Radial Disorder: Definitions, Contexts, and Implications

• Radial disorder is a field-dependent concept defined by disorder indexed through a radial variable, manifesting in reciprocal space, physical dimensions, or shell indices.
• It is distinct from angular disorder, as systems often retain sharp radial features while losing orientational coherence in contexts like skyrmion lattices and amorphous elasticity.
• Applications span from enhancing light absorption in nanowire photovoltaics to inducing multifractal states in complex graphs, highlighting its design and diagnostic utility.

Searching arXiv for papers on “radial disorder” and closely related usage across domains. Radial disorder is a field-dependent term used for disorder defined with respect to a radial variable, a radial coordinate, or radial symmetry. In the cited literature it does not denote a single universal mechanism. Instead, it can mean disorder in the magnitude of a scattering wavevector in reciprocal space, disorder in nanowire radii, disorder in interatomic distances encoded by a radial distribution function, rotationally invariant disorder in the complex-energy plane, shell-correlated onsite energies on rooted graphs, or symmetry breaking generated by disordered solids under purely radial loading. Across these contexts, the common structure is that the disordered degree of freedom is indexed by a radius-like quantity, while the physical consequences depend strongly on whether the disorder affects translational order, orientational order, localization, optical resonances, or elastic response (Dhital et al., 2018, Sturmberg et al., 2014, Dangić et al., 2022, Longhi, 2021, Logan et al., 21 Aug 2025).

1. Terminological scope and common structure

The literature uses “radial disorder” in several technically distinct senses. The table summarizes the principal meanings represented in current arXiv usage.

Context	Radial variable	Meaning
Skyrmion-lattice SANS	$|\mathbf{q}|$	Width or sharpness of the scattering radius
Nanowire photovoltaics	nanowire radius $a$	Arrays containing multiple wire radii
GeTe phase transition	bond distance $r$	Disorder in interatomic distances via the RDF
Non-Hermitian lattices	$R=|V|$	Radially symmetric disorder in the complex plane
Rooted graphs	graph distance $r$	Same onsite energy within each shell
Radial loading of amorphous solids	radial boundary displacement	Disorder-induced excitation of angular modes

What unifies these usages is not a shared microscopic model but a shared geometrical indexing. The “radial” quantity may be a reciprocal-space radius, a physical size, a bond length, a modulus in the complex plane, a shell index on a graph, or a cylindrical coordinate in elasticity. A plausible implication is that the phrase is best treated as a family of domain-specific constructions rather than a single concept with fixed semantics.

A second cross-cutting feature is that radial disorder is often contrasted with angular or orientational disorder. In chiral magnets, radial order can remain sharp while orientational order is lost; in shell-disordered graphs, amplitudes localize along the radial coordinate while remaining distributed over exponentially many sites within a shell; in amorphous elasticity, a purely radial load can generate non-radial angular response through disorder-enabled mode coupling (Dhital et al., 2018, Logan et al., 21 Aug 2025, Kumar et al., 2023).

2. Reciprocal-space radial disorder in chiral magnets

In the skyrmion-lattice literature on MnSi-based B20 chiral magnets, radial disorder is defined in reciprocal space. The relevant observable is the magnitude of the scattering wavevector in SANS, not the azimuthal distribution of intensity. For a skyrmion lattice with spacing $a$, the characteristic scattering wavevector is $q = 2\pi/a$, close to the helical wavevector $Q$; “sharp radial order” means a narrow peak in intensity versus $q=|\mathbf{q}|$, with small radial full width at half maximum. In MnSi$_{0.992}$Ga$a$0, the helix wavevector remains $a$1, and the radial FWHM of the skyrmion-lattice peak or ring is approximately the instrument resolution, about $a$2, with little variation near the conical–SKL boundary (Dhital et al., 2018).

This usage is important because the paper emphasizes that substitutional disorder from Ga does not primarily create radial disorder. Instead, it weakens pinning of the skyrmion lattice to the crystalline lattice, so that the system can display a ring-like SANS pattern with sharp radial width but compromised azimuthal order. The quoted interpretation is that the ring-like structure indicates “the lack of an orientationally ordered SKL with the retention of translational order,” and the small variation in radial width excludes formation of a skyrmion glass phase. The experimentally relevant distinction is therefore between preserved radial order and lost long-range orientational coherence (Dhital et al., 2018).

The same study extends the discussion to helical domains in MnSi$a$3Ga$a$4 and Mn$a$5Ir$a$6Si. Substitution of heavier elements on either Mn or Si sites creates a higher energy barrier for reorientation of helical order and domain formation. Here again, the main disorder effect is not broadening of the radial wavevector magnitude but altered angular/domain population under field cycling. A common misconception in this setting is to equate a ring in SANS with radial disorder; the paper argues instead that the ring can represent preserved short-range translational order together with orientational or domain disorder (Dhital et al., 2018).

3. Size, distance, and positional radial disorder

In nanowire-array photovoltaics, radial disorder has a literal geometric meaning: disorder in the radii of nanowires within an otherwise periodic array. The lattice period $a$7, height, and material are fixed, while distinct sublattices or randomly selected wires have different radii $a$8. For two sublattices at fixed fill fraction, the paper writes

$a$9

The physical mechanism is spectral superposition: different radii support different Key Modes with different cut-off wavelengths, and the absorptance of a multi-radius array is approximately the sum of the sublattice spectra. Quantitatively, a homogeneous film of thickness $r$0 nm has ultimate efficiency $r$1, a uniform-radius array at $r$2 nm and $r$3 reaches $r$4, a designed four-radius array reaches $r$5, and a designed sixteen-radius array reaches $r$6 with $r$7 Si, nearly $r$8 percentage points above the homogeneous film (Sturmberg et al., 2014).

In GeTe, radial disorder refers instead to disorder in interatomic distances, especially Ge–Te nearest-neighbor bond lengths, as represented by the radial distribution function

$r$9

The key result is that the nearest-neighbor RDF peak remains strongly non-Gaussian above the simulated critical temperature $R=|V|$0 K and is better fitted by two Gaussians than by one. However, the paper argues that this asymmetry is not necessarily evidence for an order–disorder ferroelectric transition. Similar non-Gaussian nearest-neighbor RDFs also occur in PbTe and MgO, and the GeTe study concludes that strong anharmonicity alone can produce distorted bond-length distributions. The displacement distributions of individual Ge and Te atoms at $R=|V|$1 K are Gaussian and centered at zero, supporting a predominantly dynamical, anharmonic origin of the radial disorder rather than persistent static off-centering (Dangić et al., 2022).

A third usage appears in mass–spring networks, where positional disorder is generated by randomly displacing lattice sites,

$R=|V|$2

with $R=|V|$3 and $R=|V|$4. This preserves coordination number $R=|V|$5 but destroys positional order. In that setting, the boson peak develops from the transverse van Hove singularity, and with increasing positional disorder the boson-peak frequency $R=|V|$6, transverse Ioffe–Regel frequency $R=|V|$7, and longitudinal Ioffe–Regel frequency $R=|V|$8 all decrease. For positional disorder alone, the study finds $R=|V|$9 and $r$0, showing that positional or “radial” disorder by itself does not make the boson peak equivalent to the transverse Ioffe–Regel limit (Nie et al., 2017).

4. Radially correlated disorder in polymers, complex spectra, and high-dimensional graphs

In polymer theory, radial disorder appears as spatially correlated structural disorder whose correlation function decays with distance as

$r$1

For star-branched polymers in a good solvent, this is analyzed with direct polymer renormalization in the double expansion $r$2, $r$3. The central experimentally measurable quantity is

$r$4

The paper finds that $r$5 increases with increasing $r$6, meaning that stronger disorder correlations increase the size of a star relative to a linear polymer of the same molecular weight. Here “radial” refers to distance-dependent disorder correlations in the medium, not to a radial coordinate intrinsic to the polymer itself (Blavatska et al., 2012).

In the non-Hermitian Hatano–Nelson model with unidirectional hopping, radial disorder denotes a rotationally invariant distribution of complex onsite potentials,

$r$7

with uniformly distributed phase $r$8 and bounded support $r$9. For a uniform radial distribution inside a disk, the paper derives a disorder-driven sequence of spectral phases. Under OBC the spectrum exactly reproduces the onsite disorder realization. Under PBC, if $a$0, the extended spectrum is locked to the clean circle of radius $a$1 and is fully insensitive to disorder; for intermediate disorder $a$2 the spectrum consists of a shrinking loop plus localized OBC energies; and for strong disorder $a$3 the PBC spectrum collapses completely onto the OBC spectrum. For the uniform disk distribution, the bulk-localization threshold is $a$4, and the winding number at base point $a$5 changes from $a$6 to $a$7 across the transition (Longhi, 2021).

A distinct but related notion is introduced for rooted high-dimensional graphs, where radial disorder means shell-wise correlated onsite energies,

$a$8

All sites at graph distance $a$9 from the root carry the same onsite energy, while different shells are independent. On both Cayley trees and hypercubes this induces an exact fragmentation into effective one-dimensional chains in a symmetrized or Krylov basis. Along the chain, disorder is conventional and produces exponential localization, but in the original graph basis the exponentially growing shell multiplicity yields multifractal states. The paper shows that the inverse participation ratios have an exceptionally broad distribution, the mean IPR scales only as $q = 2\pi/a$0, and the typical IPR scales as $q = 2\pi/a$1, making the typical rather than the mean IPR the appropriate multifractality diagnostic (Logan et al., 21 Aug 2025).

5. Radial fields, radial loading, and apparent radial order

In athermal crystals, radial disorder is realized microscopically as the displacement field generated by an isotropic defect. The disorder variable is local particle size, expanded as $q = 2\pi/a$2, and the response is developed order by order in a perturbation expansion around the crystalline ground state. For a single enlarged particle in a triangular lattice of soft disks, the far-field displacement generated by the defect is radial and obeys

$q = 2\pi/a$3

The first-order continuum-limit displacement has the explicit form

$q = 2\pi/a$4

and higher-order corrections are reported to be self-similar in spatial form. In this usage, radial disorder means a defect-induced field with radial decay and radial symmetry, not stochastic disorder in a radial coordinate (Acharya et al., 2021).

A different elastic meaning appears in amorphous solids subjected to infinitesimal purely radial loading in annular geometry. The inner radius is inflated by $q = 2\pi/a$5 with $q = 2\pi/a$6, and classical homogeneous elasticity would predict a purely radial displacement

$q = 2\pi/a$7

Instead, the simulations show strong symmetry breaking: radial loading excites non-radial Michell modes through disorder-induced nonlinear mode coupling. The response remains elastic and reversible, with no plasticity involved, yet low-order angular components $q = 2\pi/a$8–$q = 2\pi/a$9 are quantitatively fit by Michell’s classical solution for the biharmonic elastic problem. This suggests that under radial loading, structural disorder acts as a source of angular mode conversion even at minute strain (Kumar et al., 2023).

A third example concerns polarization in young shell-type supernova remnants. There, “radial disorder” refers to the case in which a completely turbulent magnetic field appears radially ordered in polarization maps because quasi-parallel acceleration concentrates cosmic-ray electrons where the local magnetic field is radial to the shock normal: $Q$0 The observed radial pattern therefore need not imply an intrinsically radial field. The paper proposes fractional polarization and dust polarization as diagnostics for distinguishing genuinely radial order from a turbulent field that only appears radial due to cosmic-ray-electron selection effects (1711.02176).

6. Recurring distinctions, misconceptions, and controversies

Several recurrent distinctions organize the modern usage of radial disorder. The first is radial versus angular disorder. In the skyrmion-lattice case, broad angular scattering does not imply radial disorder if the scattering ring retains a sharp radius; conversely, loss of sixfold Bragg spots can coexist with preserved local translational order. In amorphous elasticity, a purely radial boundary condition does not prevent the emergence of angular modes once disorder is present. In supernova-remnant polarization, an apparently radial pattern may be generated by selection effects acting on a disordered field rather than by true radial magnetic order (Dhital et al., 2018, Kumar et al., 2023, 1711.02176).

The second is radial disorder versus disorder-driven enhancement. In nanowire photovoltaics, radial disorder in wire radii is not treated as a fabrication nuisance but as a design degree of freedom that broadens absorption by combining multiple narrow resonances. In rooted graphs, shell-wise radial disorder does not merely localize states; it induces robust multifractality because one-dimensional localization along shells competes with exponential growth of shell multiplicity. In the Hatano–Nelson problem, radial symmetry of complex disorder yields analytically solvable spectral locking and a disorder-driven topological transition (Sturmberg et al., 2014, Logan et al., 21 Aug 2025, Longhi, 2021).

The third is diagnostic ambiguity. A non-Gaussian RDF in GeTe does not by itself establish an order–disorder phase transition; the authors argue that strong anharmonicity can generate the same radial-distribution signature. In mass–spring networks, the claim that the boson peak is equivalent to the transverse Ioffe–Regel limit is found not to be general; both local coordination number and positional disorder are required for that equivalence to hold. In supernova remnants, radial polarization vectors do not prove intrinsic radial field geometry. These cases collectively suggest that radial observables often require complementary probes before they can be assigned a unique microscopic interpretation (Dangić et al., 2022, Nie et al., 2017, 1711.02176).

Taken together, these studies establish radial disorder as a broadly useful but intrinsically context-sensitive concept. Its technical meaning depends on which radial variable is disordered, whether the disorder is local or shell-correlated, whether it acts in real space or reciprocal space, and whether the physically relevant distinction is between radial and angular order, between mean and typical observables, or between intrinsic radial structure and radial-looking response.