Skin Effect Driven Localization

• Skin-effect driven localization is a phenomenon where eigenstates in non-Hermitian systems localize at boundaries due to asymmetric hopping or engineered gain/loss, distinct from disorder-induced localization.
• The effect exhibits tunable, scale-free localization lengths with regimes such as divergent skin depths at Bloch points and transitions influenced by defects and many-body interactions.
• Extensions to higher dimensions reveal corner and geometry-dependent skin effects, with experimental validations in photonics, electric circuits, ultracold atoms, and acoustic metamaterials.

Skin-effect driven localization refers to the phenomenon where eigenstates of non-Hermitian systems, under open boundary conditions (OBC), become spatially localized at boundaries or specific interfaces. This process is mediated not by disorder, but by non-Hermitian terms such as asymmetric hopping, local gain/loss, or engineered reciprocity-violating elements. The skin effect transcends Anderson localization, giving rise to robust and sometimes extensive boundary accumulation of both bulk and topological states in a wide variety of classical and quantum platforms. Its highly tunable character encompasses scale-free localization, nonlinear and multifractal profiles, higher-dimensional variants (including corner and geometry-dependent skin effects), non-Markovian generalizations, and even critical and dynamically emergent forms.

1. Fundamental Mechanism and Non-Hermitian Band Structure

The canonical mechanism underlying skin-effect driven localization is the non-Hermitian skin effect (NHSE). This effect manifests when, under OBCs, all (or a macroscopic fraction of) bulk eigenstates become exponentially localized at one edge of a 1D system due to asymmetric (non-reciprocal) hopping or non-trivial gain/loss profiles. For single-band tight-binding models, the right eigenstates $\psi_n$ display the general form: $\psi_n \propto \beta^n$ where $\beta = e^{ik-\kappa}$ is a complex deformation of the usual Bloch phase factor. The inverse localization length $\kappa = -\ln|\beta|$ sets the exponential decay. This boundary-pinning does not arise in Hermitian systems, for which $|\beta|=1$ and all bulk eigenstates are extended.

In paradigmatic models such as the Hatano–Nelson chain, the onset of the skin effect tracks the winding of the complex PBC spectrum $H_{HN}(k)$ in the complex energy plane. When the PBC spectrum winds non-trivially (winding number $W(E)\neq0$ for some reference $E$), the OBC spectrum collapses to a set of eigenvalues with eigenmodes that localize at the system edge (Ma et al., 2020, Li et al., 2023, Gliozzi et al., 14 Apr 2025, Liu et al., 2023).

2. Universal, Critical, and Scale-free Skin Effect

The conventional NHSE yields an $N$-independent localization length $\lambda\sim O(1)$ in a chain of length $N$. Recently, it has been recognized that near certain spectral singularities—termed "Bloch points" (where OBC and PBC spectra intersect in the complex plane, $\left| \beta_B \right| = 1$)—the localization length itself may diverge sub-extensively or extensively with system size: $\lambda \sim N^j$ with integer $j$. This "scalefree NHSE" occurs universally in systems with well-defined generalized Brillouin zones (GBZ) and Bloch points, without the need for special disorder or parameter fine-tuning (Li et al., 2023). The exponent $j$ is dictated by the local geometry of the GBZ at the Bloch-point crossing: linear scaling ($j=1$) for a generic crossing, quadratic ($j=2$) for a tangency, and so on. This blurs the distinction between traditional bulk modes and boundary-localized modes, with the skin depth diverging with $N$ and the skin region encompassing the entire sample in the thermodynamic limit.

3. Extensions: Defects, Interactions, and Multipole Conservation

Defect Competition and Phase Transitions

When a strong defect (e.g., an onsite impurity) is introduced into an NHSE system, a competition arises: if the defect strength exceeds a threshold $|d_c| = 2\sinh \gamma$ (for a simple non-reciprocal model), localized eigenstates may "unbind" from the edge and localize instead at the defect. A two-scale asymptotic analysis yields the effective envelope dynamics and the criteria for this transition. The skin effect and defect-localized regimes are thus separated by a sharp, analytically tractable boundary (Davies et al., 19 Mar 2024).

Interacting and Multipole-Conserving Chains

Disorder and many-body interactions further enrich the skin localization phase diagram. In the presence of U(1)-symmetry and finite disorder, the skin effect can compete with many-body localization (MBL), leading to a delocalization-localization transition at a critical non-reciprocity $g_c\sim1/\xi$, where $\xi$ is the Hermitian localization length. For multipole-conserving (e.g., dipole or higher moments) non-Hermitian models, the skin effect becomes always dominant under OBC, overwhelming disorder and imposing extensive delocalization under PBC irrespective of disorder strength (Gliozzi et al., 14 Apr 2025).

4. Generalizations to Higher Dimensions

A precise theorem now characterizes the skin effect in $d>1$ as universally present whenever the PBC bulk spectrum covers a finite area in the complex plane ($\mathcal{A}(S)>0$). This is generically the case for non-Hermitian Bloch Hamiltonians with nonzero spectral area. Consequences include:

• Corner-skin effect: all eigenstates localize at a single corner (Zhang et al., 2021).
• Geometry-dependent skin effect: skin localization appears or vanishes depending on the sample shape, with number of skin modes scaling as the system "volume."
• Associated bulk-boundary correspondence: the PBC spectral area maps directly onto the existence (and number) of skin modes under arbitrary OBC geometries.
• Exceptional points and lines enforce the skin effect in 2D/3D, making phenomena like corner or geometry-dependent skin states generic near these degeneracies.
• Hybridization with topology: In higher-dimensional systems (Chern insulators, quantum spin Hall, higher-order topological lattices), the NHSE leads to unique boundary/corner-pinned topological states, transporting robust topological invariants (e.g., quantized Hall conductance, Chern number) to a single surface or corner (Ma et al., 2020, Liu et al., 2023).

5. Nonlinear, Non-Markovian, and Dynamical Skin Localization

Nonlinear Skin Effect

Interplay with Kerr-type nonlinearities enables intensity-tunable skin effect: above a critical field intensity, skin localization can be switched on or off (threshold behavior), or even focused at a tunable interface. In nonlinear extensions of the Hatano–Nelson model or temporal photonic lattices, the amplitude-dependent non-reciprocal hopping induces transitions from extended to skin-localized states, algebraically localized critical points, and robust skin solitons (Padlewski et al., 30 Sep 2024, Wang et al., 29 Sep 2024, Kawabata et al., 15 May 2025). In such systems, localization length, soliton profiles, and interface selectivity become power-dependent.

Non-Markovian Open Systems

Non-Markovian environments, modeled via hierarchical equations of motion (HEOM), induce a "thick skin effect": skin-localized eigenmodes with extended penetration into the bulk, exceeding the localization length expected from Markovian theory. Genuine non-Markovian features include oscillatory relaxation, boundary coherence scaling linearly with system size, and robustness to environmental noise—properties unattainable in Markovian or rotating-wave approximations (Kuo et al., 21 Mar 2024).

6. Critical and Dynamical Skin Effects in Periodically Driven Systems

Time-dependent or Floquet engineering offers new routes to skin-effect driven localization. In periodically driven non-Hermitian quasiperiodic lattices, the skin effect (SE) may be controlled or even resurrected in parameter regimes where a static electric field would completely destroy it. Under certain conditions, the Floquet spectrum loses extended unitarity and a multifractal skin profile emerges: eigenstates are neither fully localized nor extended, as diagnosed by nontrivial generalized participation ratios ($D_2\in(0,1)$). Dynamical criteria (fraction of modes with non-real quasienergies) serve as phase markers. The phase diagram as a function of drive amplitude and non-Hermitian asymmetry reveals new boundaries for delocalized, multifractal, and skin phases (Chakrabarty et al., 16 Dec 2024).

7. Experimental Realizations and Applications

Skin-effect driven localization has been verified across diverse platforms:

• Photonic crystals and waveguide arrays: direct imaging and spectral measurements reveal boundary skinning, including Chern skin effects.
• Electric-circuit networks and topolectrical lattices: passive RLC circuits realize multi-dimensional, reciprocal, and geometry-dependent skin effects with precision.
• Ultracold atoms in optical lattices: engineered driving and loss profiles induce, tune, and reverse NHSE in 1D and ladder systems.
• Acoustic metamaterials: amplitude-driven skin modes observed in chains of active resonators, with localization sharply controlled by stimulus intensity.
• Fractal and graph systems (Bethe lattices): inner skin effects localizing modes at bulk layers, not just physical boundaries (Sun et al., 11 Sep 2024).
• Devices: chip-scale one-way circulators, orbital-angular-momentum generators in photonics, non-reciprocal sensors, robust routers, and soliton logic elements.

A unified signature across all platforms is the anomalous scaling and spatial profile of the localized modes, often extractable by inverse participation ratios, direct field imaging, and nonlocal transport spectroscopy (Payá et al., 1 Oct 2025). In combination with bulk-boundary correspondence dictated by non-Hermitian spectral topology, these signatures demarcate skin-effect driven localization as a distinct paradigm in the control, routing, and protection of energy and information in open systems.