Data for different $R$ are shifted downwards by factors of 2. For all the three values of $R$ the same conductivity exponent $\mu =1.82$ was used.

Figure 4

Phase diagram: magnetic field $\sigma /R\propto B$ vs density $n^{*}$. The isodiffusivity contours are spaced logarithmically and increase from the phase boundaries towards the inner region. The red line corresponds to the magnetic transition according to Kuzmany and Spohn [6], $n_m^{*}(R)=n_c^{*}{\sigma }^2/(\sigma +R{)}^2$. The big red dots indicate the parameters for the trajectories of Fig. 1.

Figure 5

Visualization of the weak-link scenario of the field-induced percolation transition. Filled (apricot) circular regions are obstacles, the continuous (blue) arcs form a skipping orbit, and dashed circles are loci of skipping-orbit centers. The (green) overlap area of center loci scales as $A\sim W^{3/2}$ for vanishing width $W$.