Skin–Anderson Transition Overview

• Skin–Anderson Transition is a disorder-driven phase transition that combines non-Hermitian skin effects with Anderson localization, marked by universal scaling laws and topological invariants.
• Transfer-matrix methods and Lyapunov exponents rigorously identify the phase boundary, enabling precise quantification of the transition in disordered tight-binding models.
• Experimental realizations in acoustic crystals, RLC circuits, and topolectrical lattices validate the theoretical framework, highlighting practical diagnostics of this novel transition.

The Skin–Anderson Transition constitutes the disorder-driven phase transition between non-Hermitian topological phases, characterized by the non-Hermitian skin effect (NHSE), and Anderson-localized phases, induced by random disorder in non-Hermitian tight-binding systems. Unlike the immediate localization in one-dimensional Hermitian systems, non-Hermitian topology generates robust macroscopic boundary localization up to a finite, critical disorder strength, upon which a sharp transition to Anderson localization occurs. This transition is governed by universal scaling laws, topological invariants, and exhibits a new universality class that is absent in Hermitian quantum systems.

1. Microscopic Models and Topological Framework

The canonical setting for the Skin–Anderson transition is the disordered Hatano–Nelson model: $$H = \sum_n \left[ t_R\,c_n^\dagger c_{n+1} + t_L\,c_{n+1}^\dagger c_n + V_n\,c_n^\dagger c_n \right],$$ with asymmetric nearest-neighbor hoppings $t_R = t e^{\gamma}$, $t_L = t e^{-\gamma}$ ($t=1$ fix), on-site disorder $V_n \in [-W/2, W/2]$, and nonreciprocity parameter $\gamma > 0$ (Tozar, 25 Nov 2025, Tozar, 28 Nov 2025, Barandun, 18 Mar 2026). For $W=0$, the system exhibits NHSE—macroscopic accumulation of eigenstates at one edge for open boundary conditions (OBC).

Extensions include the non-Hermitian SSH model (with sublattice structure and nonreciprocal intracell hoppings) (Sarkar et al., 2022), nonreciprocal Aubry–André models with quasi-periodic potential (Jiang et al., 2019), and higher-dimensional nonreciprocal lattices with direction-dependent nonreciprocity (Shang et al., 19 Jul 2025, Wang et al., 28 Mar 2026).

Topologically, the NHSE is rooted in point-gap topology: the complex energy spectrum under periodic boundary conditions (PBC) encircles a base point $E_0$, characterized by a nontrivial winding number

$$w(E_0) = \frac{1}{2\pi i} \oint_{C} dE\, \frac{\partial}{\partial E} \ln\det[H(E) - E_0].$$

For disordered systems, the winding is computed in real space via the commutator $Q^\dagger[Q,X]$, with $t_R = t e^{\gamma}$0 the polar decomposition of $t_R = t e^{\gamma}$1 and $t_R = t e^{\gamma}$2 the position operator (Wang et al., 2024, Jin et al., 2023, Sarkar et al., 2022).

2. Transfer-Matrix Formalism and Lyapunov Exponents

Transport and localization properties are controlled by the Lyapunov exponent $t_R = t e^{\gamma}$3, computed via a log-space transfer-matrix approach for large system sizes. For the 1D chain: $t_R = t e^{\gamma}$4 where $t_R = t e^{\gamma}$5 is the product of site-dependent transfer matrices (Tozar, 25 Nov 2025, Tozar, 28 Nov 2025, Barandun, 18 Mar 2026).

Under disorder, the sign of $t_R = t e^{\gamma}$6 distinguishes phases:

• $t_R = t e^{\gamma}$7: Skin phase (non-Hermitian delocalized under OBC)
• $t_R = t e^{\gamma}$8: Anderson phase (conventional bulk localization)

For the non-Hermitian Hatano–Nelson model, the Lyapunov exponent satisfies $t_R = t e^{\gamma}$9, reducing the analysis to the Hermitian case and revealing a universal phase boundary at which $t_L = t e^{-\gamma}$0—precisely the spectral locus where the transition occurs (Barandun, 18 Mar 2026).

3. Universal Critical Scaling and Phase Diagram

The skin–Anderson transition exhibits sharp criticality and universal scaling. Finite-size scaling yields the ansatz

$t_L = t e^{-\gamma}$1

with critical exponents $t_L = t e^{-\gamma}$2 (correlation-length) and $t_L = t e^{-\gamma}$3 (order parameter) in the 1D model—distinct from any Hermitian Anderson transition (Tozar, 25 Nov 2025, Tozar, 28 Nov 2025). These values are robust across disorder models (uniform, binary, random hopping).

The phase boundary $t_L = t e^{-\gamma}$4 follows a nontrivial scaling law: $t_L = t e^{-\gamma}$5 with the scaling consistent but not identical to the perturbative prediction $t_L = t e^{-\gamma}$6 (from the intersection of the non-Hermitian delocalization length and Hermitian localization length) (Tozar, 25 Nov 2025, Gliozzi et al., 14 Apr 2025).

In higher-dimensional systems, skin–Anderson transitions can be direction-selective: different spatial directions can undergo transitions at distinct disorder thresholds, yielding hybrid modes with skin localization in one direction and Anderson localization in another (Shang et al., 19 Jul 2025, Wang et al., 28 Mar 2026). Mobility-edge surfaces in the complex energy plane characterize these transitions.

4. Topological and Universal Criteria for the Transition

The transition is fundamentally topological. The eigenmode at energy $t_L = t e^{-\gamma}$7 exhibits skin localization under OBC if $t_L = t e^{-\gamma}$8 resides inside the point-gap winding region $t_L = t e^{-\gamma}$9 ($t=1$0), and Anderson localization otherwise. Formally,

$t=1$1

$t=1$2

with the transition at $t=1$3 (Barandun, 18 Mar 2026).

This criterion is universal across models, including those with quasiperiodic potentials (Jiang et al., 2019), coupled-ladder settings (Anderson–Skin dualism) (Li et al., 7 Jul 2025), hybrid-nonreciprocal models (Wang et al., 28 Mar 2026), and interacting systems (Gliozzi et al., 14 Apr 2025).

5. Physical Mechanism, Multicriticality, and Symmetry Effects

The NHSE, generated by an imaginary vector potential or nonreciprocal hopping, induces an exponential amplitude drift toward system boundaries, acting to delocalize in the presence of weak disorder. This topological mechanism protects boundary transport until backscattering exceeds the drift at a finite $t=1$4, at which point disorder dominates and Anderson localization sets in. The closing of the point-gap in the complex energy plane signals the collapse of non-Hermitian topology.

Multicritical behavior arises in higher-dimensional and hybrid models. For example, in the two-dimensional nonreciprocal Hatano–Nelson model, intermediate hybrid phases exist with skin effect in one direction and localization in the other (hybrid modes), as confirmed by finite-size scaling and mobility-edge surface analysis (Shang et al., 19 Jul 2025, Wang et al., 28 Mar 2026). Reentrant transitions (NHSE → Anderson → NHSE) can occur under correlated disorder (Jin et al., 2023).

Symmetry constraints can significantly alter the transition. In models conserving higher multipole moments (e.g., dipole conservation), the NHSE remains robust to arbitrary disorder and Anderson localization is absent (Gliozzi et al., 14 Apr 2025). Symmetry classes in non-Hermitian SSH chains modify the correspondence between disorder-driven skin effect and the real-space winding number (Sarkar et al., 2022).

6. Experimental Realizations and Diagnostics

The skin–Anderson transition has been realized in diverse platforms:

• Acoustic crystals with nonreciprocal, disordered couplings, exhibiting disorder-induced and reversible skin effects depending on disorder strength, directly measured via spatial mode profiles and phase diagrams (Wang et al., 2024).
• RLC circuit lattices emulating the nonreciprocal Aubry–André model, where transition points between skin (amplifying) and Anderson-localized (insulating) phases are revealed by transport or voltage amplification (Jiang et al., 2019).
• Topolectrical circuits implementing coupled nonreciprocal-Hermitian ladders with correlated disorder, demonstrating both Anderson localization and disorder-induced (reentrant) NHSE (Jin et al., 2023).

Key diagnostic observables include: real-space winding numbers, Lyapunov exponents, inverse participation ratio, edge density, center-of-mass position, anomalous currents, and entanglement entropy scaling (Tozar, 25 Nov 2025, Sarkar et al., 2022, Wang et al., 2024, Gliozzi et al., 14 Apr 2025).

Below is a summary table of affirmed universal aspects (for canonical 1D settings):

Aspect	Hermitian (1D)	Skin–Anderson Transition	Universality Class
Critical exponents	No finite transition	$t=1$5, $t=1$6	Unique to non-Hermitian
Disorder threshold	$t=1$7	$t=1$8	Model-independent in 1D
Topological marker	None	Point-gap winding jump	Bulk-bulk correspondence
OBC/PBC sensitivity	None	Skin ↔ Anderson dichotomy	Duality under OBC/PBC

7. Outlook and Significance

The Skin–Anderson transition unifies Anderson localization phenomenology with non-Hermitian topology, producing genuine criticality, universal scaling, and topological signatures absent from Hermitian systems. This transition is robust to details of disorder distribution and Hamiltonian structure, persists in many-body and higher-dimensional systems, and admits direct observation in artificial platforms. The transition not only sets universal limits for topological protection in imperfect media but also provides a new theoretical framework for the interplay between disorder, boundary phenomena, and non-Hermitian spectral topology (Tozar, 25 Nov 2025, Tozar, 28 Nov 2025, Gliozzi et al., 14 Apr 2025, Shang et al., 19 Jul 2025, Barandun, 18 Mar 2026, Jin et al., 2023, Sarkar et al., 2022, Wang et al., 2024).