Anisotropic Scaling Localization (ASL)

• Anisotropic-Scaling Localization (ASL) is a higher-dimensional non-Hermitian boundary localization phenomenon defined by decay lengths that scale anisotropically with system geometry.
• It involves mechanisms such as effective bulk couplings and edge junctions, leading to scaling laws like ξₓ ∝ Lₓ/L_y in 2D systems and similar patterns in 3D.
• ASL is distinct from ordinary localization and the non-Hermitian skin effect, as evidenced by its geometry-sensitive scaling, complex eigenenergy spectra, and crossover behaviors.

Searching arXiv for papers on anisotropic-scaling localization and closely related non-Hermitian localization. arxiv_search(query="all:\"anisotropic-scaling localization\" OR all:\"anisotropic scaling localization\" OR all:\"higher-dimensional non-Hermitian systems\" OR all:\"anisotropic Anderson localization\" OR all:\"non-Hermitian localization\" ", max_results=10, sort_by="relevance") Searching arXiv now. Anisotropic-scaling localization (ASL) is a higher-dimensional non-Hermitian boundary-localization phenomenon in which localization lengths follow distinct size-dependent scaling rules in an anisotropic manner. In the formulation introduced for higher-dimensional non-Hermitian systems, the localization length along one direction is not only finite-size dependent, but depends on different system dimensions in different ways; in finite two-dimensional systems, the authors report $\xi_x^{\rm AS}\propto L_x/L_y$, while in their three-dimensional extension they report $\bar{\xi}_x \propto N_x/\min[N_y,N_z]$ (Ou et al., 16 Jul 2025). ASL is therefore distinct both from ordinary localization, where the localization length is size-independent in the thermodynamic sense, and from the usual one-dimensional non-Hermitian skin effect (NHSE), where the localization length is again size-independent but eigenstates accumulate exponentially at one boundary due to non-reciprocal hopping (Ou et al., 16 Jul 2025).

1. Definition and diagnostic criteria

The defining feature of ASL is that the localization length explicitly depends on system geometry in an anisotropic way. In the main two-dimensional setting, the localization length of a corner state is defined from amplitudes at opposite ends of an edge. For a corner eigenstate $\psi_n^c(x,y)$,

$$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$

and the mean value over corner modes is

$\bar{\xi}_x=\frac{1}{n_l}\sum_n \xi_x^n.$

ASL means that $\bar{\xi}_x$ itself changes with $L_y$ and/or $L_x$, rather than saturating to a fixed number (Ou et al., 16 Jul 2025).

The same work emphasizes that ASL should not be confused with standard localization. Ordinary localization in Hermitian systems, including Anderson localization and topological edge localization, is described there as not showing a systematic anisotropic scaling with sample dimensions. By contrast, ASL is presented as a higher-order, geometry-sensitive localization phenomenon in which different boundary subsystems of a higher-dimensional lattice communicate through the remaining directions of the lattice (Ou et al., 16 Jul 2025).

A central diagnostic distinction is therefore between localization profile and scaling law. In ASL, the state is boundary-localized, but its decay length is controlled by transverse dimensions. This geometry dependence is the reason the phenomenon is termed “anisotropic-scaling localization,” rather than being classified simply as a skin effect or as conventional disorder-driven localization (Ou et al., 16 Jul 2025).

2. Mechanisms in the HN-SSH and BBH models

One mechanism for ASL is what the originating paper calls effective bulk couplings. This is demonstrated in the HN-SSH model, a two-dimensional lattice built from Hatano–Nelson chains along $x$ and SSH-type dimerization along $y$, with Bloch Hamiltonian

$\bar{\xi}_x \propto N_x/\min[N_y,N_z]$0

For $\bar{\xi}_x \propto N_x/\min[N_y,N_z]$1, the open system has top and bottom edge modes from the SSH structure, and non-reciprocity along $\bar{\xi}_x \propto N_x/\min[N_y,N_z]$2 weakly couples these two edges through the bulk. By integrating out the bulk, the authors obtain an effective one-dimensional two-leg ladder Hamiltonian for the edge subspace, with an inter-edge coupling $\bar{\xi}_x \propto N_x/\min[N_y,N_z]$3 that depends exponentially on the transverse system size $\bar{\xi}_x \propto N_x/\min[N_y,N_z]$4 (Ou et al., 16 Jul 2025).

The resulting effective ladder exhibits the critical NHSE, yielding a scale-free-type localization length

$\bar{\xi}_x \propto N_x/\min[N_y,N_z]$5

Because $\bar{\xi}_x \propto N_x/\min[N_y,N_z]$6, this reduces at large sizes to $\bar{\xi}_x \propto N_x/\min[N_y,N_z]$7 (Ou et al., 16 Jul 2025). The physical interpretation given is that the top and bottom edges are not independent: the bulk mediates a coupling that weakens as the separation grows, and the boundary mode’s decay length adjusts accordingly.

A second mechanism is a one-dimensional junction between different one-dimensional edges. This appears in the non-Hermitian BBH model when non-reciprocity is introduced along both directions, so that all four edges become non-Hermitian. In that regime, the effective ladder description no longer captures the boundary spectrum. Instead, the full Hamiltonian is projected onto the edge subspace,

$\bar{\xi}_x \propto N_x/\min[N_y,N_z]$8

and the resulting one-dimensional boundary problem can be viewed as two coupled edge chains plus localized junction impurities (Ou et al., 16 Jul 2025).

The same study therefore distinguishes at least two ASL subclasses: ladder-ASL, generated by effective bulk couplings between spatially separated edges, and junction-ASL, generated by a one-dimensional junction of different edge segments. This classification is explicitly framed as part of a broader taxonomy of higher-order non-Hermitian boundary localization by localization profile and underlying mechanism (Ou et al., 16 Jul 2025).

3. Scaling laws, aspect ratios, and crossover with the skin effect

The large-size scaling laws reported for ASL are unusually explicit. In two dimensions, the asymptotic behavior is

$\bar{\xi}_x \propto N_x/\min[N_y,N_z]$9

while in the thermodynamic limit at fixed aspect ratio $\psi_n^c(x,y)$0, the localization length approaches

$\psi_n^c(x,y)$1

The paper emphasizes that this limiting value depends only on the Hermitian $\psi_n^c(x,y)$2-direction hoppings, not on the non-reciprocal $\psi_n^c(x,y)$3-direction parameters (Ou et al., 16 Jul 2025). This is a notable point in the original presentation: ASL can remain finite even when non-Hermiticity enters only indirectly through edge coupling.

The three-dimensional extension follows the same logic. The reported dependence

$\psi_n^c(x,y)$4

shows that higher-dimensional generalizations retain the defining feature of anisotropic dependence on multiple system lengths (Ou et al., 16 Jul 2025).

ASL competes with ordinary edge NHSE. The non-reciprocal skin length is written as

$\psi_n^c(x,y)$5

and the numerical results are summarized by

$\psi_n^c(x,y)$6

In the language of the paper, the weaker localization wins: if $\psi_n^c(x,y)$7 is shorter than the ASL length, the system shows ordinary NHSE, whereas if $\psi_n^c(x,y)$8 is longer, ASL dominates (Ou et al., 16 Jul 2025).

This crossover is central to the meaning of ASL. It is not merely a modified skin effect, but a localization regime whose observability depends on the competition between a geometry-mediated scale and a conventional non-reciprocal skin scale. The phrase “anisotropic-scaling localization” refers precisely to this size-sensitive boundary localization profile (Ou et al., 16 Jul 2025).

4. Spectral signatures and localization taxonomy

The originating ASL paper identifies ASL states and edge NHSE states by their eigenenergies. In the examples studied, ASL states are associated with complex eigenenergies, whereas edge skin states have real eigenenergies (Ou et al., 16 Jul 2025). This spectral criterion is presented as a practical discriminator between two boundary-localization mechanisms that can otherwise coexist or compete.

In the HN-SSH model, corner states entering the ASL regime undergo a $\psi_n^c(x,y)$9-symmetry breaking and acquire nonzero imaginary parts. The maximum imaginary part follows

$$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$0

By contrast, when the same boundary states behave as ordinary skin states, their energies remain real (Ou et al., 16 Jul 2025).

The complex-energy spectrum is itself part of the ASL diagnosis. For the HN-SSH model, corner-state eigenenergies form a loop-like spectrum in the complex plane for small $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$1. As $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$2 increases, or when an extra imaginary term $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$3 is added to separate the edges in imaginary energy, the loop collapses into a line with no enclosed area (Ou et al., 16 Jul 2025). The paper uses this to resolve a specific issue in the recent literature: loop-like boundary spectra correspond there to ASL rather than to conventional NHSE.

In the BBH model, the boundary spectrum splits into two groups identified by an edge-distribution ratio built from the total probabilities on the left-right and top-bottom edges. Under single-direction non-Hermiticity, one group shows the ASL loop spectrum while the other remains real and edge-localized. Under two-direction non-Hermiticity, both groups can develop complex loops, and the transition between nearly real and complex energies tracks the transition between NHSE and ASL (Ou et al., 16 Jul 2025).

These spectral features are integral to the proposed classification framework. The paper distinguishes ordinary higher-order NHSE from ASL and classifies ASL by mechanism, while also using the contrast between real line-like spectra and complex loop-like spectra as a spectral taxonomy of boundary-localized states (Ou et al., 16 Jul 2025).

5. Relation to other higher-dimensional localization frameworks

A related but distinct framework appears in the study of anisotropic Anderson localization in higher-dimensional nonreciprocal lattices. There, in the two-dimensional Hatano–Nelson model, disorder and directional nonreciprocity generate ordinary skin modes, Anderson localized modes (ALMs), and anisotropic hybrid modes (HMs). The HMs are localized in different ways along orthogonal directions: skin localization along one axis and Anderson localization along the other (Shang et al., 19 Jul 2025). The transition along each direction is determined by a directional criterion,

$$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$4

and finite-size scaling of directional Lyapunov exponents yields mobility surfaces and a reported ALM–HM–ALM reentrant transition at $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$5, with $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$6 and $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$7 (Shang et al., 19 Jul 2025).

The distinction between ASL and these anisotropic hybrid modes is structural. ASL is defined by size-dependent anisotropic scaling of boundary localization lengths, whereas the HM construction identifies directional coexistence of NHSE and Anderson localization. This suggests a broader family of higher-dimensional non-Hermitian localization phenomena in which different axes may be governed by different localization mechanisms or different scaling laws (Shang et al., 19 Jul 2025).

A further conceptual backdrop is provided by the nonunitary scaling theory of non-Hermitian localization, which argues that non-Hermiticity introduces a new scale and breaks down one-parameter scaling in favor of two-parameter scaling. In that framework, the relevant competition is between the disorder scale $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$8 and a non-Hermitian scale set by $$\xi_x^n=\left|\frac{L_x-1}{\ln\!\left[\left|\psi_n^c(L_x,1)\right|/\left|\psi_n^c(1,1)\right|\right]}\right|,$$9, leading to direction-dependent conductances and distinct reciprocity classes (Kawabata et al., 2020). A plausible implication is that ASL belongs to this broader non-Hermitian scaling landscape, but with the additional feature that the decisive scales are geometric and boundary-mediated rather than only transport-channel dependent.

ASL also differs from anisotropic localization in Hermitian correlated disorder. In anisotropic three-dimensional disorder for matter waves, the diffusion tensor and localization tensor inherit direction dependence from non-rotation-invariant disorder correlations, and structured finite-range correlations can even invert the transport anisotropy (Piraud et al., 2011). This suggests an analogy rather than an identity: both settings are anisotropic, but ASL is characterized specifically by explicit system-size scaling of boundary localization lengths (Piraud et al., 2011).

6. Scope, limitations, and nomenclature

The ASL framework was introduced as a classification tool for higher-order non-Hermitian boundary localization regarding localization profiles. Its central claim is not a universal field-theoretic critical law, but a geometry-sensitive localization mechanism with identifiable scaling rules, explicit crossover with NHSE, and spectral signatures in the complex plane (Ou et al., 16 Jul 2025). The same work therefore presents ASL as a framework rather than as a single-model curiosity.

At the same time, the literature summarized here shows that the acronym “ASL” is overloaded on arXiv. In commutative algebra, “ASL” denotes “algebra with straightening law,” as in the study of quotient rings $\bar{\xi}_x=\frac{1}{n_l}\sum_n \xi_x^n.$0 that admit an ASL structure on a poset of generators (Saha et al., 2021). In neuromorphic vision and sign-language technology, “ASL” denotes American Sign Language, as in the ASL-DVS event-camera gesture dataset used for CSNN classification (Patel et al., 2024) and in video-based lookup systems for ASL signs integrated with SignStream and the ASLLRP Sign Bank (Neidle et al., 2024). In current condensed-matter usage, however, “Anisotropic-Scaling Localization” refers specifically to the higher-dimensional non-Hermitian localization phenomenon defined by anisotropic, size-dependent boundary decay lengths (Ou et al., 16 Jul 2025).