Non-Hermitian effective \mathcal{P}\mathcal{T}-symmetry restoration from structural disorder

Abstract

In non-Hermitian systems, disorder-induced localization is generally regarded as a competitive mechanism to the highly non-local influence of the non-Hermitian skin effect (NHSE). In this work, we reveal a more intricate interplay between these two phenomena by investigating structural disorder that modifies asymmetric hopping amplitudes in multi-orbital lattices. Through a comparative analysis under periodic and open boundary conditions, we show that disorder can counterintuitively enhance the NHSE, restoring the effectiveness of \mathcal{P}\mathcal{T} symmetry with suppression of imaginary energies. This intriguing observation can be attributed to the translation-breaking clustering of displaced atoms, which can strengthen the NHSE overall while suppressing state amplification simultaneously. Our findings are substantiated through extensive numerical simulations and are expected to hold qualitatively in generic amorphous setups harboring the NHSE, as can be readily simulated in present-day metamaterials and quantum setups.

Graphical abstract

Keywords

non-Hermitian skin effect / structural disorder / non-Hermitian systems

1 Introduction

Non-Hermitian systems exhibit a wide range of intriguing phenomena due to their complex energies and modified notion of band structure. Complex energy bands can intersect at branch cuts, leading to defective exceptional points [1–9] with unconventional spectral and quantum information properties [10, 11]. A distinctive phenomenon of non-Hermitian systems is the non-Hermitian skin effect (NHSE) [12–71], whose energy spectrum is highly sensitive to boundary conditions, breaking conventional bulk–boundary correspondences by complex-deforming the Bloch lattice description into so-called generalized Brillouin zones [15–19, 22, 41]. In recent years, much experimental progress have also been made in investigating non-Hermitian phenomena across different platforms, including photonic crystals [2–4, 41, 72, 73], electronic circuits [11, 16, 42, 43, 74, 75], mechanical systems [44, 45, 76] and quantum many-body setups [77–83].

Yet, these above-mentioned theoretical and experimental progress have by and large centered around ordered systems and their associated notions, such as momentum-space band structure. But most recently, studies have uncovered that disorder can intriguingly induce non-local or ultra-sensitive behavior with far greater implications than their Hermitian counterparts [38, 84–93]. Based on historical precedence, Anderson disorder have been studied most actively, with disorder-induced non-Hermitian topological Anderson insulators found to exhibit unique localized behavior and topological phase transitions [38, 84–86]. However, the effects of structural disorder has not been so well understood in non-Hermitian systems, despite being particularly important in real materials, which can exist in an amorphous form due to the unavoidable presence of impurities and defects during fabrication [94–97]. Given the distinct nature of these two types of disorder, a natural question arises: Can structural disorder, when combined with non-Hermitian couplings, give rise to emergent phenomena beyond those expected from either ingredient alone?

In this work, we provide a strongly affirmative answer to this question: Surprisingly, we find that structural disorder, which is usually thought to weaken physical responses, does not simply hinder the NHSE. Instead, weak to moderate structural disorder is found to enhance heightened NHSE amplification, as well as effectively restore parity–time (\mathcal{P}\mathcal{T}) symmetry by suppressing imaginary energy components. This intriguing observation can be attributed to the translation-breaking clustering of displaced atoms, which strengthens the NHSE overall while suppressing state amplification simultaneously. It also leads to more pronounced differences between periodic boundary conditions (PBC) and open boundary conditions (OBC) spectra, due to increased effective NHSE amplification pathways. This finding is unexpected, since disorder-localized states should rightly diminish the effects of faraway boundaries. The dynamic implications are significant, as complex energies suggest asymptotic gain over extended periods. Our findings further extend our understanding of the interplay between structural disorder and non-Hermitian physics.

2 Model and results

2.1 Representative 2D amorphous model

To explore the interplay between structural disorder and non-Hermitian physics, we propose a representative amorphous model with atoms that are randomly perturbed about their lattice positions. A key distinction from previous studies is that the position perturbation affects the non-Hermitian hopping amplitude asymmetry rather than the on-site energies. Since the asymmetric hoppings themselves control the emergent non-locality from the non-Hermiticity [19, 20, 27, 41, 98], we expect such disorder to fundamentally modify the global properties of the system. Motivated by models commonly employed in materials science, we consider a multi-orbital lattice framework. The corresponding clean Hamiltonian is given by

(1)H={\displaystyle \sum _{i,\alpha }{ϵ}_{\alpha }|i,\alpha ⟩⟨i,\alpha |+{\displaystyle \sum _{⟨i,\alpha ;j,\beta ⟩}{t}_{\alpha ,\beta }({\mathit{r}}_{ij})|j,\beta ⟩⟨i,\alpha |,}}

where i and j label lattice sites, \alpha ,\beta denoting orbitals (s,p_x,p_y) on each site. The first term describes the on-site energy of the \alpha orbital. The second term is the hopping integral term t_{\alpha ,\beta }({\mathit{r}}_{ij}), which is given by

(2)t_{\alpha ,\beta }({\mathit{r}}_{ij})={\displaystyle \frac{\mathrm{\Theta }(R-r_{ij})}{r_{ij}^2}}\mathrm{S}\mathrm{K}\left[\begin{array}{c}{V}_{\alpha \beta \delta },{\hat{\mathit{r}}}_{ij}\end{array}\right],

where atom i interacts with other atoms j within the cutoff radius R=1.9a, as shown in the yellow circle in Fig.1(a), it depends on the orbital type (\alpha and \beta) and the intersite vector {\mathit{r}}_{ij}. The \mathrm{\Theta }(R-r_{ij}) is a step function. The distance dependence of the bond parameters is assumed to be approximately captured by the r_{ij}^{-2} Harrison relation [99] and r_{ij} is set to a constant 2r_0 with r_0=0.2 if the intersite distance is shorter than the effective atomic diameter 2r_0. The \mathrm{S}\mathrm{K}[\cdot ] represents Slater−Koster (SK) parametrization for three orbitals, V_{\alpha \beta \delta } are the bond parameters, and {\hat{\mathit{r}}}_{ij} is the unit direction vector. The matrix elements of SK parameterization take the form

(3)\begin{array}{rl}& \mathrm{S}\mathrm{K}\left[\begin{array}{c}{V}_{\alpha \beta \delta },{\hat{\mathit{r}}}_{ij}\end{array}\right]\\ & =\left(\begin{array}{ccc}{V}_{ss\sigma }& l{V}_{sp\sigma }& m{V}_{sp\sigma }\\ -l{V}_{sp\sigma }& l^2{V}_{pp\sigma }+(1-l^2)V_{pp\pi }& lm{V}_{pp\sigma }-lm{V}_{pp\pi }\\ -m{V}_{sp\sigma }& lm{V}_{pp\sigma }-lm{V}_{pp\pi }& m^2{V}_{pp\sigma }+(1-m^2)V_{pp\pi }\end{array}\right).\end{array}

Each bond integral V_{\alpha \beta \delta } primarily depends on the interatomic distance, while the directional cosines (l,m,n) encode the spatial orientation of the bond. Without loss of generality, we adopt the parameter set used in the original two-dimensional triangular lattice model [100]: {ϵ}_s=1.8 eV, {ϵ}_p=-6.5 eV, V_{ss\sigma }=-0.256 eV, V_{sp\sigma }=0.576 eV, V_{pp\sigma }=1.152 eV, V_{pp\pi }=0.032 eV. For simplicity, the lattice constant is set to unity, a=1.

We next introduce non-reciprocity by modifying the hopping term t_{\alpha ,\beta }({\mathit{r}}_{ij}). Physically, it corresponds to unbalanced couplings between atoms, which can also be interpreted as imaginary flux. Taking site i as the origin, we define rightward hopping (x_j-x_i>0) as t_R=t_{\alpha ,\beta }{\mathrm{e}}^{\kappa } and leftward hopping () as t_L=t_{\alpha ,\beta }{\mathrm{e}}^{-\kappa }, where \kappa characterizes the non-reciprocal strength. The resulting non-Hermitian Hamiltonian can thus be written as

(4)\begin{array}{rl}{H}_{NH}=& \sum _{i,\alpha }{ϵ}_{\alpha }|i,\alpha ⟩⟨i,\alpha |\\ & +\sum _{⟨i,\alpha ;j,\beta ⟩}{t}_{\alpha ,\beta }({\mathit{r}}_{ij}){\mathrm{e}}^{\mathrm{s}\mathrm{g}\mathrm{n}(\mathrm{\Delta }x)\kappa }|j,\beta ⟩⟨i,\alpha |,\end{array}

with \mathrm{\Delta }x=x_j-x_i.

Second, we introduce structural disorder by applying a random displacement \delta \mathit{r} to each atomic site. To emulate the quenched disorder characteristic of amorphous materials [95, 100–103], we assume that the displacement vectors are drawn from a Gaussian probability distribution with mean \mu and variance {\sigma }^2. the position of the ith atom is given by {\mathit{r}}_i=(x_i,y_i). After the amorphization process, the new position becomes {\mathit{r}}_i^{\prime }={\mathit{r}}_i+\delta {\mathit{r}}_i, where \delta {\mathit{r}}_i=(\delta x_i,\delta y_i), as illustrated in Fig.1(b). The probability density function governing the displacement is given by

(5)f(\delta \mathit{r})={\displaystyle \frac{1}{2\pi {\sigma }^2}}\exp \left(-{\displaystyle \frac{|\delta \mathit{r}{|}^2}{2{\sigma }^2}}\right),

where \sigma represents the structural disorder strength, and for simplicity, we take \mu =0.

2.2 Skin-dominated and Anderson-dominated regions

Before presenting our main results, we highlight a key distinction in the nature of eigenstates in the presence of disorder. Fig.2 shows the energy spectra of the disordered system under PBC (orange) and OBC (blue) for a representative disorder realization. In accordance with the well-known behavior of the NHSE [18, 20, 22], OBC eigenenergies are generally expected to lie within the interior of the PBC spectrum. However, as evident in Fig.2(a), this holds only within a central energy window (shaded light blue), corresponding to the range E_2\in (-9.54,2.47], where the imaginary components of the PBC eigenvalues are significantly larger than those of the OBC spectrum. Notably, this energy window aligns precisely with the spectrum of the clean system, as shown in Fig.3. This observation is further supported by the inverse participation ratio (IPR) [34, 104, 105], defined for the ith eigenstate as the normalized sum of the fourth power of the wavefunction amplitude across all lattice sites

(6){\mathrm{I}\mathrm{P}\mathrm{R}}_i={\displaystyle \frac{{{\displaystyle \sum }}_x{\left|{\psi }_i^x\right|}^4}{({{\displaystyle \sum }}_x{\left|{\psi }_i^x\right|}^2{)}^2}},

where x denotes the lattice site. The IPR values plotted in Fig.2(b) reveal that OBC eigenstates in the skin-dominated region exhibit markedly higher localization than their PBC counterparts, consistent with strong skin accumulation. These skin-localized states concentrate near the upper right corner of the lattice, as illustrated by the spatial probability density summed over this energy range in Fig.2(d).

In contrast, the spectral regions at both high and low energies [unshaded areas in Fig.2(a) and (b)] remain largely insensitive to the choice of boundary conditions. These are the Anderson-dominated regimes, where disorder-induced localization suppresses the skin effect. The eigenstates in these regions appear as isolated, highly localized states, as shown in Fig.2(c) and (e). The IPR data confirms that these states are already localized by disorder, with little to no distinction between PBC and OBC results.

In what follows, we focus primarily on the skin-localized eigenstates, as they are the ones most susceptible to non-local effects and exhibit rich spectral features arising from the interplay between non-Hermiticity and structural disorder.

3 \mathcal{P}\mathcal{T}-symmetry breaking and effective restoration from non-Hermitian and disorder

In this section, we systematically investigate the impact of structural disorder \sigma and non-Hermiticity \kappa, first by introducing each individually, and then by considering their combined effects. These three scenarios are illustrated in Fig.3 for a two-dimensional lattice comprising N\times N atoms. Remarkably, we find that when disorder is introduced in conjunction with non-Hermiticity, it can counterintuitively enhance the non-local signatures associated with the NHSE, rather than suppressing them as might be expected from conventional localization arguments.

3.1 Non-Hermiticity or disorder, one at a time

For the original clean Hermitian lattice, (\sigma =0,\kappa =0), Hermiticity guarantees a real energy spectrum, as shown in Fig.3(a). Representative eigenstates (indicated by the orange and blue arrows) exhibit extended Bloch-like character and remain essentially insensitive to the choice of boundary conditions.

Upon introducing non-reciprocity (\kappa >0) in the absence of disorder, as shown in Fig.3(b), the energy spectrum expands significantly into the complex plane. The asymmetric hopping terms induce a pronounced NHSE, wherein all eigenstates become exponentially localized at one boundary or corner of the system. While this behavior qualitatively resembles that of the well-known Hatano−Nelson model [106] H(k)={\mathrm{e}}^{\kappa +\mathrm{i}k}+{\mathrm{e}}^{-\kappa -\mathrm{i}k}, our system exhibits richer spectral features due to its multi-orbital nature in two dimension. Compared with single-orbital models, complex multi-orbital models can provide more feedback loops when introducing disorder, and exhibit obvious and rich physical phenomena. In other words, states can choose to take the most efficient route towards amplification, even for OBCs (where the feedback loops are local, until the global PBC loop around the whole system). Indeed, there exists a mapping between complex OBC spectra and the structure of non-Hermitian hoppings [107].

In contrast, structural disorder in the absence of non-Hermiticity (\kappa =0) leads exclusively to Anderson-type localization, which appears to suppress non-local phenomena altogether. As illustrated by the representative eigenstates in Fig.3(c), the wavefunction amplitudes are confined to isolated regions dictated by the local disorder landscape and exhibit no sensitivity to boundary conditions. Consistently, the energy spectra under PBC and OBC are nearly indistinguishable. The spectrum remains real and broadens along the real axis, as structural disorder modifies local bond strengths and consequently shifts eigenenergies. In general, eigenstates at the spectral edges tend to be more strongly localized due to larger energy deviations induced by disorder. However, as we will demonstrate below, this conventional picture of disorder-induced localization must be revisited when structural disorder interacts nontrivially with non-Hermitian couplings.

3.2 Enhanced NHSE from the interplay of non-Hermiticity and disorder

Thus far, the transition from real to complex spectra — indicative of \mathcal{P}\mathcal{T}-symmetry breaking — is anticipated in systems with non-reciprocal couplings. This arises from directional amplification enabled by feedback loops, which in turn leads to spectral broadening into the complex plane. However, a more intriguing question concerns how it responds to structural disorder.

We observe the intriguing phenomenon: the enhancement of the NHSE when both non-reciprocal hopping (\kappa) and structural disorder (\sigma) are present. As shown in Fig.3(d), the imaginary parts of the eigenenergies under OBC are significantly smaller than those under PBC, more so than in the clean non-Hermitian case [Fig.3(b)]. While it is generally expected that OBCs reduce amplification by interrupting global feedback loops, our results suggest that this suppression is more pronounced in the presence of disorder.

To quantify this effect, we introduce the boundary growth suppression ratio, defined as

(7)⟨R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}⟩=⟨{\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}[\mathrm{I}\mathrm{m}(E)]{|}_{\mathrm{P}\mathrm{B}\mathrm{C}}}{\mathrm{m}\mathrm{a}\mathrm{x}[\mathrm{I}\mathrm{m}(E)]{|}_{\mathrm{O}\mathrm{B}\mathrm{C}}}}⟩.

This ratio equals 1 for systems without the NHSE and diverges (R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}\to \mathrm{\infty }) for ideal cases where OBC completely suppresses amplification — i.e., when the OBC spectrum remains entirely real while the PBC spectrum becomes complex. Such behavior is characteristic of the Hatano−Nelson model and other related classes of non-Hermitian systems [20, 21, 108].

To evaluate this effect in our model, we compare the PBC and OBC spectra for \kappa =0.15 in both clean and disordered cases, as shown in panels (b) and (d) of Fig.3. For the clean case (\sigma =0), we find \mathrm{m}\mathrm{a}\mathrm{x}[\mathrm{I}\mathrm{m}(E){]}_{\mathrm{P}\mathrm{B}\mathrm{C}}=0.62 and \mathrm{m}\mathrm{a}\mathrm{x}[\mathrm{I}\mathrm{m}(E){]}_{\mathrm{O}\mathrm{B}\mathrm{C}}=0.20, yielding a suppression ratio of ⟨R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}⟩\approx 3.1. In contrast, for the disordered case (\sigma =0.3), we obtain \mathrm{m}\mathrm{a}\mathrm{x}[\mathrm{I}\mathrm{m}(E){]}_{\mathrm{P}\mathrm{B}\mathrm{C}}=0.21 and \mathrm{m}\mathrm{a}\mathrm{x}[\mathrm{I}\mathrm{m}(E){]}_{\mathrm{O}\mathrm{B}\mathrm{C}}=0.04, giving ⟨R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}⟩\approx 5.25. These results indicate a more pronounced suppression in the presence of disorder — counter to the naive expectation that disorder alone would diminish non-local effects.

Interestingly, as the disorder strength \sigma increases, the influence of boundary conditions — specifically, the distinction between PBC and OBC spectra — becomes more pronounced in controlling amplification, as quantified by the state growth suppression ratio ⟨R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}⟩. This is counterintuitive: stronger structural disorder typically enhances localization and is expected to diminish sensitivity to boundary conditions. However, as shown in Fig.4(a), our numerical results across a broad range of disorder strengths reveal that ⟨R_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}}⟩ nearly doubles as \sigma increases from weak to moderate values (\sigma \sim 0.05 to 0.3). This implies that the spectral distinction between PBC and OBC — manifested in the extent of imaginary energy components — is greater in disordered non-Hermitian systems than in their clean counterparts. Thus, rather than merely suppressing amplification, disorder in the presence of non-reciprocity amplifies the boundary sensitivity intrinsic to the NHSE, and this trend is robust and not accidental.

To understand why structural disorder can enhance, rather than suppress, non-local boundary effects such as the NHSE, it is instructive to examine the broader implications of disorder beyond Anderson localization. A key observation is that structural disorder inherently breaks translational invariance in the local atomic density. That is, regardless of model-specific details, disorder generically induces inhomogeneities that manifest as regions of locally increased atomic density or clustering (Fig.2 and Fig.3), which are expected to be realized in more models and spatial dimensions.

Such clustering leads to a higher number of neighboring atoms within a fixed interaction cutoff R, effectively increasing the number of non-Hermitian (asymmetric) couplings. As a result, the strength of the NHSE is enhanced, even though the average atomic density remains unchanged. Intuitively, this occurs because atomic clusters not only localize wavefunctions but also host a denser network of direction-dependent hoppings, thereby enabling more efficient amplification. Provided the clusters are not spatially isolated, percolating paths through the system remain accessible, facilitating the global propagation and accumulation of amplification effects.

Importantly, this amplification mechanism is generic to structurally disordered non-Hermitian lattices where asymmetric couplings are present. In Fig.4(b), we empirically verify that even weak structural disorder leads to a statistically significant increase in the number of effective hoppings, consistent with enhanced NHSE arising from disorder-induced clustering.

4 Examining spectral transitions with non-Hermiticity and disorder

To elucidate the aforementioned trends more explicitly, we systematically examine the evolution of the energy spectrum of the 2D lattice as a function of structural disorder \sigma and non-Hermiticity \kappa. In each case, eigenvalues are color-coded according to the IPR of their corresponding eigenstates [Eq. (6)], enabling a clear visual distinction between localized states (high IPR, bright colors) and extended states (low IPR, dark red). This representation provides direct insight into how disorder and non-Hermitian effects modify the spatial structure of the eigenstates across the spectrum.

The energy spectra in the absence of structural disorder (\sigma =0) are presented in Fig.5(a) and (b). As \kappa increases, the energy spectrum broadens along the imaginary axis, consistent with the emergence of non-reciprocal amplification. Upon switching from PBC to OBC, extended bulk states undergo a transition to skin-localized states, accompanied by a noticeable increase in localization, as reflected by the brighter IPR coloring — an effect that becomes more pronounced at larger \kappa. Fig.5(c) and (d) display the corresponding spectra for a representative structurally disordered configuration with \sigma =0.3. Under PBC and at \kappa =0, the central spectral region (associated with the clean-limit Bloch band) remains weakly localized (dark coloring), while highly localized states induced by disorder appear near the spectral edges (yellow). With increasing \kappa, the central skin-dominated region expands significantly along the imaginary axis, whereas the isolated Anderson-localized states at the edges remain largely insensitive to the non-Hermiticity. Under OBC, the imaginary extent of the spectrum in the skin-dominated region likewise increases with \kappa, and the corresponding eigenstates exhibit enhanced localization. Crucially, the effects of boundary conditions are predominantly manifested within the skin-dominated region, while the Anderson-localized states remain essentially unaffected.

Fig.6 illustrates the evolution of the energy spectrum as a function of increasing disorder strength \sigma, for both the Hermitian (\kappa =0) and representative non-Hermitian (\kappa =0.15) cases. In the Hermitian limit (\kappa =0), the spectrum remains confined to the real axis but broadens with increasing \sigma, reflecting the disorder-induced smearing of the band edges. The small band gap near the center gradually closes, and localized states with high IPR (yellow) emerge near the spectral extremities, characteristic of Anderson localization. The bulk states, however, remain weakly localized (dark), retaining a resemblance to extended Bloch states. In contrast, for the non-Hermitian case (\kappa =0.15), increasing \sigma leads to a suppression of spectral weight along the imaginary axis and a concomitant broadening along the real axis. This reflects the competition between non-reciprocal amplification and disorder-induced scattering. Isolated high-IPR states begin to populate the edges of the spectrum. Notably, while the overall growth of \mathrm{I}\mathrm{m}(E) is reduced with increasing disorder, the spectrum does not fully fragment into isolated boundary-localized skin states; rather, a substantial fraction of the spectrum remains composed of delocalized or weakly localized states.

4.1 Skin condensation fraction

Thus far, our discussion has primarily focused on how structural disorder alters the spectral characteristics of the system. We now turn our attention to its impact on the formation of skin states, which are preferentially localized toward the upper-right corner of the system [Fig.7(a)], consistent with the directionality imposed by non-reciprocal hopping. To quantify the occurrence of skin states in a manner that is robust against disorder-induced localization (which can artificially inflate metrics like the IPR), we introduce a geometric criterion: an eigenstate is classified as a skin state if more than 20% of its total amplitude resides within a corner sector of radius L/2 centered in the NHSE-directed corner. While the 20% threshold is heuristic, we find that moderate variations of this value do not alter the qualitative trends described below.

The fraction of eigenstates within the skin-dominated region that satisfy this criterion is shown in Fig.7(b) and (c), plotted as a function of the non-Hermitian asymmetry \kappa and disorder strength \sigma, respectively. As expected, increasing \kappa enhances the skin-state fraction, with nearly all eigenstates exhibiting skin behavior at \kappa \gtrsim 0.4, independent of the degree of disorder. In contrast, increasing \sigma tends to suppress the skin effect by inducing disorder-localized states that do not exhibit directed amplification, even though residual skin states may still give rise to enhanced non-Hermitian amplification effects, as discussed earlier. These competing trends are encapsulated in the phase diagram of Fig.7(d), which maps the skin-state fraction across the (\sigma ,\kappa ) parameter space. As seen, larger disorder strengths require correspondingly stronger nonreciprocity to sustain a high density of skin states.

It is important to emphasize that this corner-based metric may also capture eigenstates localized by other mechanisms, such as disorder-induced localization within the same sector. Therefore, the reported skin-state fraction should be interpreted in a relative sense. For example, in the Hermitian limit \kappa =0, no NHSE is expected, yet a nonzero skin-state fraction can still arise due to the incidental localization of a subset of states within the selected region.

4.2 Phase diagrams for spectral vs. localization behavior

Finally, we present in Fig.8 phase diagrams of three spectral observables — ⟨max[\mathrm{I}\mathrm{m}(E)]⟩, ⟨\mathrm{I}\mathrm{P}\mathrm{R}⟩, and ⟨[\mathrm{I}\mathrm{m}(E){]}^2⟩ — as functions of non-Hermitian asymmetry \kappa and disorder strength \sigma, each averaged over 50 independent disorder realizations. In agreement with prior discussions, both ⟨max[\mathrm{I}\mathrm{m}(E)]⟩ and ⟨[\mathrm{I}\mathrm{m}(E){]}^2⟩ generally increase with larger \kappa and decrease with increasing \sigma, consistent with the amplification-driven nature of the NHSE and the localization tendency of structural disorder.

However, notable distinctions emerge between results computed from eigenstates restricted to the skin-dominated region (top row) versus those obtained from the full energy spectrum (bottom row). In the former case, the average IPR remains high in the large-\kappa regime, reflecting the enhanced spatial localization due to the skin effect [Fig.8(b)]. In contrast, when all eigenstates are included, disorder-induced (Anderson) localization becomes the dominant contributor to the IPR beyond \kappa \gtrsim 0.15, masking the spectral signature of the NHSE.

Another key contrast lies in the behavior of ⟨[\mathrm{I}\mathrm{m}(E){]}^2⟩ versus ⟨max[\mathrm{I}\mathrm{m}(E)]⟩ in the full-spectrum case. Specifically, at large \kappa and \sigma, ⟨max[\mathrm{I}\mathrm{m}(E)]⟩ remains elevated, while ⟨[\mathrm{I}\mathrm{m}(E){]}^2⟩ does not increase correspondingly. This indicates the presence of a small subset of highly amplifying eigenstates — likely localized in rare amplification-prone configurations — whose large imaginary components elevate the spectral upper bound without significantly influencing the mean-square value. These findings underscore the importance of distinguishing between bulk trends and extreme behaviors in strongly non-Hermitian, disordered systems.

5 Conclusion and discussion

Starting from the spectral separation between Anderson-localized and NHSE states, we have systematically investigated how the interplay between non-Hermiticity (\kappa) and structural disorder (\sigma) governs both spectral characteristics and eigenstate localization in a two-dimensional non-Hermitian lattice. Our results reveal that structural disorder — unlike conventional on-site disorder — not only induces localization but also modulates the distribution of effective asymmetric hoppings, leading to competing effects that can effectively restore \mathcal{P}\mathcal{T} symmetry that suppress the imaginary components of the spectrum. Importantly, we find that weak to moderate structural disorder can enhance the NHSE.

Although our numerical analysis is based on a representative multi-orbital tight-binding model, the physical insights are expected to extend broadly to other multi-component systems where structural disorder correlates with direction-dependent non-Hermitian couplings. Such conditions are experimentally realizable in a wide range of synthetic platforms, including photonic crystals, topolectrical circuits, mechanical lattices, and other classical metamaterials [2, 4, 16, 41–43, 72, 73, 109–118], as well as well as quantum circuits [11, 16, 42, 43, 119–128] and ultracold atomic systems with synthetic dimensions and anisotropic orbital couplings [77–79, 81, 129, 130]. In topolectrical circuits, asymmetric hopping can be implemented using non-reciprocal elements such as operational amplifiers [16], with structural disorder introduced via circuit component variation. In photonic coupled-resonator arrays, asymmetric coupling can be realized through spatially separated gain and loss regions [41], while structural disorder can be simulated by perturbing waveguide positions or modulating gain/loss contrast locally. These approaches enable direct realization and control of spatially inhomogeneous non-Hermitian couplings, rendering our predictions experimentally accessible. In addition, the enhanced non-Hermitian sensitivity due to disorder can also be potentially utilized in the design of high-precision sensors [131], which are holding increased importance in practical applications.

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