Scale-Free Localization in Non-Hermitian Systems

• Scale-free localization (SFL) is defined by a localization length proportional to the system size, leading to boundary- or impurity-centered accumulation.
• SFL arises when boundary perturbations impose a scaling eigenstate envelope that contrasts with the size-independent localization found in NHSE and Anderson models.
• Experimental and theoretical studies using modified Hatano–Nelson chains and circuit implementations validate SFL’s finite-size scaling and boundary sensitivity.

Scale-free localization (SFL) denotes a localization regime in which the characteristic decay length grows proportionally with system size, so that finite systems display boundary- or impurity-centered accumulation without a size-independent localization length. In the contemporary non-Hermitian literature, this is commonly expressed as $\xi(L)\propto L$ or, equivalently, $\kappa(L)\propto 1/L$, with spatial envelopes that remain invariant when plotted against a normalized coordinate (Li et al., 2020, Yılmaz et al., 2024, Wang, 2 Apr 2026). This behavior is distinct from the conventional non-Hermitian skin effect (NHSE), where the localization length is size-independent, and from Anderson localization, where disorder sets a finite localization length independent of $L$ (Yılmaz et al., 2024). The term has also been used in network science for localization phenomena tied to scale-free topology, but the most developed modern usage concerns non-Hermitian, boundary-sensitive lattice systems [(Ódor, 2014); (Kuptsov et al., 2014)].

1. Conceptual definition and distinguishing features

A standard parametrization of SFL is

$$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$

so that a mode envelope takes the form $|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$ and therefore depends on the intensive coordinate $x/L$ rather than on an intrinsic decay length (Yılmaz et al., 2024). Closely related formulations appear in non-Bloch analyses, where the generalized Bloch factor satisfies $|\beta|=1-\kappa/L+O(1/L^2)$ or $|\beta|\approx \exp(A/L)$, yielding $\xi\approx L/|A|$ (Wang, 2 Apr 2026).

This immediately separates SFL from three more familiar regimes. In standard NHSE, the decay length is determined by microscopic nonreciprocity and remains finite as $L\to\infty$; in the Hatano–Nelson model, for example, skin states have $\kappa(L)\propto 1/L$0 (Yılmaz et al., 2024). In Anderson localization, the envelope is exponential but the decay length is controlled by disorder strength rather than by boundary coupling or system size (Yılmaz et al., 2024). In ordinary extended states, the effective localization length diverges and no boundary accumulation persists.

Several authors use the term “scale-free skin effect” (SFSE) for the same underlying phenomenon: boundary localization with $\kappa(L)\propto 1/L$1 and spectra that converge to the periodic-boundary-condition (PBC) spectrum in the thermodynamic limit (Wang, 2 Apr 2026). The central point is therefore not algebraic decay, but exponential localization with a decay exponent that vanishes as $\kappa(L)\propto 1/L$2.

2. Boundary impurities, local non-Hermiticity, and reversed accumulation

One of the clearest SFL constructions is the Hatano–Nelson chain with a single modified boundary coupling,

$\kappa(L)\propto 1/L$3

with $\kappa(L)\propto 1/L$4, where $\kappa(L)\propto 1/L$5 gives PBC and $\kappa(L)\propto 1/L$6 gives OBC (Li et al., 2020). In the strong-impurity branch, the continuous eigenstates take the form

$\kappa(L)\propto 1/L$7

so the decay exponent is explicitly proportional to $\kappa(L)\propto 1/L$8 (Li et al., 2020). The sign of $\kappa(L)\propto 1/L$9 determines the accumulation direction: $L$0 gives ordinary scale-free accumulation along the non-reciprocal direction, while $L$1 gives reversed scale-free accumulation against it; at $L$2, the continuous spectrum coincides with the PBC spectrum and the eigenstates are spatially uniform except for vanishing amplitude at the impurity site (Li et al., 2020).

A complementary generic mechanism starts from an otherwise Hermitian lattice with a local non-Hermitian perturbation. In that setting, the bulk ansatz still satisfies $L$3, but the boundary and defect matching conditions force the continuous-spectrum solutions to obey $L$4, which produces scale-free envelopes $L$5 and imaginary parts of eigenenergies scaling as $L$6 (Li et al., 2023). When the perturbation is moved a finite distance $L$7 from the boundary, the scale-free modes are promoted to exponentially localized modes, and the number of such bound states is exactly $L$8 (Li et al., 2023).

A third boundary-based formulation uses a single non-Hermitian bond in an otherwise Hermitian ring. There the asymmetric bond imposes a loop constraint

$L$9

so that

$$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$0

Under PBC in the PT-broken regime this produces SFL, whereas under OBC the same model has real spectra and no SFL (Yılmaz et al., 2024).

3. Spectral, non-Bloch, and finite-size characterizations

A recurring interpretation of SFL treats generalized boundary conditions as a finite-rank perturbation of a periodic chain. If a PBC eigenstate acquires an energy correction $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$1, then the generalized Bloch factor shifts from $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$2 to

$$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$3

so that $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$4 and $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$5 (Wang, 2 Apr 2026). In this picture, SFL is a finite-size boundary-sensitivity effect: the spectrum approaches the PBC spectrum as $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$6, but finite systems exhibit clear boundary accumulation.

The same logic underlies the formula-agnostic characterization of SFSE as the regime in which the boundary modification forces $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$7 to stay near the unit circle with only $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$8 corrections (Wang, 2 Apr 2026). This also explains why SFL can arise even without point-gap winding: the mechanism is boundary driven rather than a direct consequence of the bulk non-Bloch deformation that produces conventional NHSE (Wang, 2 Apr 2026).

A more explicit finite-size diagnostic was introduced for the non-reciprocal Aubry–André–Harper chain with a tunable impurity bond. If $$\xi(L)=\alpha L \quad \Leftrightarrow \quad \kappa(L)=\frac{1}{\alpha L},$$9 denotes the matrix of right eigenvectors, the condition number

$|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$0

and the boundary-sensitive ratio

$|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$1

distinguish SFL from NHSE and from extended states (Saito et al., 11 Feb 2026). Under an exponential-boundary-wavefunction assumption, $|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$2; hence NHSE yields exponential growth in $|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$3, while SFL yields only subexponential, numerically logarithmic growth because $|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$4 (Saito et al., 11 Feb 2026). Real-space fitting in the same work uses

$|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$5

thereby separating scale-free, skin, and bulk-localized contributions within a single eigenstate profile (Saito et al., 11 Feb 2026).

4. Disorder, quasiperiodicity, and localization transitions

Disorder can convert SFL into ordinary Anderson localization, but in a way that differs sharply from the Hatano–Nelson case. In the single-impurity model

$|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$6

with $|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$7 uniformly distributed in $|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$8, the clean PT-broken phase supports SFL under PBC, including the special point $|\psi|\propto e^{-x/\xi(L)}=e^{-x/(\alpha L)}$9 where all eigenstates are SFL (Yılmaz et al., 2024). Adding disorder drives a size-dependent Anderson transition with

$x/L$0

so that $x/L$1 in the thermodynamic limit (Yılmaz et al., 2024). This is the opposite of the usual NHSE scenario, where the Anderson threshold is typically size independent.

Quasiperiodic lattices provide a second transition mechanism. In the unidirectional model

$x/L$2

the boundary equation reduces to

$x/L$3

This yields a scale-free regime when the boundary fixes the Lyapunov exponent to

$x/L$4

and an Anderson-localized regime when the bulk fixes $x/L$5 (Zhang et al., 20 Feb 2025). The threshold is

$x/L$6

Below it, the spectrum is complex and lies on an $x/L$7-controlled ellipse; above it, the eigenvalues are real and boundary sensitivity becomes exponentially small (Zhang et al., 20 Feb 2025).

A different quasiperiodic effect appears in the non-reciprocal lattice with a tunable impurity bond,

$x/L$8

with $x/L$9 and bulk transition point $|\beta|=1-\kappa/L+O(1/L^2)$0 (Saito et al., 11 Feb 2026). There, quasiperiodicity destroys the SFL regime and drives it either into NHSE or into an extended regime, depending on the generalized boundary condition. For fixed $|\beta|=1-\kappa/L+O(1/L^2)$1 and $|\beta|=1-\kappa/L+O(1/L^2)$2, the crossover length at $|\beta|=1-\kappa/L+O(1/L^2)$3 is $|\beta|=1-\kappa/L+O(1/L^2)$4, and for $|\beta|=1-\kappa/L+O(1/L^2)$5 the SFL–NHSE crossover scales as $|\beta|=1-\kappa/L+O(1/L^2)$6 (Saito et al., 11 Feb 2026).

5. Dynamical consequences, anomalous variants, and experimental realization

SFL is not confined to static band structure. In a dissipative cross-stitch lattice mapped to an effective non-Hermitian Su–Schrieffer–Heeger model, the impurity-generated localization is anomalous because the Lyapunov exponent depends explicitly on the eigenenergy:

$|\beta|=1-\kappa/L+O(1/L^2)$7

This contrasts with conventional impurity-induced SFL, where all eigenstates share the energy-independent exponent

$|\beta|=1-\kappa/L+O(1/L^2)$8

(Liu et al., 20 May 2026). The anomalous eigenmode structure produces an impurity-induced loss burst: for a single impurity at $|\beta|=1-\kappa/L+O(1/L^2)$9, the burst region is $|\beta|\approx \exp(A/L)$0, and the effect occurs without imaginary-gap closing (Liu et al., 20 May 2026).

A related dynamical conclusion emerges from a lossy ladder built from two weakly coupled non-Hermitian chains. Weak inter-chain coupling destroys the skin modes of the uncoupled chains and replaces them by bipolar SFL, yet the system still exhibits an edge burst effect in the local decay distribution (Sen et al., 10 Jun 2025). This shows that NHSE is not a necessary condition for boundary-concentrated loss: non-Hermitian funneling plus scale-free boundary localization is sufficient (Sen et al., 10 Jun 2025).

Floquet systems furnish another route. In a two-step drive with shift operators $|\beta|\approx \exp(A/L)$1 and $|\beta|\approx \exp(A/L)$2,

$|\beta|\approx \exp(A/L)$3

the Floquet Hamiltonian is Hermitian under PBC but acquires non-Hermitian boundary terms under OBC through the Baker–Campbell–Hausdorff expansion,

$|\beta|\approx \exp(A/L)$4

with

$|\beta|\approx \exp(A/L)$5

PT symmetry breaks when the PBC bandwidth spans the full frequency Brillouin zone, $|\beta|\approx \exp(A/L)$6, and the PT-broken phase exhibits SFL with $|\beta|\approx \exp(A/L)$7, $|\beta|\approx \exp(A/L)$8, and

$|\beta|\approx \exp(A/L)$9

(Li et al., 24 Mar 2026).

Experimental realization has been achieved most directly in electrical circuits. One circuit implementation realizes a disordered Hatano–Nelson ring with a single non-Hermitian impurity bond,

$\xi\approx L/|A|$0

and observes anomalous scale-free accumulation opposite to the bulk hopping direction (Wang et al., 15 Jan 2025). The measured localization length scales linearly with $\xi\approx L/|A|$1, and the circuit Laplacian shares right and left eigenvectors with the effective non-Hermitian Hamiltonian, allowing direct reconstruction of spatial mode profiles from resonance voltages (Wang et al., 15 Jan 2025). Earlier circuit work had already proposed topolectrical detection of ordinary and reversed scale-free accumulation in the impurity-modified Hatano–Nelson chain (Li et al., 2020).

6. Other uses of the term and terminological ambiguity

Outside non-Hermitian lattice physics, “scale-free localization” has been used for localization phenomena associated with scale-free networks. In the susceptible–infected–susceptible model on networks, quenched mean-field theory identifies a localization transition at degree exponent $\xi\approx L/|A|$2: for $\xi\approx L/|A|$3, the inverse participation ratio

$\xi\approx L/|A|$4

remains finite, whereas for $\xi\approx L/|A|$5 it tends to zero (Ódor, 2014). That work connects localization of the principal eigenvector to rare-region effects and Griffiths-phase-like slow dynamics on heterogeneous networks (Ódor, 2014).

A different network usage appears in covariant Lyapunov vectors on scale-free networks of Hénon maps. There, localization is pinned to specific nodes by full and phase cluster synchronization, is nonwandering, and is predictable from dynamical and topological diagnostics in the parameter window $\xi\approx L/|A|$6 (Kuptsov et al., 2014). Related structural studies of adjacency eigenvectors in Barabási–Albert networks report critical-like probability distributions of amplitudes, normalized participation ratio

$\xi\approx L/|A|$7

and multifractal ranked amplitude series (Zhu et al., 2011).

The abbreviation “SFL” is also established in software engineering for Statistical Fault Localization, where it denotes ranking program entities by their correlation with failing tests; that usage is explicitly unrelated to scale-free localization in physics (Smytzek et al., 29 Jun 2026). The coexistence of these meanings makes context essential. Within non-Hermitian condensed-matter and wave-physics research, however, SFL now refers specifically to localization with $\xi\approx L/|A|$8, boundary-sensitive spectra, and finite-size scaling distinct from both NHSE and Anderson localization (Yılmaz et al., 2024, Wang, 2 Apr 2026).