Anomalous Skin Effects in Disordered Systems with a Single non-Hermitian Impurity

We explore anomalous skin effects at non-Hermitian impurities by studying their interplay with potential disorder and by exactly solving a minimal lattice model. A striking feature of the solvable single-impurity model is that the presence of anisotropic hopping terms can induce a scale-free accumulation of all eigenstates opposite to the bulk hopping direction, although the nonmonotonic behavior is fine tuned and further increasing such hopping weakens and eventually reverses the effect. The interplay with bulk potential disorder, however, qualitatively enriches this phenomenology leading to a robust nonmonotonic localization behavior as directional hopping strengths are tuned. Nonmonotonicity persists even in the limit of an entirely Hermitian bulk with a single non-Hermitian impurity.

A unique feature of NH systems, which has no counterpart in the Hermitian domain, is the accumulation of an extensive number of eigenstates at the boundaries, a phenomenon coined as the non-Hermitian skin effect (NHSE) [9].The phenomenology and origin of this remarkable effect -including the connection to the existence of non-trivial topological invariants -remains an active field of current research [26][27][28][29][30][31][32][33][34][35][36].Crystal defects in NH systems [37] have also turned out to be an alternative platform to induce the NHSE, even when it is not present under open boundary conditions (OBC) [38][39][40][41].This extensive theoretical research has effusively led to the exploration of different experimental platforms on which NHSE could be observed [42,43].Underpinning the NHSE lies an extreme sensitivity of the spectrum to boundary conditions [8,44,45], which opens up new potential avenues for sensor applications [46][47][48][49][50].
On the other hand, the impact of single local impurities on physical properties have also been the subject of much current interest in many other areas of physics [70][71][72][73][74][75].
In Ref. 76, it was first observed how non-reciprocal impurities in a non-Hermitian Hatano-Nelson chain induced scale-free localized (SFL) states.It was also discussed how the variation of the impurity strength could pro- duce transitions between NHSE and SFL, including the counter-intuitive behavior of localization in the direction opposite to the predominant hopping term.Moreover, solutions for an on-site impurity in NH Hatano-Nelson and SSH chains were discussed in [86], showing how the impurity effectively can act as an open boundary condition for the system.Similarly, tight connections between the NHSE and the presence of an impurity in onedimensional lattices were also studied in [17,87], where an on-site infinite impurity (site vacancy) was considered to develop a method to calculate the eigenstates of the system based on the Green's function method.Further work includes the study of the effect of NH impurities on the properties of Dirac systems [88] or the introduction of topological defects in NH electrical circuits [89].
Here we combine the aforementioned notions of localization by considering a single NH impurity in a onedimensional tight-binding model subject to on-site disor-  c)).We refer to these as anomalous skin effects for several reasons.First, the scale-free localization in the presence of impurities is qualitatively distinct from the NHSE occurring at open boundaries both in that its localization is not dictated by the bulk (Fig 1(b)) and that the localization length is not fixed but instead proportional to the system size [76].Second, in presence of disorder, there is a non-monotonic localization behavior as a function of the hopping terms (Cf.Fig 1(b)).Third, the key features including a scale-free skin localization and non-monotonic dependence of localization as a function of hopping strength persist even in the limit of a NH impurity in an otherwise Hermitian bulk (Fig 1(c)).To elucidate these points we begin by fully analytically solving a special limit of this system that nevertheless contains key ingredients that, together with the standard phenomenology of Anderson localization, explains our numerically-obtained phase diagrams (Fig 1(b) and (c)).

II. MODEL
Throughout this work we consider an extension of the Hatano-Nelson model [77][78][79], a paradigmatic example of a one-dimensional chain in which the non-reciprocity of nearest neighbour hoppings leads to the NHSE under OBC.The Hatano-Nelson Hamiltonian reads where t r and t l indicate nearest-neighbor hopping respectively to the right and to the left, V n is an on-site disorder potential that we take from a random uniform distribution [−V, V ] [80], and γ r and γ l are hopping strengths between the first and last site, which parametrize an impurity.This model is sketched in Fig. 1(a).In general Eq. ( 1) is non-Hermitian.However, we consider t r , t l , γ r , γ l , V to be real, thus H has a real matrix representation, H = H * .Eigenvalues are consequently real or appear in complex conjugate pairs.

A. Solvable limits
Key insights can already be understood through the study of simple particular cases.Solving the Hamiltonian Eq. (1) in the simplest non-trivial case when t r = t l = V = γ l = 0 but γ r ̸ = 0, we find an order two exceptional point and N -fold energy degeneracy at E = 0.The N − 1 distinct right eigenstates can be taken as localized at sites n = 1, . . ., N − 1.This solution, however, is highly unstable towards hopping.
As we turn on the hopping, t r , while keeping t l = V = γ l = 0, we get a more non-trivial case in which the Hamiltonian reads as Again the eigenspectrum E n and the (right) eigenvectors |Ψ R,n ⟩ can be found exactly as [79] E with some normalization constant N and where n = 1, . . ., N labels the eigenstates and their corresponding eigenvalues, which occur on a circle in the complex plane with radius γ . Here it is important to note that the t r → 0 limit is not smooth as can be seen by comparing with the first solvable example described at the beginning of the previous paragraph.
In Eq. ( 4) one can readily see that in the limit γ r → 0 all right eigenvectors are completely localized on the right boundary: T .Dual to this limit, when γ r → ∞, i.e. the boundary coupling is infinitely strong, there also exists perfect boundary localization.In the dual limit, however, all right eigenvectors are completely localized at the left boundary: . Note that this is more general: interchanging t r and γ r changes the localization to the opposite side.
In this toy model we can thus control the localization of the eigenstates through the strength of the impurity γ r .Since the expression Eq. ( 4) is only valid for a finite t r (analogous results and conclusions would hold for finite t l , γ l ), this clearly implies that adding finite t l/r immediately localizes all eigenstates, even if the added hopping is directed opposite to the direction of localization.The reason for this is essentially that the sites need to be connected in order to make the localization possible.The nature of the connections are however not crucial, hence allowing for the counter intuitive phenomenon of localization in the opposite direction compared to the added terms.This counter-intuitive effect highlights a different behavior compared to the NHSE, where the localization is towards the leading unbalanced hopping.
We remark that while biorthogonality and simultaneously considering both left and right eigenstates is fundamental for understanding some aspects such the appearance of boundary modes [8,50], the phenomenology of the skin effect is readily highlighted by considering the localization properties of (either left or) right states as we do in this work.
Analytical investigation of the toy model in Eq. ( 2) and its experimental realization based on a non-Hermitian circuit platform has been performed in Ref. 82.

B. Phenomenology beyond the solvable limits
From the above exact solutions it is possible to understand the full behavior of the generic model in Eq. ( 1) in detail.To begin with, we keep the maximum nonreciprocity in both the hopping and impurity terms, setting γ l = t l = 0, while introducing disorder in the form of a random uncorrelated on-site potential of maximal strength ±V .We then scan the t r -γ r parameter space and report the behavior of the eigenstates, as shown in Fig. 1(b).
For small values of the impurity strength γ r , making the hopping t r larger (i.e. the non-Hermiticity of the bulk) will gradually increase the localization of the eigenstates to the hopping direction, as expected in the NHSE regime.This behavior can be easily quantified by calculating the average eigenstate localization in the form of the mean center of mass (mcom) of the amplitude squared of all eigenvectors |Ψ R,n ⟩, averaged over many disorder realizations N r , i.e.

⟨A(ℓ)⟩
where ⟨•⟩ V indicates disorder averaging.We plot the mcom in Fig. 2(a) over six orders of magnitude for both t r and γ r and disorder strength V = 0.1.From this plot, we can clearly see that in the limit of small γ r and large t r , all eigenstates pile up on the right end of the chain.
We remark that there are other quantities that could be used to probe the localization, but the information they provide should be essentially equivalent to that of the mcom because of the noninteracting nature of the system.In appendix B, for instance, we show results for the disordered-averaged inverse participation ratio (IPR).However, while the IPR performs very well in detecting Anderson localization, its measure of the NHSE localization is inferior to that obtained via the mcom.In appendix D, we show plots of biorthogonal quantities, calculated using both left and right eigenvectors.Since the left and right eigenstates are localized on opposite boundaries in the NHSE phases, biorthogonal quantities are not useful in determining localization properties and can at most only discriminate whether a phase is Anderson-localized or not.We also remark that the information provided by the eigenspectrum, such as the complex eigenvalue fraction, is also redundant for our purposes of determining NHSE localization.
More interestingly, increasing the impurity strength causes the system to exhibit a different behavior.For small values of the right hopping t r , the disorder introduced through the on-site potential will always dominate, leading the eigenstates to localize on the basis of Anderson localization.This corresponds to a mcom ≈ N/2 in Fig. 2(a) (purple region).However, at sufficiently large values of t r , we can distinguish two different behaviors of the system when γ r is ramped up to progressively larger values.
At first, when t r ≪ γ r , the increase of the right hopping t r causes the eigenstates to pile up to the left of the chain, following the counterintuitive picture already observed in the toy model at V = 0.In Fig. 2(a), this is signalled by mcom ≈ 1 (dark-colored region).The localization is exponential, i.e. the disorder-averaged mean amplitude squared of all eigenvectors can be very well fitted by ⟨A(ℓ)⟩ with the localization length ξ L and some amplitude A L .
Upon increasing the right hopping beyond t r ⪆ γ r , however, the localization is rapidly inverted and all eigenstates pile up again towards the right end of the chain (next to the impurity) as observed for small γ r and large t r .Again, the localization is exponential, with the form . The progression from Anderson localized, to left-localized, to right localized skin effects is depicted explicitly in panels (b)-(e) of Fig. 2, where the mean amplitude squared of all eigenvectors is plotted for N r = 1000 disorder realizations.
In both skin localized phases, the localization length ξ L/R is proportional to the system size N , as can be seen in Fig. 2(f).We find that ξ L ≃ ξ R over many orders of magnitude in N , indicating that the underlying scaling is the same for both left-and right-localization.Furthermore, the scaling matches perfectly with what we can obtain exactly for the clean limit of the toy model (2), i.e. when V = 0.0, t l = r l = 0.0.This agreement demonstrates that the core mechanism for the localization physics lies within the interplay between hopping and impurity, but is stabilized by the disorder to a proper phase.This linear N -dependence is to be starkly contrasted with the standard NHSE occurring at γ r = 0.0, where the localization length remains instead constant for any value of N (green markers and line).This unique phenomena of scale-free eigenstates, representing an anomalous skin effect are always accompanied by the emergence of complex eigenspectrum, as observed in [76].
We remark that the nonmonotonicity in the localization arises not from the NHSE phase, but rather from the Anderson localized phase.The disorder element is therefore crucial: at low values of disorder the nonmonotonicity namely disappears completely and the same behaviour is observed in presence of a periodic potential, see the supplementary material for details [83].A heuristic explanation for the lack of nonmonotonicity in the periodic potential case is that the eigenstates remain delocalized in absence of the NH impurity, and hence immediately feel its presence when turned on.We also note that larger values of V will lead to qualitatively similar phase diagrams to the one shown in Fig. 2, but where the appearance of the left-localized phase is shifted to larger values of the parameters.Similarly, nonzero values of t l and γ l might shift the detailed shape of the phase diagram, but do not alter the generic features we presented.

C. Analytical mcom in the clean case
By using the exact solutions of the toy model, Eq. ( 4), it is possible to obtain an analytical expression for the mcom in the limit of zero disorder V = 0 (see supple-mentary material for detailed derivation): Fig. 3 shows the analytical mcom normalized with the chain length for various values of N .The analytical result showcases the change in localization from the left to the right of the chain as a function of the ratio t r /γ r , and is corroborated by the numerical results.Furthermore, we can appreciate how in the limit of large N , all curves fall on top of each other, indicating scale-free localization.
FIG. 3. Behavior of the mean center-of-mass as a function of the ratio tr/γr and for increasing values of chain length N in the clean case V = 0. Solid lines are calculated analytically with formula Eq. ( 7), while dashed lines correspond to numerical results.

D. Hermitian hopping case
As we have explored in the solvable model, large values of the right impurity strength γ r dominate the system behavior, localizing eigenstates to the left.This means that the increase of the hopping in the direction of the localization (i.e.t l ) will have no effect on the localization behavior, i.e. the scale-free localization to the left will prevail.Thus adding such hopping to the solvable model Eq. ( 2) to make it Hermitian in the bulk, i.e. t l = t r preserves the localization property of Eqs. ( 3)-( 4).This sheds light on the provenance of the scale-free localization induced by local non-Hermitian impurities in otherwise Hermitian systems as noticed very recently in Refs.84 and 85.In the Hermitian case, however, neither scale-free localization to the right nor NHSE is observed, as would be expected.
The sketch of the phase diagram in the Hermitian case is displayed in Fig. 1(c).A more detailed description of the localization properties is instead presented in Fig. 4. In panel (a), the phase diagram is obtained by means of the mcom, where we can clearly see the absence of localization at right boundary.Note that from the perspective of the mcom, the Anderson localized phase and the phase where the hopping dominates (for t r > γ r > V ) cannot be distinguished from each other.In this case, the inverse participation ratio could be used as additional tool (see appendix B).However, here we are mainly interested in scale-free localization behavior.
Panels (b)-(e) show the mean amplitude squared of all eigenvectors N r = 1000 disorder realizations, and increasing value of t r corresponding to the points displayed in panel (a).While the left localization is achieved upon increasing the right hopping to values comparable to but smaller than γ r , further increasing t r simply delocalizes the eigenstates across the entire chain.Finally, panel (f) shows the localization length ξ L obtained by fitting the disorder-averaged mean amplitude squared with an exponential function.Again, we observe a linear behavior with the system size N indicating scale-free localization, and indeed with the same slope observed for the non-Hermitian hopping case.

IV. CONCLUSIONS
We have explored anomalous skin localization phenomena induced by non-Hermitian impurities.On a basic level this showcases fundamental properties of NH spectra [45] and their associated novel eigenvector properties [90].At the same time it relates directly to experimental realities in a variety of systems ranging from robotic metamaterials [25] and electrical circuits [23,24] to optical systems [22,91,92] in which the standard NHSE has already been observed.
While the effect of impurities in NH one-dimensional tight-binding models has been explored in previous works, here we have added several important aspects.First, we have identified a minimal analytically solvable model that exhibits anomalous skin effects.The solution of our model do not involve any approximations and yields all eigenenergies and eigenstates at any finite size.This model highlights a previously overlooked nonanalytic weak hopping limit and a counterintuitive nonmonotonic relation between (directional) hopping and localization.Second, we have established that, by adding bulk disorder, the aforementioned non-monotonic behavior is promoted from a highly fine-tuned point in parameter space into a generic and stable phenomenon.Third, we have shown that adding bulk hopping in the direction of the scale-free localization cannot undo it (in fact it leaves it unchanged) hence explaining why a single non-Hermitian impurity can also induce a scale-free skin localization in an otherwise Hermitian system.
These results corroborate the potential for harnessing impurities for local sensing and control of a large class of effectively non-Hermitian systems.That the phenomenology also extends to systems that are Hermitian in the bulk, which follows transparently from perturbing away our solvable model, further extends the scope of these insights.
Finally, very shortly before posting this work, several pre-prints pointed out the possibility of inducing scalefree localization through NH impurities in a Hermitian bulk based on tight-binding models distinct from ours [84,85], as well as by solving dissipating spin chains in the thermodynamic limit using Bethe ansatz [93].In this section, we summarize the behavior of the mean center of mass (mcom) Eq. ( 6) as a function of different on-site potential strengths.This quantity was used in the main text as an effective order parameter to classify the system into phases with different localization properties.We have calculated the mcom as a function of the parameters t r and γ r for N = 50 sites and t l = γ l = 0.0.The results are shown in Fig. 5. Panel (a) shows the results for the clean limit given by the toy model of Eq. ( 2).As discussed in the main text, the nonmonotonic behavior in the eigenstate (left) localization is not present in this limit: for any value of γ r > t r , increasing t r will only localize the eigenstates more to the right.
The same behavior is observed also when we add an alternating on-site potential to the toy model, i.e. for V 2n−1 = V , V 2n = −V , n = 1, . . ., N in Eq. ( 1).Remarkably, the localization properties are identical to those of the toy model, shown in Fig. 5(b).
In order to introduce a nonmonotonic behavior in the eigenstate localization, we need to add a disordered potential as explained in the main text.This leads to the emergence of an Anderson localized phase at small values of the right hopping t r .For large enough values of the impurity hopping γ r , it is then possible to left-localize the eigenstates by increasing the hopping to the right.We find that this qualitative picture persists for all values of the disorder potential strength V , whereas V controls the onset of the left-localization as seen in Fig. 5(c) and  (d).
In the Hermitian hopping regime t l = t r , it is still possible to control the eigenstate localization by increasing t r and large values of γ r , much like in the non-Hermitian case.This feature is shown in Fig. 5(e) and (f).Here, however, the right-localized phase disappears completely and the localization is only possible on the left end of the chain.Again, increasing the value of V will not change this qualitative picture, but will simply shift the leftlocalized phase to larger values of t r and γ r .
A comprehensive analysis of the localization behavior as a function of all system parameters (t r , γ r , and V ) is shown in Fig. 6.The picture that comes out of these phase diagrams is consistent with the expectation that when V is the dominant energy scale, the states are Anderson localized with a mcom of around N/2; when t r is the dominant energy scale, the states localize on the right end of the chain; when γ r is the dominant energy scale, the states localize on the left end of the chain.
The strength of the disorder can also impact how the localization length scales with system size.More precisely, we find that when V becomes comparable to the hopping strength t r , the slope of ξ(N ) begins to increase and deviate from the clean limit of the toy model already at values N ∼ 50, as shown in Fig. 7.This can be understood as a consequence of the shift of the localized phases towards larger values of t r and γ r induced by increasing values of V , already seen in Fig. 5(c) and (d).When γ r is kept fixed at a large enough value such that the system lies in the left-localized phase (dark area in Fig. 5), increasing V will progressively shift the mcom towards more central values (dark purple region in Fig. 5), affecting the scaling of the localization length in the process.This behavior indicates that disorder might be used as an additional lever in controlling the scale-free localization observed in the generalized Hatano-Nelson model.

FIG. 1 .
FIG. 1.(a) Sketch of the system considered in this work, consisting of a chain of sites with hopping tr (to the right) and t l (to the left), on-site potentials Vn, and impurity hopping between first and last site γr (|N ⟩ → |1⟩) and γ l (|1⟩ → |N ⟩).(b)-(c) Sketch of the phase diagram of the model as a function of right hopping tr and impurity strength γr.In (b) t l = 0 and in (c) t l = tr (Hermitian hopping).
der (Fig 1(a)).A rich interplay between different phases of the system is observed, including the NHSE, Anderson localization and the appearance of scale-free skin localization (Fig 1(b) and (

FIG. 2 .
FIG. 2. (a) mcom as a function of right impurity hopping γr and right bulk hopping tr for V = 0.1, showing the three different phases discussed in the main text.(b)-(d) mean eigenvector amplitude squared for Nr = 1000 disorder realizations (colored lines) and disorder-averaged (thick black line) for the four points indicated in panel (a).The vertical dashed lines indicate the mcom in each case.(f) Localization length ξ L/R extracted from the scale-free left-and right-localized phases (red squares and yellow dots, respectively), compared to the same results obtained from the toy model (green crosses and cyan triangles).We can observe perfect agreement between the scaling of the toy model and that of the disordered model.The scaling of the standard NHSE phase at γr = 0.0 is also depicted as blue pluses and clearly shows no N -dependence.The colored dashed lines indicate linear fits to the data points.

FIG. 4 .
FIG. 4. (a) mcom as a function of right impurity hopping γr and Hermitian bulk hopping t l = tr for V = 0.1.(b)-(d) mean eigenvector amplitude squared for Nr = 1000 disorder realizations (colored lines) and disorder-averaged (thick black line) for the four points indicated in panel (a).The vertical dashed lines indicate the mcom in each case.(f) Localization length ξL extracted from the scale-free left-localized phase.The dashed line indicate a linear fit to the data points.

FIG. 7 .
FIG. 7. Scaling of the (left) localization length as a function of chain length N up to N = 50 for increasing values of the disorder strength V .The other parameters are t l = 0.0, tr = 10 3 , γr = 10, γ l = 0.0.