Electronic Lévy Glasses

• Electronic Lévy glasses are disorder-engineered media with heavy-tailed scattering geometries that yield anomalous, superdiffusive transport behavior.
• They translate the optical Lévy glass paradigm into electronics by replacing transparent inclusions with impurity-free intervals or spin-orbit clusters, affecting localization regimes.
• These systems enable tunable conduction and improved spintronic functionality in platforms like graphene nanoribbons through the interplay of disorder correlations and finite-size effects.

Electronic Lévy glasses are disorder-engineered electronic media in which the geometry of scattering regions, impurities, or ballistic segments is distributed according to a heavy-tailed law, so that transport departs from ordinary diffusion and acquires strong finite-size, correlation, and localization effects. The concept is inherited from optical Lévy glasses, where a power-law step-length distribution $p(l)\propto l^{-(\alpha+1)}$ yields a diffusive regime for $\alpha\ge 2$ and a superdiffusive regime for $0<\alpha<2$ (Burresi et al., 2011). In the electronic literature, the term covers both direct realizations—most prominently one-dimensional tight-binding systems with Lévy-distributed impurity spacings and graphene nanoribbons containing power-law-distributed spin-orbit clusters—and broader analogies to disordered open systems with heavy-tailed fluctuations (Sepehrinia, 2022, Fonseca et al., 2024, Fonseca et al., 14 May 2026, Gomes et al., 2015).

1. Optical origin and defining transport idea

The original Lévy-glass paradigm is optical rather than electronic. In the optical construction, jammed-packed glass spheres with diameters spanning $5$ to $230\,\mu\mathrm{m}$ are embedded in an index-matched polymeric matrix with dispersed $\mathrm{TiO_2}$ nanoparticles acting as point scatterers; because the spheres are nearly transparent and the scatterers are excluded from their interiors, photons undergo long ballistic traversals through the spheres and scattering in the interstitial medium. The sphere-size distribution is $p(\phi)\propto \phi^{-(\beta+1)}$, and the resulting step-length distribution is $p(l)\propto l^{-(\alpha+1)}$ with $\alpha=\beta-1$ under exponential sampling of diameters. The superdiffusive regime is $0<\alpha<2$, and finite-size truncation is central because the slab thickness is only about $\alpha\ge 2$0 larger than the largest sphere diameter (Burresi et al., 2011).

That optical background fixes the central meaning of a Lévy glass: anomalous transport generated by heavy-tailed geometric disorder, not merely by strong disorder in the Anderson sense. It also motivates the electronic translation in which transparent optical inclusions are replaced by extended low-scattering regions, impurity-free intervals, or spin-orbit-active clusters.

A crucial correction to overly simple Lévy-walk intuition comes from simulations of optical Lévy-glass transmission. In slab geometry, the transmission obeys $\alpha\ge 2$1, but diffusive scaling with $\alpha\ge 2$2 can coexist with a step-size distribution $\alpha\ge 2$3 even when $\alpha\ge 2$4, because the quenched geometry correlates successive flights. The fixed arrangement of spheres or discs therefore distinguishes a Lévy glass from an ideal Lévy walk with independent steps, and this distinction carries directly into electronic analogies (Groth et al., 2011).

2. Disorder architectures used in electronic realizations

A minimal electronic realization is the one-dimensional tight-binding alloy with correlated impurity positions. Its governing equation is

$\alpha\ge 2$5

with binary on-site potential $\alpha\ge 2$6. The disorder is generated by a renewal process: after $\alpha\ge 2$7 clean sites there is one impurity, and the spacings $\alpha\ge 2$8 are i.i.d. random integers drawn from a prescribed distribution $\alpha\ge 2$9. In the Lévy-glass case, $0<\alpha<2$0 with $0<\alpha<2$1. The key structural distinction is between $0<\alpha<2$2, where the mean spacing is finite, and $0<\alpha<2$3, where the mean spacing diverges and the impurity density vanishes in the thermodynamic limit (Sepehrinia, 2022).

A more direct mesoscopic implementation is the graphene-nanoribbon platform with circular regions of enhanced Rashba spin-orbit coupling. In that system, the active region of an armchair or zigzag graphene nanoribbon is populated by randomly positioned, non-overlapping circular SOC clusters whose radii follow a power-law distribution,

$0<\alpha<2$4

with $0<\alpha<2$5. The histogram fit reported for the cluster radii gives $0<\alpha<2$6, and the numerics impose a cutoff by limiting the maximum radius to one-eighth of the ribbon width. For the representative AGNR sample used to characterize the radius histogram, the SOC clusters occupy $0<\alpha<2$7 of the lattice area. The same geometry is carried into the four-terminal spin Hall study, where the radii obey $0<\alpha<2$8 with $0<\alpha<2$9, $5$0, and $5$1 (Fonseca et al., 2024, Fonseca et al., 14 May 2026).

Because the cluster positions are fixed, these graphene systems inherit the defining Lévy-glass feature of correlated scattering. They are not annealed random walks in which every step is independently redrawn from a heavy-tailed law; rather, the broad path-length statistics emerge from a quenched, spatially organized disorder landscape.

3. Localization theory and one-dimensional Lévy-correlated disorder

For the one-dimensional tight-binding model, localization is formulated through transfer matrices or, equivalently, through the Riccati variable

$5$2

The Lyapunov exponent $5$3 is the asymptotic growth rate of random transfer-matrix products, and the localization length is $5$4. The methodological advance of the electronic Lévy-glass treatment is a perturbative expansion carried out in impurity index rather than site index, which makes the spacing variables $5$5 explicit through the averages

$5$6

This yields the Lyapunov exponent up to fourth order for arbitrary spacing distribution $5$7, exposes the impurity-density factor $5$8, and avoids explicit construction of the site-disorder correlator (Sepehrinia, 2022).

Several physical consequences follow. First, heavy-tailed spacing modifies localization relative to uncorrelated Anderson disorder because the structural correlations enter through the nontrivial $5$9-dependence of $230\,\mu\mathrm{m}$0. Second, transparent states can occur not only in periodic spacing but also in random binary or ternary spacing distributions when the transfer matrices phase-match appropriately. Third, the perturbative expansion develops extra divergences at energies determined by $230\,\mu\mathrm{m}$1 and $230\,\mu\mathrm{m}$2, producing internal-gap and Kappus-Wegner-type anomalies. Most importantly for the Lévy regime, the perturbative formulas indicate

$230\,\mu\mathrm{m}$3

so when $230\,\mu\mathrm{m}$4 and $230\,\mu\mathrm{m}$5, the Lyapunov exponent vanishes. The interpretation given is divergence of the localization length due to vanishing impurity density, together with the explicit caution that this does not necessarily imply ordinary ballistic transport (Sepehrinia, 2022).

A closely related wave-localization prototype is the one-dimensional harmonic chain with Lévy-distributed impurity spacing. There the impurity spacing again obeys $230\,\mu\mathrm{m}$6, but the analysis is carried out for classical lattice waves. In the regime $230\,\mu\mathrm{m}$7, the disorder power spectrum is singular at small wave number,

$230\,\mu\mathrm{m}$8

and the low-frequency localization law becomes

$230\,\mu\mathrm{m}$9

instead of the standard $\mathrm{TiO_2}$0 scaling recovered for $\mathrm{TiO_2}$1. This is not an electronic calculation, but it establishes a transferable point: Lévy-type positional disorder modifies coherent localization through the low-$\mathrm{TiO_2}$2 structure factor of the disorder, not merely through its variance (Zakeri et al., 2014).

4. Graphene nanoribbons with Rashba-cluster Lévy disorder

The graphene realization is modeled by a nearest-neighbor tight-binding Hamiltonian with proximity-induced Rashba SOC,

$\mathrm{TiO_2}$3

with $\mathrm{TiO_2}$4. The Rashba term is switched on only inside the circular SOC regions, so $\mathrm{TiO_2}$5 inside a given cluster and $\mathrm{TiO_2}$6 outside. Transport is computed with the Landauer-Büttiker formalism using KWANT’s Green-function-based scattering-matrix implementation. The characteristic scaling observable is the disorder-averaged transmission,

$\mathrm{TiO_2}$7

where $\mathrm{TiO_2}$8 corresponds to ordinary diffusion and $\mathrm{TiO_2}$9 identifies superdiffusive or Lévy-like transport (Fonseca et al., 2024).

The main result is a Fermi-energy-controlled crossover from superdiffusive to diffusive transport. The control variable is the number of open channels,

$p(\phi)\propto \phi^{-(\beta+1)}$0

Low $p(\phi)\propto \phi^{-(\beta+1)}$1 corresponds to low Fermi energy, and high $p(\phi)\propto \phi^{-(\beta+1)}$2 to higher energy. For AGNR with $p(\phi)\propto \phi^{-(\beta+1)}$3, the reported scaling exponent is $p(\phi)\propto \phi^{-(\beta+1)}$4 at $p(\phi)\propto \phi^{-(\beta+1)}$5 and $p(\phi)\propto \phi^{-(\beta+1)}$6 at $p(\phi)\propto \phi^{-(\beta+1)}$7; for AGNR with $p(\phi)\propto \phi^{-(\beta+1)}$8, $p(\phi)\propto \phi^{-(\beta+1)}$9 at $p(l)\propto l^{-(\alpha+1)}$0 and $p(l)\propto l^{-(\alpha+1)}$1 at $p(l)\propto l^{-(\alpha+1)}$2. For ZGNR, the corresponding values are approximately $p(l)\propto l^{-(\alpha+1)}$3 at $p(l)\propto l^{-(\alpha+1)}$4 and $p(l)\propto l^{-(\alpha+1)}$5 at $p(l)\propto l^{-(\alpha+1)}$6, for both $p(l)\propto l^{-(\alpha+1)}$7 and $p(l)\propto l^{-(\alpha+1)}$8. The practical criterion adopted is therefore $p(l)\propto l^{-(\alpha+1)}$9 for the superdiffusive regime and $\alpha=\beta-1$0 for the diffusive regime (Fonseca et al., 2024).

Spin transport is intertwined with this crossover. The spin-resolved transmission matrix yields polarizations $\alpha=\beta-1$1, $\alpha=\beta-1$2, and $\alpha=\beta-1$3, with the paper mainly presenting $\alpha=\beta-1$4. Finite spin polarization appears only in the superdiffusive regime; in the diffusive regime, the spin polarization is approximately zero regardless of edge type, device length, or Rashba strength. The same study also applies Multifractal Detrended Fluctuation Analysis to energy-dependent transmission and polarization series. Charge-transmission series are multifractal in the superdiffusive regime and tend toward monofractality in the diffusive regime, with enhanced multifractality near the crossover—around $\alpha=\beta-1$5 for AGNR and $\alpha=\beta-1$6 for ZGNR. By contrast, the spin-polarization series remain multifractal in both regimes (Fonseca et al., 2024).

5. Spin Hall response and spintronic functionality

The four-terminal extension of the graphene Lévy-glass platform uses the same clustered Rashba geometry but adds transverse probes and, optionally, a cluster-confined on-site potential $\alpha=\beta-1$7. The device Hamiltonian combines graphene hopping, cluster-confined Bychkov-Rashba SOC, and an on-site term with $\alpha=\beta-1$8 inside the clusters and $\alpha=\beta-1$9 outside. Transport is evaluated with KWANT and the Landauer-Büttiker formalism, ensemble-averaged over $0<\alpha<2$0 independent disorder realizations, for parameter sweeps over

$0<\alpha<2$1

The spin Hall angle is defined as the ratio of transverse spin current to longitudinal charge current, $0<\alpha<2$2 (Fonseca et al., 14 May 2026).

Within this geometry, the low-energy regime is identified as superdiffusive, especially for $0<\alpha<2$3, with $0<\alpha<2$4 used as a representative point; the high-energy diffusive regime is represented by $0<\alpha<2$5. The central quantitative finding is that the spin Hall angle can reach roughly $0<\alpha<2$6 in the superdiffusive regime, whereas in the diffusive regime it is only $0<\alpha<2$7. The detailed edge-resolved values reported in the main text are about $0<\alpha<2$8 for AGNR and about $0<\alpha<2$9 for ZGNR when $\alpha\ge 2$00, with the possibility of reaching about $\alpha\ge 2$01 in AGNR when a cluster-local electrostatic potential is also used (Fonseca et al., 14 May 2026).

The current response makes the regime contrast explicit. In the superdiffusive regime, the average charge current stays approximately constant at $\alpha\ge 2$02 over the full $\alpha\ge 2$03 range for both edge terminations. In the diffusive regime, the charge current is strongly suppressed as $\alpha\ge 2$04 increases, falling from about $\alpha\ge 2$05 to roughly $\alpha\ge 2$06. For AGNR, the spin Hall current saturates near $\alpha\ge 2$07 in the superdiffusive regime and near $\alpha\ge 2$08 in the diffusive regime. For ZGNR, both regimes saturate near $\alpha\ge 2$09, but the superdiffusive regime achieves that spin current with a much smaller charge current, so the conversion efficiency remains markedly higher. The proposed mechanism is SOC-induced spin-dependent scattering at the circular islands embedded in a Lévy-glass transport environment with reduced resistive and magnetoresistive losses (Fonseca et al., 14 May 2026).

6. Conceptual limits, misconceptions, and relation to other glassy systems

Electronic Lévy glasses should not be treated as a synonym for any system with large disorder or with a divergent step-size variance. The optical weak-localization literature shows that interference survives in superdiffusive media, but the appropriate theory is a finite-size fractional transport equation rather than ordinary diffusion with a renormalized mean free path. In optical Lévy glasses, standard diffusion fits fail for coherent backscattering cones, whereas fractional-diffusion theory with a finite slab reproduces the cusped, non-diffusive line shape. This implies, conservatively, that an electronic Lévy glass should be expected to exhibit boundary-sensitive interference governed by a nonlocal propagator rather than by standard diffusive Cooperon intuition (Burresi et al., 2011).

A second recurring misconception is that heavy-tailed free-flight statistics alone determine the transport exponent. The optical transmission study explicitly demonstrates that diffusive scaling $\alpha\ge 2$10 can coexist with $\alpha\ge 2$11 for $\alpha\ge 2$12 because quenched geometry correlates successive steps (Groth et al., 2011). In the electronic context, this means that a power-law distribution of cluster sizes or impurity spacings does not, by itself, guarantee ideal Lévy-walk transport.

A third boundary concerns the word “glass.” The random-laser work on Nd$\alpha\ge 2$13:YBO$\alpha\ge 2$14 does not study electrons, does not realize a Lévy glass in the optical-transport sense, and does not demonstrate electronic Lévy transport. What it does show is a threshold-tuned crossover from Gaussian fluctuations to Lévy-stable intensity statistics and then back to Gaussian behavior, together with a replica-symmetry-breaking transition diagnosed by the overlap distribution

$\alpha\ge 2$15

In that photonic system, the Lévy interval $\alpha\ge 2$16 coincides with the narrow critical region of the spin-glass transition. This suggests a broader conceptual bridge in which broad-tailed fluctuations and glassy ordering arise from the same disordered nonlinear couplings, but the connection to electronic Lévy glasses is explicitly conceptual and statistical rather than literal (Gomes et al., 2015).

Taken together, these works define electronic Lévy glasses not as a single Hamiltonian class but as a family of electronic and mesoscopic-wave systems in which heavy-tailed geometric disorder, quenched correlations, and finite-size effects jointly control transport. The strongest direct realizations are the one-dimensional renewal-disorder localization models and the graphene nanoribbon platforms with power-law-distributed Rashba clusters. Their shared message is that Lévy-type structural disorder modifies transport at multiple levels: it can generate superdiffusive scaling, reshape localization laws, create transparent or anomalous energies, preserve interference in a nonlocal form, and, in spin-orbit graphene, strongly enhance charge-to-spin conversion (Sepehrinia, 2022, Fonseca et al., 2024, Fonseca et al., 14 May 2026).