Anomalous diffusion in multichannel systems without a Lévy distribution of disorder

TL;DR

The paper reveals a channel-asymmetric mechanism for anomalous diffusion (CAAD) in multichannel quantum systems without the need for Lévy-type disorder. A minimal two-channel model shows a tunable crossover between normal diffusion ($\alpha\approx 2$) and anomalous diffusion ($\alpha<2$) controlled by interchannel coupling and channel-disorder asymmetry, with quantum interference between coexisting ballistic and localized modes driving long-tailed transmission statistics. This CAAD behavior persists in realistic quasi-1D geometries, such as edge-disordered graphene and quartic nanoribbons, where edge-localized channels compete with bulk quasi-ballistic channels over a broad energy range, yet normal localization eventually dominates at very long lengths, in violation of the Thouless relation in the CAAD regime. The work provides a distinct quantum transport mechanism beyond classical Lévy paradigms and suggests practical routes for disorder engineering and transport control, including four-probe measurements as a diagnostic tool and potential platforms like metallic nanowires encapsulated in CNTs.

Abstract

We show that multichannel quantum systems with uncorrelated but asymmetric Anderson-type disorder can exhibit anomalous diffusion, even in the absence of heavy-tailed disorder. Using a minimal two-channel model with channel asymmetry, we demonstrate a crossover from normal to anomalous transport tuned by interchannel coupling. Applied to quasi-one-dimensional lattices with edge disorder, this leads to long-tailed transmission statistics characterized by ballistic segments interspersed with localized ones, reminiscent of Lévy flights. This channel-asymmetric anomalous diffusion (CAAD) emerges from quantum interference between channels with differing disorder strengths. While CAAD governs transport at intermediate lengths, conventional localization prevails asymptotically, violating the Thouless relation. These results highlight a distinct quantum mechanism for anomalous diffusion beyond classical paradigms.

Figures (15)

• Figure 1: Two-channel model of CAAD demonstrating transition from normal to anomalous diffusion by varying $t_\perp/t_\parallel$. Length dependent transmission averaged over 1000 configurations is plotted for and $E/t_\parallel=0$ (a and d). Strong interchannel coupling ($t_\perp/t_\parallel{=}1$) gives rise normal diffusion (a). The contribution of each channel to transmission is resolved for each energy on the energy-band diagram with colors referring to the channel resolved transmission probabilities (b--c, colorbar is shown on the left). The same quantities are plotted for weak interchannel coupling ($t_{\perp}/ t_{\parallel}{=}1/20$) and shown in the same order in (d--f). The comparison of transmission in normal and anomalous cases can be observed in (d). The contrast in channel-resolved transmission probabilities is striking as shown in panels (e) and (f).
• Figure 2: Multichannel anomalous diffusion in quasi-one dimensional hexagonal lattices. The average transmission (a,d), channel-resolved transmission probabilities (b,e), and edge/bulk character of channels (c,f) are shown for GNRs (a--c) and QNRs (d--f). The average transmission values depending on the length are plotted for 0.84 eV for GNR (a), and for -0.51 eV for QNR (d). The band-resolved transmission probabilities (b and e) display variations for different channels. The contribution of the edge and bulk states to the transmission process is represented via an energy band diagram.
• Figure 3: Channel-resolved probability distributions in edge-disordered GNR for $L\sim\lambda_\mathrm{sp}$ reveal the same type of asymmetry in channel resolved transport regimes as in the two-channel model.
• Figure 4: Onset of normal localization and the Thouless relation. Geometric average of the conductance depending on the system length for (a) the two-channel model with Anderson disorder, QNR and GNR with $20\%$ edge defect density (b and c). The linear increase of $-\langle\ln\mathcal{T}\rangle$ with system length indicates normal localization behavior. The validity of Thouless relation is inspected for systems showing normal and anomalous diffusion (d and e). For $\alpha=2$, $\eta\simeq1$ is satisfied for various $N_\mathrm{ch}$ in different geometries (e), whereas for $\alpha<2$, $\eta$ is always larger than 1 (d).
• Figure S1: The average transmission versus system length for $20\%$ edge vacancy averaged over 1500 different disorder configurations for GNR at (a) $E =0.28$ eV (b) $E = 0.69$ eV (c) $E = 1.09$ eV, and for QNR at (d) $E =-0.63$ eV, (e) $E =-0.59$ eV, (f) $E =-0.51$ eV. The spread in the numerical data is calculated by using standard deviation.