Scale-Free Localization in Complex Systems

• Scale-Free Localization is a framework where wavefunctions or eigenmodes lack a fixed decay length, showing size-dependent exponential behaviors.
• It encompasses mechanisms from non-Hermitian boundary perturbations, impurity-induced effects, to topology-driven localization in scale-free networks.
• This concept underpins applications ranging from circuit experiments and Floquet systems to multi-scale, adaptive RSS-based device-free localization.

Searching arXiv for relevant papers on scale-free localization and closely related usages of the term. Scale-free localization denotes several distinct but technically precise notions of localization that appear in different research traditions. In the non-Hermitian lattice literature, it usually refers to states whose envelope remains exponential at finite size but whose decay length scales with system size, as in $\psi_x\sim e^{-Cx/L}$ or $\xi\propto L$, so the state has no intrinsic finite localization length (Li et al., 2020). In work on scale-free networks, the phrase instead refers to localization generated or pinned by heterogeneous graph structure, such as nonwandering covariant Lyapunov vectors or localized principal eigenvectors (Kuptsov et al., 2014). In RSS-based device-free localization, the expression is used in a different and explicitly nonliteral sense: localization is “scale-adaptive,” because each radio link has its own spatial impact area rather than sharing a single global support scale (Kaltiokallio et al., 2013).

1. Terminological scope

A common thread is the absence of a single universal localization scale, but the mathematical content differs sharply across fields.

Usage	Representative meaning	Defining scale
Non-Hermitian lattices	Boundary or impurity accumulation with $\xi\propto L$	System size
Scale-free networks	Localization tied to heterogeneous graph topology	Degree hierarchy or cluster structure
RSS device-free localization	Per-link adaptive spatial support	Fade-level- and sign-dependent measurement support

In the non-Hermitian setting, scale-free localization is not ordinary exponential localization and is also not standard NHSE. Several papers emphasize that the profile is still exponential at finite size, but with an $L$-dependent exponent, so the localization weakens as the system grows rather than retaining a fixed skin depth (Li et al., 2020). This usage also differs from critical multifractality: the states are generally discussed through $\xi\propto L$, not through power-law envelopes or multifractal scaling (Zhang et al., 20 Feb 2025).

A recurring misconception is that “scale-free” implies mathematical scale invariance in the strict sense. That is not generally how the term is used. In circuit experiments on defect-induced localization, for example, the defining observation is that profiles collapse after rescaling position by the total system size and that $\text{IPR}\propto 1/N$, even though the finite-size envelope is still fitted by $Ae^{-|x-x_0|/\xi}$ with $\xi\propto N$ (Xie et al., 2024). In RSS localization, the paper itself is best understood as replacing a constant-size ellipse by a multi-scale model, not as proving literal scale invariance (Kaltiokallio et al., 2013).

2. Boundary and impurity mechanisms in non-Hermitian lattices

The most developed meaning of scale-free localization arises in non-Hermitian one-dimensional lattices with tunable boundaries or local non-Hermitian impurities. In the boundary-impurity Hatano–Nelson chain, a single impurity strength $\mu$ interpolates between periodic and open chains and produces several regimes: conventional NHSE, Bloch-like delocalized states despite broken translational invariance, ordinary scale-free accumulation, and reversed scale-free accumulation against the non-reciprocal direction (Li et al., 2020). There the defining profile is

$\psi_x\sim e^{-Cx/L},$

so the localization length grows proportionally to system size. The same work stresses that this is not algebraic localization; it is an exponential with an $\xi\propto L$0-dependent exponent. It also identifies special scales such as $\xi\propto L$1, where the continuous states become spatially uniform on $\xi\propto L$2, and $\xi\propto L$3, where the system crosses into OBC-like NHSE behavior (Li et al., 2020).

A complementary line of work shows that a strictly local non-Hermitian perturbation in an otherwise Hermitian 1D lattice generically induces scale-free localization for continuous-spectrum states (Li et al., 2023). In that formulation, the bulk factor satisfies $\xi\propto L$4, so $\xi\propto L$5. If the impurity is placed a finite distance from the boundary, the scale-free states on that segment are promoted to exponentially localized modes, and the number of such localized modes is proportional to the impurity–boundary distance (Li et al., 2023). The same paper ties this mechanism directly to $\xi\propto L$6 symmetry: in the $\xi\propto L$7-exact phase the relevant states remain extended, while $\xi\propto L$8-broken states are necessarily scale-free or exponentially localized (Li et al., 2023).

A model-independent perturbative explanation is given by the general mechanism for the scale-free skin effect. Starting from extended PBC Bloch states, a finite-support boundary perturbation produces an energy correction of order $\xi\propto L$9. Because the bulk dispersion remains $\xi\propto L$0, that shift forces

$\xi\propto L$1

and therefore

$\xi\propto L$2

In this view, scale-free localization is a finite-size boundary-perturbation effect on extended PBC states, not an OBC deformation of ordinary skin modes (Wang, 2 Apr 2026). The same paper is explicit about its scope: it does not explain the Hermitian chain with a purely imaginary edge impurity, because its perturbative framework assumes non-degenerate PBC bands (Wang, 2 Apr 2026).

3. Quasiperiodicity, Anderson localization, and finite-size crossover

Once quasiperiodicity or disorder is added, scale-free localization becomes a crossover phenomenon between boundary-controlled and bulk-controlled localization. In the unidirectional quasiperiodic lattice with generalized boundary conditions, the bulk Lyapunov exponent and boundary equation can be solved exactly. The weak-disorder scale-free regime is defined by

$\xi\propto L$3

valid for $\xi\propto L$4. The strong-disorder Anderson-localized regime instead has

$\xi\propto L$5

with exact phase boundary

$\xi\propto L$6

The paper therefore distinguishes a boundary-controlled scale-free regime from a bulk-controlled Anderson-localized regime, and even constructs a mosaic generalization with exact energy edges separating the two (Zhang et al., 20 Feb 2025).

In the non-Hermitian Aubry–André/Harper chain with a tunable impurity bridge, scale-free localization is explicitly identified as a finite-size effect. The localization length obeys $\xi\propto L$7, while the thermodynamic-limit bulk threshold remains

$\xi\propto L$8

Quasiperiodicity destroys the scale-free regime in two distinct ways. Far from PBC and close to effective OBC, increasing $\xi\propto L$9 drives

$L$0

Near PBC, increasing $L$1 instead yields

$L$2

The paper introduces the non-normality ratio $L$3 to separate these regimes and concludes that the quasiperiodicity-induced NHSE window collapses to $L$4 in the thermodynamic limit because SFL itself is finite-size in nature (Saito et al., 11 Feb 2026).

Disorder added to a single-impurity non-Hermitian chain produces a related but distinct size dependence. In that model, clean periodic-boundary states in the $L$5-broken regime are scale-free localized with $L$6. Random onsite disorder converts them into ordinary Anderson-localized states, but the critical disorder strength is itself size dependent: $L$7 For $L$8, the reported fit is $L$9 (Yılmaz et al., 2024). The contrast with the Hatano–Nelson model is central: skin states there have size-independent localization length, whereas scale-free states make the SFL–Anderson crossover itself size dependent (Yılmaz et al., 2024).

A closely related circuit realization combines weak disorder, bulk nonreciprocal hopping, and a single non-Hermitian impurity in a Hatano–Nelson chain. The measured profiles collapse versus normalized position $\xi\propto L$0, the fitted localization length scales as $\xi\propto L$1, and the accumulation direction can be opposite to the bulk hopping direction because it is controlled by the impurity rather than by bulk asymmetry alone (Wang et al., 15 Jan 2025).

4. Floquet, dissipative, coupled-ladder, and experimental realizations

Direct experimental observation of scale-free localized states was first reported in electric circuit lattices with local non-Hermitian defects. For a finite 1D Hermitian chain with a single lossy defect, the circuit measurements showed an extensive number of scale-free localized states with $\xi\propto L$2 and $\xi\propto L$3. For the device parameters used in the single-defect realization, the critical frequency was

$\xi\propto L$4

above which the extra state is scale-free and below which one conventional localized state appears (Xie et al., 2024). In the companion $\xi\propto L$5-symmetric defect model, all states are extended in the $\xi\propto L$6-unbroken phase, scale-free localized states emerge only after

$\xi\propto L$7

and two additional conventional localized states appear for stronger non-Hermiticity satisfying $\xi\propto L$8 (Xie et al., 2024).

Floquet driving supplies another mechanism. A two-step drive can produce a Floquet Hamiltonian that is Hermitian under PBC but acquires non-Hermitian boundary terms under OBC because $\xi\propto L$9 and $\text{IPR}\propto 1/N$0 fail to commute once the end links are removed. The $\text{IPR}\propto 1/N$1-breaking threshold occurs when the quasienergy bandwidth spans the full Floquet Brillouin zone,

$\text{IPR}\propto 1/N$2

and in the broken phase the imaginary quasienergy scales as $\text{IPR}\propto 1/N$3, implying $\text{IPR}\propto 1/N$4 and

$\text{IPR}\propto 1/N$5

The resulting localization is therefore scale free rather than a conventional NHSE with $\text{IPR}\propto 1/N$6-independent skin depth (Li et al., 24 Mar 2026).

Dissipative lattices add a dynamical consequence. In the cross-stitch model mapped to an effective non-Hermitian SSH chain, a local impurity acts as a tunable effective boundary. For finite $\text{IPR}\propto 1/N$7, the eigenstates exhibit anomalous scale-free localization with energy-dependent Lyapunov exponent

$\text{IPR}\propto 1/N$8

so the localization strength varies from state to state (Liu et al., 20 May 2026). The same paper shows that this anomalous scale-free structure generates an impurity-induced loss burst localized at the impurity site and its adjacent effective-boundary site, $\text{IPR}\propto 1/N$9, and that the effect occurs without imaginary-gap closing (Liu et al., 20 May 2026).

Weakly coupled non-Hermitian ladders and lossy dual-chain systems extend the picture further. In weakly coupled lossy lattices with bipolar NHSE in the decoupled limit, the coupling destroys bipolar skin localization and replaces it with bipolar scale-free localization, while a pronounced edge burst still appears in the local decay probabilities. The authors use this to argue that NHSE is not a necessary condition for the edge burst effect (Sen et al., 10 Jun 2025). Under Möbius boundary conditions, two weakly coupled Hatano–Nelson chains realize an even richer coexistence: the dominant chain of a given eigenstate exhibits SFL with $Ae^{-|x-x_0|/\xi}$0, while the less-occupied chain exhibits ordinary NHSE, and the chain carrying SFL can exchange with eigenenergy (Long et al., 8 Jul 2025).

5. Localization on scale-free networks and graph-induced state concentration

In network science, “scale-free localization” refers not to $Ae^{-|x-x_0|/\xi}$1 but to localization generated or pinned by scale-free topology. The most rigid example is the localization of covariant Lyapunov vectors in random scale-free trees of coupled Hénon maps. There, localization is both nonwandering and predictable: one family of CLVs is exactly confined to transverse invariant subspaces of fully synchronized clusters, while another family localizes preferentially on low-degree nodes that remain outside the two large phase-synchronized clusters. These localization sites stay fixed in time, and most of the nodes carrying the second mechanism have degree $Ae^{-|x-x_0|/\xi}$2 (Kuptsov et al., 2014).

A different network usage appears in the quenched-mean-field analysis of SIS dynamics on heterogeneous graphs. There the principal eigenvector of the quenched operator localizes for sufficiently strong topological heterogeneity, and the paper identifies a localization transition on scale-free networks at degree exponent $Ae^{-|x-x_0|/\xi}$3. For $Ae^{-|x-x_0|/\xi}$4 the principal eigenvector is delocalized, for $Ae^{-|x-x_0|/\xi}$5 the inverse participation ratio tends to a finite constant, and at $Ae^{-|x-x_0|/\xi}$6 the IPR distribution becomes broad and crossover-like rather than sharply critical (Ódor, 2014).

The adjacency-eigenvector approach to complex networks gives yet another formulation. Using the representative eigenvector closest to the band center, the paper “Localizations on Complex Networks” studies Barabási–Albert scale-free networks through participation ratio, structural entropy, nearest-neighbor level statistics, and multifractal analysis of rank-ordered node probabilities. It concludes that the localization properties of BA networks are captured by critical states with extremely significantly small values of $Ae^{-|x-x_0|/\xi}$7, that the Brody parameter saturates near $Ae^{-|x-x_0|/\xi}$8 rather than reaching the Wigner value $Ae^{-|x-x_0|/\xi}$9, and that the ranked probability series is generally multifractal (Zhu et al., 2011).

For deterministic scale-free trees, the localization problem becomes exactly recursive. In the normalized tight-binding model on the tree $\xi\propto N$0, the spectrum obeys the renormalization map

$\xi\propto N$1

and the inverse participation ratio is expressed analytically in terms of the associated eigenvalue chain. For constant chain tails, the asymptotic scaling is

$\xi\propto N$2

while more general periodic tails yield exponents determined by the corresponding chain products (Xie et al., 2017). This exact construction makes localization a property of spectral ancestry on a scale-free tree rather than of Euclidean distance.

An adjacent but distinct usage appears in inverse-designed disordered materials. There the strongest scale-free claim is not that the eigenmodes themselves obey a scale-free law, but that machine-learned disorder realizations that reproduce target localization spectra acquire heavy-tailed, scale-free structural statistics with hub atoms (Yu et al., 2020). This suggests a broader semantic drift in which “scale-free localization” can also refer to localization supported by scale-free structure rather than to a scale-free wavefunction envelope.

6. Scale-adaptive RSS-based device-free localization

In RSS-based device-free localization, “scale-free localization” is best read as scale-adaptive localization. Classical radio tomographic imaging assumes that each wireless link is affected inside a fixed-width ellipse with a single global excess path-length parameter $\xi\propto N$3. The multi-scale spatial model replaces that constant-size assumption by link-, channel-, and sign-dependent ellipses whose size is determined by fade level

$\xi\propto N$4

and by whether RSS increases or decreases (Kaltiokallio et al., 2013). The sign-specific ellipse widths are fitted as

$\xi\propto N$5

and the measurement model converts RSS changes into inside-ellipse probabilities

$\xi\propto N$6

The system then reconstructs the RTI image from a sign-separated, multi-channel weight matrix $\xi\propto N$7 rather than from the classical constant-ellipse matrix (Kaltiokallio et al., 2013).

The substantive point is that the spatial support of each measurement is not fixed. Anti-fade links and deep-fade links respond differently, positive and negative RSS excursions have different spatial scales, and deep-fade channels are not discarded but used with larger or differently calibrated support regions (Kaltiokallio et al., 2013). In that sense the method avoids committing to one global spatial scale, but it is not mathematically scale invariant.

Experimentally, the model was derived from over 26 million RSS measurements and tested in an open indoor environment, a one-bedroom apartment, and a through-wall scenario. The reported average tracking error in the open environment was $\xi\propto N$8 for multi-scale RTI, compared with $\xi\propto N$9 for fade-level-ranked RTI and $\mu$0 for channel-diversity RTI. In the apartment, the combined average error was $\mu$1, compared with $\mu$2 and $\mu$3. In the through-wall experiment, the corresponding averages were $\mu$4, $\mu$5, and $\mu$6, and the parameters learned in the first two environments were reused without retuning in the third (Kaltiokallio et al., 2013). The expression “scale-free localization” in this context therefore names a non-constant-scale, multi-scale, link-adaptive localization framework rather than the boundary-sensitive finite-size localization studied in non-Hermitian lattice physics.