Skin-Anderson Transition in Non-Hermitian Models

• Skin-Anderson transition is the phenomenon where non-Hermitian skin accumulation competes with disorder-induced Anderson localization, characterized by a sign change in a shifted Lyapunov exponent.
• The transition is diagnosed by contrasting boundary-accumulated skin modes (via low inverse participation ratios) with bulk-localized Anderson modes, using precise Lyapunov and winding number criteria.
• Finite-size scaling and universal critical exponents in disordered Hatano–Nelson models, alongside extensions to quasiperiodic, hybrid, and many-body systems, highlight the broad applicability of this transition.

The Skin-Anderson transition is a disorder-driven change in localization character that arises when non-Hermitian skin accumulation competes with Anderson localization. In the canonical one-dimensional non-reciprocal setting, weak disorder preserves a skin phase in which eigenstates accumulate at a boundary under open boundary conditions, whereas sufficiently strong disorder produces bulk Anderson-localized states. Recent work has given both an exact Lyapunov-based criterion and a finite-size scaling description for this transition in disordered Hatano–Nelson systems, while related literature has extended the idea to quasiperiodic lattices, interacting many-body systems, higher-dimensional nonreciprocal models, and hybrid geometries with reciprocal and nonreciprocal directions (Barandun, 18 Mar 2026, Tozar, 25 Nov 2025, Wang et al., 28 Mar 2026). At the same time, the term is used non-uniformly: some works describe disorder-induced skin phases or boundary-condition-induced Anderson-to-skin conversion under the same label, whereas others explicitly show that skin-like boundary accumulation can be caused by a different many-body mechanism and should not be classified as a genuine Skin-Anderson transition (Wang et al., 2024, Li et al., 7 Jul 2025, Tozar, 5 Dec 2025).

1. Canonical formulation in disordered Hatano–Nelson chains

The standard setting is the disordered Hatano–Nelson chain, with asymmetric hopping and random onsite potential,

$$H = \sum_{n=1}^{N-1}\left(t e^{\gamma} c_{n+1}^\dagger c_n + t e^{-\gamma} c_n^\dagger c_{n+1}\right) + \sum_{n=1}^{N} V_n c_n^\dagger c_n,$$

where $t = 1$, $\gamma > 0$, and $V_n \in [-W/2, W/2]$ (Tozar, 25 Nov 2025). In the clean limit, the non-reciprocity produces the non-Hermitian skin effect (NHSE): under open boundary conditions, eigenstates pile up at a boundary, while under periodic boundary conditions the complex spectrum forms a loop with nonzero spectral winding $w=1$, which is the topological signature underlying the NHSE (Tozar, 25 Nov 2025).

Disorder competes with the non-reciprocal drift responsible for skin accumulation. As $W$ increases, the complex-energy spectral loop shrinks, the point gap closes, the system loses its non-Hermitian topological winding, and eigenstates become genuinely Anderson localized (Tozar, 25 Nov 2025). A standard diagnostic is the inverse participation ratio,

$\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$

for normalized eigenstate $\psi_k$. In the skin phase, $\mathrm{IPR}\sim 1/N$ in the thermodynamic sense of extended occupancy even though states are boundary accumulated; in the Anderson phase, $\mathrm{IPR}\sim 1/\xi$ and remains finite as $t = 1$0 (Tozar, 25 Nov 2025).

This sharply contrasts with the Hermitian one-dimensional case, where standard scaling theory predicts immediate localization for any nonzero disorder. In the non-Hermitian Hatano–Nelson chain, finite disorder is required before Anderson localization overtakes the skin phase, so the disorder-driven breakdown of non-Hermitian topological transport is a bona fide critical phenomenon rather than the automatic consequence of arbitrarily weak randomness (Tozar, 25 Nov 2025).

2. Lyapunov-exponent criterion and topological winding

A precise mechanism for the transition has been given for the one-dimensional non-Hermitian disordered tight-binding model

$t = 1$1

with i.i.d. disorder $t = 1$2 (Barandun, 18 Mar 2026). Localization is analyzed through transfer matrices. For a tridiagonal operator $t = 1$3, the transfer matrix at spectral parameter $t = 1$4 is

$t = 1$5

and the Lyapunov exponent is

$t = 1$6

For the Hatano–Nelson operator, the asymmetry shifts the Hermitian Lyapunov exponent by

$t = 1$7

This shift is the key mechanism behind the transition (Barandun, 18 Mar 2026).

The physical interpretation is explicit. When $t = 1$8, transfer matrices decay in a way that produces exponentially edge-localized states, namely the NHSE. When $t = 1$9, growth in both directions forces normalizable states to localize in the bulk, namely Anderson localization (Barandun, 18 Mar 2026). The sign change of $\gamma > 0$0 is therefore the localization diagnostic.

The same paper relates this criterion to the winding region of the clean Hatano–Nelson symbol. For constant potential $\gamma > 0$1,

$\gamma > 0$2

and the winding region is

$\gamma > 0$3

The associated topological invariant is the winding number around the eigenvalue,

$\gamma > 0$4

This integer is nonzero inside $\gamma > 0$5 and changes exactly when $\gamma > 0$6 crosses $\gamma > 0$7 (Barandun, 18 Mar 2026).

For the Lloyd model with Cauchy disorder of scale $\gamma > 0$8, the Hermitian exponent is

$\gamma > 0$9

and the exact criterion becomes

$V_n \in [-W/2, W/2]$0

up to small-$V_n \in [-W/2, W/2]$1 corrections, with error $V_n \in [-W/2, W/2]$2 for $V_n \in [-W/2, W/2]$3 and $V_n \in [-W/2, W/2]$4 for $V_n \in [-W/2, W/2]$5 (Barandun, 18 Mar 2026). The paper’s central claim is that the topological change $V_n \in [-W/2, W/2]$6 coincides with the eigenvector crossover from boundary-accumulated skin mode to bulk Anderson-localized mode. A plausible implication is that the most compact formulation of the Skin-Anderson transition in one dimension is the equivalence between a sign change of a shifted Lyapunov exponent and a change of point-gap winding.

3. Critical scaling and universality in one dimension

Beyond the exact Lyapunov criterion, recent work treats the Skin-Anderson transition as a sharp continuous phase transition with universal scaling (Tozar, 25 Nov 2025). Because ordinary recursive methods overflow when skin modes grow like $V_n \in [-W/2, W/2]$7, one study introduced a gauge-invariant log-space transfer-matrix method. The transmission is computed from the Caroli formula,

$V_n \in [-W/2, W/2]$8

and in the one-dimensional chain,

$V_n \in [-W/2, W/2]$9

The finite-size Lyapunov exponent is then

$w=1$0

Its sign distinguishes phases: $w=1$1 indicates a skin-dominated topological transport phase, $w=1$2 indicates an Anderson localized phase, and $w=1$3 defines the critical point (Tozar, 25 Nov 2025).

The scaling ansatz is

$w=1$4

equivalently

$w=1$5

Using system sizes up to $w=1$6, the reported values for $w=1$7 are

$w=1$8

with essentially the same exponents at $w=1$9, indicating an energy-independent universality class (Tozar, 25 Nov 2025). The same study also finds a global phase boundary $W$0 with fitted exponent $W$1, compared with the weak-coupling expectation $W$2 (Tozar, 25 Nov 2025).

A second study tested whether this criticality is model dependent by comparing three disorder landscapes: uniform diagonal disorder, binary diagonal disorder, and off-diagonal random hopping disorder (Tozar, 28 Nov 2025). To avoid overflow from $W$3, it introduced Log-Space Non-Hermitian Scaling (LNS) and performed finite-size scaling up to $W$4. The key conclusion is that, once the singular band-center point protected by chiral symmetry is excluded, the critical exponents remain

$W$5

independent of whether disorder is continuous or discrete and whether it enters onsite or bond terms (Tozar, 28 Nov 2025). The uniform diagonal model has $W$6 at $W$7, the binary model shifts to $W$8, and the off-diagonal model at $W$9 has $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$0, but the exponents do not change (Tozar, 28 Nov 2025).

The principal exception is the off-diagonal random-hopping model at $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$1, where chiral symmetry produces a Dyson-like singularity and the system resists localization up to very large disorder strengths $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$2 (Tozar, 28 Nov 2025). This establishes an important qualification: the universality class with $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$3 and $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$4 is robust across generic one-dimensional non-Hermitian Anderson problems with skin effect, but symmetry-protected singular points can obstruct the ordinary scaling picture.

4. Quasiperiodic, ladder, and boundary-condition-dependent variants

A closely related precursor appears in the non-reciprocal Aubry–André lattice,

$\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$5

with $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$6 and $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$7 (Jiang et al., 2019). Under open boundary conditions, the model is similar to a Hermitian Aubry–André chain, which yields an exact rescaled transition point

$\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$8

Below this threshold, non-reciprocity produces a left-skin or right-skin phase depending on the sign of $\mathrm{IPR}_k = \sum_{n=1}^{N} |\psi_k(n)|^4,$9; above it, quasiperiodic localization dominates (Jiang et al., 2019). The localized states become asymmetric,

$\psi_k$0

so there are two Lyapunov exponents, $\psi_k$1 and $\psi_k$2. The same transition is characterized under periodic boundary conditions by a winding number $\psi_k$3, which the paper describes as a bulk-bulk correspondence (Jiang et al., 2019).

A more recent formulation, termed Anderson-Skin dualism, considers coupled Hatano–Nelson and Aubry–André systems in which some states are Anderson localized under periodic boundary conditions but become skin modes under open boundary conditions if their eigenvalues lie inside a point-gap loop (Li et al., 7 Jul 2025). In one dimension the transfer-matrix analysis yields asymmetric Lyapunov exponents

$\psi_k$4

where $\psi_k$5 is the Lyapunov exponent of the Hermitian or quasiperiodic sector and $\psi_k$6 is the non-reciprocity parameter (Li et al., 7 Jul 2025). Outside the point-gap loop, both $\psi_k$7 and $\psi_k$8 are nonzero and the state is an ordinary Anderson-localized mode with two-sided decay; inside the point gap, one side ceases to decay and the same state becomes a skin mode under open boundary conditions (Li et al., 7 Jul 2025). This suggests a boundary-condition-induced conversion rather than a simple monotonic disorder-driven suppression of the skin effect.

Another non-standard variant is the reentrant NHSE obtained by coupling a strongly disordered Hatano–Nelson chain to a disordered Hermitian chain with anti-symmetrically correlated disorder (Jin et al., 2023). In the strong-coupling regime, the effective disorder strength satisfies

$\psi_k$9

so increasing the inter-chain coupling $\mathrm{IPR}\sim 1/N$0 can suppress effective disorder, induce Anderson delocalization, and then restore the NHSE, producing a localized $\mathrm{IPR}\sim 1/N$1 delocalized $\mathrm{IPR}\sim 1/N$2 skin-localized sequence (Jin et al., 2023).

A further analogue is the size-dependent transition from scale-free localization to Anderson localization in a Hermitian Anderson chain with a single non-Hermitian impurity (Yılmaz et al., 2024). Here the prelocalized states are not ordinary skin modes but scale-free localized states whose localization length scales proportionally with system size. The disorder threshold obeys

$\mathrm{IPR}\sim 1/N$3

so $\mathrm{IPR}\sim 1/N$4 in the thermodynamic limit (Yılmaz et al., 2024). This is best regarded as a size-dependent analogue rather than the canonical Hatano–Nelson Skin-Anderson transition.

5. Higher-dimensional, hybrid, and many-body generalizations

In higher-dimensional nonreciprocal lattices, disorder and non-reciprocity can localize different spatial directions differently. In the two-dimensional Hatano–Nelson model with nonreciprocity $\mathrm{IPR}\sim 1/N$5, one study identified hybrid modes (HMs) that are skin-localized along one direction and Anderson-localized along the orthogonal direction (Shang et al., 19 Jul 2025). The transition criterion is derived by similarity-transforming to a reciprocal Anderson model, which shifts the Lyapunov exponents by the nonreciprocity. Along $\mathrm{IPR}\sim 1/N$6,

$\mathrm{IPR}\sim 1/N$7

marks the transition between normal skin behavior and Anderson localization, with an analogous condition along $\mathrm{IPR}\sim 1/N$8 (Shang et al., 19 Jul 2025). This produces mobility surfaces in complex energy space separating normal skin modes, Anderson localized modes (ALMs), and hybrid modes. A reported example is an $\mathrm{IPR}\sim 1/N$9 reentrant transition at $\mathrm{IPR}\sim 1/\xi$0, with critical disorders

$\mathrm{IPR}\sim 1/\xi$1

and at $\mathrm{IPR}\sim 1/\xi$2 the finite-size scaling gives $\mathrm{IPR}\sim 1/\xi$3 and $\mathrm{IPR}\sim 1/\xi$4 (Shang et al., 19 Jul 2025).

A distinct higher-dimensional construction is the hybrid-nonreciprocal system, where one direction remains reciprocal and the others are nonreciprocal (Wang et al., 28 Mar 2026). In the under-nonreciprocal regime, the Hamiltonian is similar to a Hermitian one, and the inverse similarity transformation injects a universal skin profile with skin depth

$\mathrm{IPR}\sim 1/\xi$5

Disorder then produces a two-stage sequence

$\mathrm{IPR}\sim 1/\xi$6

with the first transition identified as the Skin-Anderson transition proper (Wang et al., 28 Mar 2026). Because the transfer matrix along the reciprocal direction is similar to that of the Hermitian counterpart, the first critical point inherits the same criticality as the ordinary Anderson transition in the reciprocal model. For the hybrid-nonreciprocal Rashba model at $\mathrm{IPR}\sim 1/\xi$7, $\mathrm{IPR}\sim 1/\xi$8, and $\mathrm{IPR}\sim 1/\xi$9, the reported values are

$t = 1$00

from transfer-matrix scaling, and

$t = 1$01

from participation-ratio scaling (Wang et al., 28 Mar 2026). The second transition occurs when the Anderson localization length $t = 1$02 falls below the skin depth, $t = 1$03, and is reported to exhibit a Berezinskii–Kosterlitz–Thouless-like correlation length

$t = 1$04

In interacting many-body systems, the picture depends strongly on conservation laws. For charge-conserving non-Hermitian systems with Hamiltonian

$t = 1$05

the same similarity transformation as in the Hatano–Nelson problem still works under open boundary conditions,

$t = 1$06

so the competition is again between disorder-induced localization and non-Hermitian amplification (Gliozzi et al., 14 Apr 2025). The paper finds a genuine disorder-driven transition between a skin-effect phase and a many-body localized phase when only $t = 1$07 charge is conserved; in dynamics, the same transition coincides with an area-law to volume-law entanglement transition (Gliozzi et al., 14 Apr 2025). Under periodic boundary conditions, the skin phase becomes a delocalized phase with a unidirectional current. By contrast, when dipole or higher multipole moments are conserved, the appropriate similarity transformation grows super-exponentially, and the multipole skin effect remains stable to arbitrary disorder; under periodic boundary conditions the system is then always delocalized regardless of disorder strength (Gliozzi et al., 14 Apr 2025). A plausible implication is that the Skin-Anderson transition is not a universal consequence of non-Hermiticity alone, but depends on which conserved quantities determine the similarity-transformed growth law.

6. Disorder-induced skinning, symmetry effects, experimental realizations, and misidentifications

Not all literature uses the phrase to mean “skin phase destroyed by disorder.” In a one-dimensional chain with reciprocal hopping $t = 1$08 and disordered nonreciprocal corrections,

$t = 1$09

with $t = 1$10, disorder itself can induce non-Hermitian point-gap topology and boundary localization (Wang et al., 2024). The relevant invariant is the real-space winding number

$t = 1$11

constructed from the doubled Hermitian Hamiltonian. As $t = 1$12 changes, the mean mode position moves from the bulk to either boundary, with $t = 1$13 matching bulk-like, left-skin, and right-skin regimes (Wang et al., 2024). In an acoustic crystal, increasing disorder can even reverse the direction of boundary accumulation, and next-nearest-neighbor disorder can produce a bipolar skin effect (Wang et al., 2024). This is a disorder-induced skin phase rather than a disorder-destroyed one.

In the non-Hermitian SSH chain, disorder produces a sequence under open boundary conditions from skin effect to partial skin effect to no skin effect, while under periodic boundary conditions it produces an extended phase, a mobility-edge phase, and a fully localized phase (Sarkar et al., 2022). The real-space winding number

$t = 1$14

remains nonzero through the partially localized regime and jumps to zero only at full localization (Sarkar et al., 2022). The same work also describes the non-Hermitian Anderson skin effect (NHASE), in which a system without clean-limit skin effect develops disorder-induced skin accumulation at intermediate disorder (Sarkar et al., 2022). This reinforces that the relation between disorder, point-gap topology, and skin accumulation can be non-monotonic and symmetry dependent.

Several experimental platforms have been proposed or realized. The non-reciprocal Aubry–André model has an electronic-circuit implementation using only linear passive RLC devices, where transport switches between amplifying and insulating across the rescaled transition $t = 1$15 (Jiang et al., 2019). The reentrant NHSE of a disordered Hatano–Nelson/Hermitian ladder has been observed in topolectrical circuits through steady-state voltage profiles and quench dynamics (Jin et al., 2023). The disorder-induced boundary localization driven by point-gap topology has been demonstrated in a non-Hermitian acoustic crystal composed of 21 resonators and 20 externally connected amplifiers (Wang et al., 2024). These realizations show that the Skin-Anderson vocabulary spans both electronic and classical-wave platforms.

A final qualification concerns apparent skin-like boundary accumulation in interacting disordered systems. In a one-dimensional disordered Fermi–Hubbard chain with open boundary conditions,

$t = 1$16

strong repulsion $t = 1$17 drives spin segregation toward opposite boundaries, with order parameter

$t = 1$18

and a pronounced peak in

$t = 1$19

near $t = 1$20 (Tozar, 5 Dec 2025). However, the segregation persists in the Hermitian limit $t = 1$21, and the authors therefore interpret it as interaction-induced thermodynamic segregation rather than a genuine NHSE-driven Skin-Anderson transition (Tozar, 5 Dec 2025). This establishes an important encyclopedia-level distinction: boundary accumulation is not, by itself, sufficient evidence for a Skin-Anderson transition; the mechanism must be tied to non-Hermitian topology or to the Lyapunov/winding criteria that define the competition between skin effect and Anderson localization.