Impurity induced scale-free localization

Non-Hermitian systems have been shown to have a dramatic sensitivity to their boundary conditions. In particular, the non-Hermitian skin effect induces collective boundary localization upon turning off boundary coupling, a feature very distinct from that under periodic boundary conditions. Here we develop a full framework for non-Hermitian impurity physics in a non-reciprocal lattice, with periodic/open boundary conditions and even their interpolations being special cases across a whole range of boundary impurity strengths. We uncover steady states with scale-free localization along or even against the direction of non-reciprocity in various impurity strength regimes. Also present are Bloch-like states that survive albeit broken translational invariance. We further explore the co-existence of non-Hermitian skin effect and scale-free localization, where even qualitative aspects of the system’s spectrum can be extremely sensitive to impurity strength. Specific circuit setups are also proposed for experimentally detecting the scale-free accumulation, with simulation results confirming our main findings. Non-Hermitian physics describes an open system, which is susceptible to loss or gain and has been recently used to demonstrate unusual physical phenomena in non-trivial topological systems. Here, the authors investigate the physics of impurity effects in non-Hermitian, non-reciprocal lattices and discuss how the results can be realised using an electrical circuit.

S patial inhomogeneity in physical systems is the norm rather than the exception. It can trigger a wide variety of physical phenomena, such as the Anderson localization, topological edge states, and topological defect states. In non-Hermitian systems, intriguing physics from spatial inhomogeneity encompasses not just the non-Hermitian skin effect (NHSE)  , but also impurity-induced or defect-induced topological bound states [30][31][32][33] , disorder-driven non-Hermitian topological phase transitions 34 , as well as non-Hermitian quasi-crystals and mobility edges with an incommensurate modulation [35][36][37][38] .
Due to their emergent non-locality, non-reciprocal impurities in non-Hermitian systems generate dramatic spectral flows as their strengths are varied 3,39,40 . This has even been proposed for exponentially enhanced quantum sensing in an experimentally realistic setting 41,42 . Since most of conventional condensed matter studies have been based on the assumption of locality, this nonlocality can challenge usual notions of wavefunction decay, such as the relationship between scale-free behavior and criticality. Besides, there does not exist a full framework for non-Hermitian impurity physics, with periodic and open boundary conditions (PBCs and OBCs) being special cases across a whole range of boundary impurity strengths. This work aims to fill in this important gap and reports unexpected findings of general theoretical and experimental interest.
Specifically, we discover that boundary impurities in nonreciprocal lattices can generate new types of steady-state localization behavior characterized by scale-free accumulation (SFA) of eigenstates, despite having non-power-law profile. This enigmatic behavior is possible because the correlation length does not have a fixed value but scales with the system size, despite the system being not critical. In sharp contrast to the NHSE, the SFA direction can be counter-intuitive, opposite of the non-reciprocal directionality. With varying impurity strengths, the steady state makes transitions between the NHSE behavior, Bloch-like eigenstates with broken translational invariance, ordinary SFA, and reversed SFA. A careful inspection of these qualitatively rich transitions reveals fascinating duality relations between weak and strong inhomogeneities, yielding a big picture of non-Hermitian impurity physics. Known NHSE properties are thus revealed as only one of the many impurity-induced consequences in nonreciprocal non-Hermitian systems. Drastically different steadystate behaviors can even co-exist when next-nearest hoppings are present, a useful phenomenon that can benchmark the hypersensitivity of non-Hermitian systems to boundary/impurity effects due to their emergent non-locality.

Results
Impurity-induced SFA. We consider impurities in the simplest 1D Hatano-Nelson chain 43 , which already exhibits nearly the full scope of impurity-induced phenomena in more generic lattices. An impurity is represented as a modified coupling between the first and last sites: with μ ± = μe ±α , μ controlling the local impurity, α > 0 and x = 0, 1, . . ., L labeling the lattice sites (Fig. 1a). Physics of impurity in a non-Hermitian lattice was also studied in ref. 33 . However, ref. 33 investigated a topological nontrivial lattice with on-site impurity, with a focus on the few topological edge modes isolated from the continuous bands. Of central interest here is the impact of hopping impurity on the bulk eigenmodes. Because of our setting, PBCs are recovered at μ = 1, where translational symmetry is restored and the system can be described by a Bloch Hamiltonian H(z) = e α z + e −α /z with z = e ik , k the quasi-momentum. Perfect OBCs yielding the NHSE are recovered at μ = 0, although a finitesize system behaves like OBCs when μ ≲ e −α(L+1) 26,40 . Cases with 0 < μ < 1 may be interpreted as interpolations between PBCs and OBCs, but a full picture with new physics emerges only if the whole range of 0 < μ < ∞ is investigated. Beyond μ ∈ [0, 1], eigenstates can exhibit weaker boundary accumulation toward either direction, even unexpectedly against the direction of nonreciprocity and NHSE (Fig. 1b). Furthermore, this intriguing localization phenomenon is dubbed the SFA because the eigenstates display a scale-free spatial profile, decaying as e −Cx/L with constant C, as elaborated later. Unlike in the NHSE, the spectrum of these SFA eigenstates forms a loop that can be deformed away from, or even enclosing the PBC spectrum (Fig. 1c). Different accumulation regimes exist for μ ranging from 0 to ∞, and notably similar behaviors are seen in both the small and large μ limits (Fig. 1d). As detailed in our concrete examples later, we find two types of dualities between μ and~1/μ, which allow us to probe the large μ regime from the small μ regime, and vice versa.
Ordinary and reversed SFA. To understand why SFA occurs, we analytically solve for the eigenstates Ψ n ¼ P L x ψ x;nĉ y x 0 j i, n = 0, . . . , L via HΨ n = E n Ψ n , under reasonable approximations. In the large-μ limit with μ ≫ e ±α , two isolated eigenstates strongly localize at x = 0, L, with eigenenergies E iso ≈ ±μ (see the "Methods" section). The other eigenstates are exponentially decaying: with ψ 0,n ≈ 0 (see the "Methods" section], n = 1, 2, . . . , L−1 yielding L−1 different eigenstates. These L−1 eigenstates have the common decay constant κ L . Physically, the vanishing amplitude . ., L the number of lattice sites, e ±α the non-reciprocal nearest-neighbor couplings, and impurity couplings μe ±α between the sites 0 and L. Periodic and open boundary conditions (PBCs and OBCs) correspond to μ = 1 and 0, respectively, but other μ support qualitatively different phenomena. b Spatial eigenstate distributions ρ(x) = |ψ x | 2 with different accumulating behaviors (normal and reversed scale-free accumulations, and the non-Hermitian skin effect (NHSE)), with ψ x the wave-function value at x. c Complex spectra with Re½E and Im½E the real and imaginary parts of the energies, distinguishing the four types of eigenstates in (b). d Different regimes across the whole range μ are marked by different accumulation phenomena, with dualities relating strong and weak μ. at x = 0 can be partially appreciated by the physics underlying electromagnetic field-induced transparency 44 . That is, the much stronger coupling between sites L and 0 effectively makes the rest of the lattice more "transparent", and hence suppresses the population pumping from the rest of the lattice to site 0. In a more restricted parameter regime with e α ≫ e −α and μ ≪ e α(L+1) , the corresponding eigenenergies can be further approximated by with k n := (2n + 1)π/(L−1) and ϵ(k) the eigenenergy function at μ = 1 (i.e. PBCs) (see the "Methods" section). Remarkably, here the spectrum is obtainable via a complex deformation of the PBC quasi-momentum, similar to the approach of generalized Brillouin zone (GBZ) for OBC systems 1-3 , yielding an exponential decay profile of the eigenstates. Yet, the associated decay exponent κ L in Eq. (1) is inversely proportional to the size of the lattice, namely, κ L / 1 LÀ1 % 1 L , indicating weaker accumulation for a larger system. In fact, because of this, the overall decay of the eigenstates from one end of the lattice to the other end, given by |ψ L,n /ψ 1,n | = e 2α /μ, is independent of L, thus giving rise to a scalefree decay profile from x = 1 to L. The dependence of κ L on μ (and L) also differs from that of impurity-induced topological localization (see Supplementary Note 1 and Fig. 1).
Counter-intuitively, reversed accumulation with negative κ L can occur when μ < e 2α , which still falls into a valid sub-regime if μ ≫ e α ≫ e −α , as confirmed by the agreement between our approximate solutions and numerical results in Fig. 2a. For the peculiar borderline case of μ = e 2α between ordinary and reversed SFA, κ L = 0 and the eigenstates are uniformly distributed (except at x = 0) and hence resemble Bloch states [Eq. (1) and Fig. 2a], even though translational invariance is broken. Indeed, the continuous part of the associated spectrum also coincides with the PBC spectrum [Eq. (2) and Fig 2b]. This curious case of quasi-PBC delocalized states is elaborated in the "Methods" section. While we have considered a strong non-reciprocity of α = 4 in Fig. 2 for a better illustration, more examples with weaker α are found in Supplementary Note 2 and Fig. 2.
Duality between strong and weak impurity couplings. The discussions above imply a duality between PBCs at μ = 1 and quasi-PBCs at μ = e 2α . This motivates us to seek duality relations for the whole range of μ. For e −2α ≪ μ ≪ 1, another set of exponentially decaying eigenfunctions are found, i.e., provided that e α ≫ e −α , where k 0 n :¼ 2nπ=ðL þ 1Þ (see the "Methods" section of the weak impurity). Taking κ L and κ 0 L as functions of μ, we have κ L ðμÞ % κ 0 L ðe 2α =μÞ for a sufficiently large system, suggesting a duality between μ ¼ μ ± α with μ ± α ¼ e α A ± 1 parametrized by a variable A, with μ þ α ¼ μ À α at A = 1. This duality can be seen in both the spectrum and the eigenstate accumulation, which can be characterized by the inverse participation ratio (IPR) defined as I n = ∑ x |ψ x,n | 4 for a given eigenstate. The IPR approaches 1 for a perfectly localized state, and 1/(L + 1) for a spatially homogeneous one. To further characterize the different directions of the SFA states, we define a directed IPR as I d,n = ∑ x (x c − x)|ψ x,n | 4 /(L/2), with x c = L/2 being the center of the system. By definition, I d takes positive (negative) values for states accumulating at x = 0 (x = L), and I d = 0 for a spatially homogeneous state.
In Fig. 3a, we take averages over all continuous states for the IPRs and directed IPRs ( Iðμ ± α Þ and I d ðμ ± α Þ), and present them as functions of A. Note that for μ ≫ 1, the continuous eigenstates have vanishing amplitude at x = 0, analogous to a system with L, not L + 1 sites. Therefore, to properly compare the averaged IPRs between large and small μ, they are rescaled as ð I; I d Þ ! ð I; I d ÞL=ðL þ 1Þ for μ > e α , and the system's center is redefined as x c = (L − 1)/2 for the directed IPR. We can see from Fig. 3a that the quasi-PBCs and PBCs are recovered at A = e α for μ ¼ μ ± α respectively, where IðμÞ ¼ 1=ðL þ 1Þ and I d ðμÞ ¼ 0 as all eigenstates are fully delocalized. The IPR profiles agree well between the dual values of μ in the regime close to PBCs and quasi-PBCs (A~e α ), but begin to diverge when A gets larger.
To understand this divergence, we unveil a second duality between μ~e (L+1)α and μ~e −(L+1)α , the latter corresponds to a transition between the qualitative spectral properties found for PBCs (loops) and OBCs (lines) 26,40 . In Fig. 3b, we illustrate both IPRs for μ ¼ μ ± 0 as functions of the variable A, i.e. μ ± 0 ¼ A ± 1 . The above PBC-OBC transition is seen as Iðμ À 0 Þ and I d ðμ À 0 Þ become constant for ln A ⩾ ðL þ 1Þα, reflecting the OBC skin modes. Interestingly, a similar transition also occurs at large μ = e (L+1)α , characterized by the constant IPRs in Fig. 3b when μ exceeds the critical value, indicating a second duality between μ ¼ μ ± 0 in the large A limit. These IPRs take different saturation values mainly because of the rescaling in the large μ regime with effectively different number of sites. The critical value for this transition can also be identified from our approximation of Eq.
(1), where the decay exponent κ L = α at μ = e (L+1)α , recovering the decay exponent (and the imaginary flux) κ OBC for NHSE under OBCs. In Fig. 3c, we illustrate the spectra with several pairs of dual parameters, clearly showing the two types of dualities and the transition to a OBC-like spectrum.
Co-existence of different regimes. The decay exponents κ L (μ) of SFA states, as induced by the impurity, are insensitive to the exact configuration of non-reciprocal hoppings in the bulk. By contrast, skin modes under OBCs may have k-dependent decay exponents κ OBC (k) if the system has hoppings beyond nearest neighbors 45 . That is 1-3 , the GBZ describing the NHSE under OBCs is given by z :¼ e i½kþiκ OBC ðkÞ , and may not be a prefect circle on the complex plane. On the other hand, the decay exponents κ L (μ) also suggests an effective GBZ given by z SFA :¼ e i½kþiκ L ðμÞ , which is always a  Fig. 2 Average distribution of anomalously accumulating eigenstates and their spectra. a Average distribution of all the anomalously accumulating eigenstates ρðxÞ ¼ P LÀ1 n¼1 ρ n ðxÞ=ðL À 1Þ with ρ n (x) = |ψ x,n | 2 , ψ x,n the wavefunction value of the nth eigenstate at x, and α = 4, L = 20. The blue, orange, and green cases correspond to reversed scale-free accumulation (SFA), quasi-periodic boundary conditions (PBCs) and SFA. b Spectra for these anomalously accumulating eigenstates, excluding the two isolated eigenstates induced by the strong impurity. The black dashed curve is the true PBC spectrum (μ = 1), which overlaps with the quasi-PBC orange curve at μ = e 2α = e 8 . In both panels, the circles, squares, and triangles are numerical data points, and the colored solid curves are approximations from Eqs. (1) and (2).
prefect circle with its radius determined by μ. Requiring κ L (μ c ) = κ OBC (k), one finds a k-dependent critical value of μ c (k), where the GBZ of z and the effective one of z SFA coincide at the corresponding k. A remarkable consequence then arises from this competition, namely, the co-existence of the SFA and NHSE for different eigenstates at a fixed μ. Physically, this coexistence reflects that at different wavenumbers k, an eigenstate effectively experiences couplings across different distances.
Consider a system with different forward and backward couplings ranges and an impurity between x = 0 and L: The decay exponents for the SFA at μ ≪ 1 and the NHSE under OBCs can be obtained as 45 (see the "Methods" section) with j = ⌊(k + π/3)/(2π/3)⌋. In Fig. 4a-c we illustrate these two quantities versus k for different μ. Together with the spectra in Fig. 4d-f, we see that an eigenstate always obeys the localization behavior with the smaller decaying exponent. That is, all eigenstates exhibit the SFA when κ L < κ OBC (k) in Fig. 4a and d, and the NHSE when κ L > κ OBC (k) in Fig. 4c and f. In the intermediate regime of Fig. 4b and e, the SFA and NHSE co-exist for different k, as the spectrum follows the prediction of SFA when k ∈ [k 2m−1 , k 2m ] (m = 1, 2, 3) where κ L is smaller, and the prediction of the NHSE otherwise, with k 2m and k 2m−1 being the six special momentum values marked on Fig. 4b for which κ L = κ OBC . As also seen from Fig. 4, due to the possibility of coexistence of the SFA and NHSE accumulation, even the qualitative spectral features are extremely sensitive to boundary impurity parameter μ, an observation of general interest when it comes to build a sensing platform based on non-Hermiticity.
Proposed experimental demonstration. As steady-state phenomena, the SFA can be most easily demonstrated in an electrical circuit setting. In place of the Hamiltonian, we consider the circuit Laplacian J which governs its steady-state response via I = JV, where the components of V and I are, respectively, the electrical potentials and input currents at each node. The eigenspectra and eigenstates of J can be directly resolved by measuring the voltage profile 46,47 viz.
where ψ L=R λ E are the left/right eigenvectors of J corresponding to eigenvalue ϵ λ , and V α , hαjψ R λ i are, respectively, the potential and x (x c − x)|ψ x,n | 4 /(L/2), ψ x,n the wave-function value of the nth eigenstate at x, N and L the total number of states and lattice sites, and x c = L/2 being the center of the system. The impurity strength μ is parametrized by a variable A, i.e. μ ¼ μ ± α ¼ e α A ± 1 and μ ¼ μ ± 0 ¼ A ± 1 for (a) and (b), respectively. In (a), μ þ α and μ À α form a pair of duality, whereas in (b) with much larger A values, μ þ 0 and μ À 0 form a pair of duality. The summation of n runs over all continuous eigenstates, and N = L−1 (L + 1) is their total number in the presence (absence) of the pair of isolated states E iso ≈ ± μ. Colors of the curves indicate IPRs for different choices of μ. Blue and orange curves are almost identical in (a). c Spectra with dual parameters, as indicated by the gray arrows. A phase transition to OBC-like line-spectrum at μ = e ±(L+1)α = e ±21 , with the parameters L = 20 and α = 1. The duality indicated in (a) is further confirmed by the identical spectra (blue circles and red stars) of μ ¼ μ ± α (with α = 1), at A = 0.5 and A = 2 for the left two panels, respectively. Similarly, the two panels on the right depict the duality between μ ¼ μ ± 0 indicated in (b), for the two points of A = 20.5 and A = 21. ψ R λ values at the αth node. To isolate a particular λ 0 th eigenmode, we tune the circuit until ϵ λ 0 % 0, either by adjusting its variable components or by varying the AC frequency ω 46 . V α is then dominated by ϵ À1 λ 0 hαjψ R λ 0 ihψ L λ 0 jIi. If we further connect an input current I 0 to a fixed node β 0 (the current leaves via the ground), hψ L λ 0 jIi ¼ I 0 hβ 0 jψ L λ 0 i Ã and the eigenstate profile hαjψ R λ 0 i across all nodes α becomes approximately proportional to the measured potential profile V α i.e.
In other words, ψ R λ 0 can be approximately measured through V when it is topolectrically resonant (ϵ λ 0 % 0).
A circuit Laplacian J with a similar form as H of Eq. (1) can be realized with the L + 1-node LC circuit of Fig. 5a. Adjacent nodes acquire asymmetric non-Hermitian couplings through a negative impedance converter with current inversion (INIC) 48,49 in series with a capacitor C 1 , which together contribute an admittance of iωC 1 À1 1 À1 1 to the Laplacian 48 (see the "Methods" section).
The extent of asymmetry is regulated by another parallel capacitor C 2 , such that C 1 =C 2 ¼ tanh α (see the "Methods" section). To implement an "impurity" coupling between nodes L and 0, we connect an extra variable inductor l in series with the parallel INIC + capacitors configuration, such that the admittance in both directions is uniformly scaled by a factor of μðωÞ ¼ μðωÞ ¼ ð1 À ω 2 lðC 2 À C 1 ÞÞ À1 (see the "Methods" section). To measure the profile of a desired eigenmode ψ R λ 0 , we first need to shift its eigenvalue ϵ λ 0 maximally close to 0. This can be achieved with additional identical grounding inductors l g at each bulk node, together with more carefully designed grounding circuits j 0 , j L at the impurity nodes (see the "Methods" section). In all, our circuit Laplacian takes the form where C 1 ¼ C sinh α, C 2 ¼ C cosh α and ω À2 0 ¼ l gr C, which is equal to −iωCH (Eq. (1)) up to a tunable real shift (the x j i x h j term). Plotted in Fig. 5b are simulated impedance measurements 46 across the impurity as ω is varied, for different μ(ω) = μ adjusted through the inductors l. The impedance peaks correspond to values of ω where a Laplacian eigenvalue ϵ ≈ 0. For instance, the strongest peaks belonging to ln μ ¼ 0; 1 arise from eigenvalues already on the real line (inset), while the weakest peaks from ln μ ¼ 2 are due to eigenvalues far from the real line. The entire spectrum (inset) can be reconstructed via systematic impedance measurements 28,47 .
At these impedance peaks, the potential profile approximately corresponds to the eigenstate profile of the resonant eigenmode, as verified by simulated measurements (Fig. 5c). We clearly observe reversed and non-reversed eigenstates at different μ, perfectly as predicted (Figs. 1d and 2a). Physically, the reversed voltage profile is a steady-state solution that represents a compromise between the competing non-reciprocal feedback mechanisms from the op-amps in the INICs. Scale-free behavior can be similarly detected when new nodes are introduced. More generally, we expect to measure these new forms of impurityinduced eigenstate accumulation in a variety of media whose steady-state description involve non-Hermitian asymmetric couplings [49][50][51] .

Discussion
The key and surprising finding of this work is the impurityinduced scale-free decay of steady states in a non-Hermitian nonreciprocal lattice. On the one hand the steady states are exponentially localized; on the other hand the decay exponent is inversely proportional to the system's size, hence the overall decay is free of system's size. This peculiar scale-free feature may be first observed in an electrical circuit setting. We have also found rich transitions between NHSE, Bloch-like, and SFA eigenstates along or against the direction of non-reciprocity, with stimulating duality relations between cases of weak and strong impurity strengths. Recognizing now that the well-known NHSE is only one of many impurity-induced consequences, a new basket of non-Hermitian phenomena waits to be discovered. The coexistence of SFA and NHSE can be highlighted, because of which even qualitative spectral features of a non-Hermitian system can be extremely sensitive to some system parameters. This finding may lead to new sensing-related applications of non-Hermitian lattices.
This work also hints that even more exotic phenomena can be expected with spatial inhomogeneity beyond the single impurity cases considered here. One straightforward extension is to consider multiple impurities with independent strengths, each can lead to the SFA or NHSE individually. In this scenario different segments separated by these impurities may exhibit different  8), whose asymmetric couplings are implemented through a negative impedance converter with current inversion (INICs) and capacitors. The extra variable inductor l gives rise to a coupling "impurity''. Suitably designed grounding elements (see the "Methods" section) enable desired eigenstates to be isolated at appropriate driving voltage frequency ω. b Simulated impedance measurements across the impurity at various impurity strength μ, with position ω and height of impedance peaks loosely corresponding to the real and imaginary parts of the spectrum ϵ/(iωC) (inset). Here the capacities of C 1 and C 2 are parameterized as Parameters used are L = 9, C 1 = 1 and C 2 = 3, so that C ¼ ffiffiffi 8 p and α = 0.364. c Simulated electrical potential measurements vs. the profile of the bulk eigenstate with largest ϵ/ (iωC), tuned close to resonance via Eq. (2). Not only is Eq. (7) accurate, the nature of eigenstate accumulation also agrees perfectly with the regimes of

SFA in the Hatano-Nelson model with a boundary impurity
Strong impurity. We consider the following Hamiltonian: Solving eigen-equation HΨ n = E n Ψ n with Ψ n ¼ P L x ψ x;nĉ y x 0 j i the nth eigenstate of the system, we obtain the following recursive conditions: e α ψ xþ1;n þ e Àα ψ xÀ1;n ¼ E n ψ x;n ð10Þ for x = 1, 2, . . . , L − 1, and μe α ψ 0;n þ e Àα ψ LÀ1;n ¼ E n ψ L;n ; ð11Þ e α ψ 1;n þ μe Àα ψ L;n ¼ E n ψ 0;n : Intuitively, when μ is large, two isolated solutions localized around x = 0 and x = L are expected due to the strong couplings between these two sites. Assuming these solutions decay exponentially from the two sites into the bulk, we find that they can be explicitly expressed as ψ À 0 ¼ Àψ À L e Àα ; ψ À x ¼ À e Àα μ ψ À xÀ1 for x ¼ 1; 2; :::; ψ À x ¼ À e α μ ψ À xþ1 for x ¼ L; L À 1; :::; ð14Þ whose eigenenergies are given by These solutions are valid provided that μ > e ±α , so that they indeed decay from x = 0 and x = L into the bulk; and ðe ± α =μÞ L $ 0, so that they have vanishing amplitudes in the middle of the system. In the main text, we have assumed μ ≫ e ±α , therefore the above conditions are satisfied and we have E ± iso % ± μ. For convenience, we label these two isolated eigenstates with n = 0 and n = L, respectively. The other L − 1 eigenstates of n ∈ [1, L − 1], referred to as continuous eigenstates as they have a continuous spectrum, shall mainly distribute within the rest L − 1 sites of the system with eigenenergies E n ≪ μ. Thus one expects a vanishing ψ 0,n from Eq. (11). We further consider an ansatz of exponentially decaying eigenstates given by ψ 0;n ( 1; ψ x;n ¼ e ÀM n ðxÀ1Þ for x ≠ 0: Substituting the ansatz into Eq. (12), we obtain Àe α ¼ μe Àα e ÀM n ðLÀ1Þ ; yielding However, Eqs. (10) and (11) give different eigenenergies for the exponentially decaying solutions. A consistent solution can be obtained by further requiring e α ≫ e −α and e α e ÀM n ) e Àα e M n . The first condition corresponds to a strong nonreciprocity of the system, and the second one is equivalent to μ ≪ e (L+1)α , which is generally satisfied for a large enough system. Under these conditions, Eq. (10) gives E n % e α e ÀM n ¼ e α e À ln μÀ2αÀið2nþ1Þπ ½ =ðLÀ1Þ % ϵðk n þ iκ L Þ; ð18Þ with k n : = (2n + 1)π/(L−1), n = 1, 2, . . . , L−1, and ϵ(k) ≈ e α e ik the eigenenergies under PBCs and strong non-reciprocity. On the other hand, since now we have μ ≫ e α~E n ≫ e −α , the second term of Eq. (11) can be neglected, yielding which is consistent with the vanishing ψ 0,n obtained previously.
Weak impurity. Next we consider a weak impurity limit with μ ≪ 1 and a strong non-reciprocity e α ≫ e −α , and correspondingly a different ansatz with M n > 0 (because we do not observe any cases with reversed accumulation in the regime with μ < 1). Thus Eq. (12) is simplified to Substituting the above equation into Eq. (11) with its second term being neglected, one can obtain This solution also confirms that the second term of Eq. (11) is negligible as compared with the rest two terms. The eigenenergies are thus directly given by Eq. (21).
Quasi-PBC delocalized eigenstates. To gain further insights into the quasi-PBCs at μ = 2α, let us exploit the following effective translational invariant Hamiltonian, e Àκ L e αĉy xĉxþ1 þ e κ L e Àαĉy xĉxÀ1 : In above site L + 1 is understood as site 0. According to our spectral results in Eq. (2) in the main text, H PBC , though having an extra κ L -related imaginary flux, yields the approximate eigenvalues of our lattice system for μ ¼ e ðLÀ1Þκ L þ2α . We next remove the imaginary flux in the bulk by applying a similarity transformation H 0 ¼ S À1 L H PBC S L with S L ¼ Diagf1; e κ L ; e 2κ L ; :::; e κ L L g. This gives (without changing the eigenvalues) þ e Àκ L ðLþ1Þ e αĉy Lĉ0 þ e κ L ðLþ1Þ e Àαĉy 0ĉL : ð24Þ So long as L is sufficently large, we still have κ L ðL þ 1Þ % ln μ À 2α, thus the boundary hopping in H 0 shown above becomes It is seen that at μ = e 2α , H 0 recovers the original Hamiltonian under PBCs. This is fully consistent with the observation from Eq. (1) in the main text, namely, the decay exponent κ L = 0 for μ = e 2α . The above treatment is however more stimulating to digest situations with μ ≠ e 2α , where the translational invariance of H 0 is broken at the boundary. For μ > e 2α , the hopping from x = L to x = 0 is further enhanced whereas the opposite hopping is further suppressed (as respectively, compared with the translational invariant case). The eigenstates are then expected to populate more at x = 0. Likewise, eigenstates should accumulate more at x = L when μ < e 2α , thereby exhibiting the reversed SFA.
Derivation of circuit Laplacian. Here we provide a detailed derivation of the Laplacian (Eq. (8) of the main text) of the circuit as illustrated in Fig. 5 of the main text, and also furnish more details about its grounding connections. This circuit design is inspired by previous experimental cicuit realizations of various topological and non-Hermitian states 46,[52][53][54][55][56][57][58][59] . The Laplacian J is defined as the operator that connects the vectors of input current and electrical potential via I = JV. In this work, we design a circuit array that (i) is non-Hermitian and non-reciprocal, with right/left couplings proportional to e ±α , (ii) has special impurity couplings (in both directions) that are stronger than the rest by a tunable factor of μ = μ(ω) and (iii) also contains suitable grounding elements that allows the Laplacian eigenvalue spectrum to be shifted uniformly as desired.
For (i), the unbalanced couplings ∝ e ±α can be implemented by a parallel configuration of a capacitor C 2 , and a combination of another capacitor C 1 that is connected in series with an INIC (negative INIC). As elaborated in ref. 48 , an INIC is an arrangement of operation amplifiers (op-amps) that reverses the sign of the impedance of components "in front of" it. Specifically, for a generic ideal INIC configuration as shown in Fig. 6a, the input currents and potentials at the two ends obey where Z A , Z B are the impedances of components A and B. The Laplacian matrix above is not just asymmetric and hence non-Hermitian, but is also inversely proportional to the difference between the two impedances, contrary to the usual case without the INIC. To implement the ∝ e ±α couplings, we consider parallel configurations of two capacitors C 1 , C 2 , one on its own, and the other in series with an INIC (Fig. 6b). This gives a Laplacian contribution of if we set p a reference capacitance scale. If we connect each node of a OBC linear circuit array with these parallel configuration units, we end up with the Laplacian Note that the coefficient of x j i x h j merely sums out the total outgoing hopping amplitude.
To implement (ii) the impurity couplings that are equally asymmetric, but larger than the other couplings by a factor of μ, we connect a tunable inductor l with admittance (iωl) −1 in series with the abovementioned parallel configuration (Fig. 6c). Elementary applications of Kirchhoff's law gives us which is proportional to J NN at the two nodes coupled by the impurity, up to a factor of μðωÞ ¼ 1 1Àω 2 ðC 2 ÀC 1 Þl . In other words, the impurity strength μ(ω) can be adjusted both by changing the AC frequency ω, or by tuning the inductor l itself. Note that the upper-left term of J bdry. NN reduces to the simple result that the combined impedance of components connected in series is just the sum of their impedances.
The third important feature (iii), which is the implementation of grounding components that allow for a uniform shift in Laplacian eigenvalues, is more tricky. With ground connections given by J gr ¼ P L x¼0 j x x j i x h j, the circuit Laplacian we have is given by (impurity is between the Lth and 0th nodes) Notably, the on-site terms are not even uniform. For identification with the Hatano-Nelson model with a single coupling impurity (see the main text), we need to add grounding terms such that they are not just uniform but also tunable i.e. giving rise to a tunable multiple of the (L + 1)-by-(L + 1) identity matrix. Since nodes 1 through L−1 already have the same onsite coefficient of 2iωC cosh α, we just need to ground them via identical inductors l gr , such that j x ¼ ðiωl gr Þ À1 for x = 1, . . . , L−1. The more tricky part is grounding nodes 0 and L with appropriate sets of components with combined admittance j 0 , j L such that all onsite terms are equal. We first tidy up Eq. (37) such that the NN couplings, bulk groundings and impurity groundings are grouped together: For all onsite terms to be equal, we hence require that j 0 ¼ ðiωl gr Þ À1 þ iωð1 À μðωÞÞðC 2 þ C 1 Þ; ð39Þ j L ¼ ðiωl gr Þ À1 þ iωð1 À μðωÞÞðC 2 À C 1 Þ: Recall from Eq. (36) that iωμ(ω)(C 2 ± C 1 ) are the admittances of the J imp. NN configuration with respect to the ground. The remaining admittances iω(C 2 ± C 1 ) can be realized by the configuration of J NN (Eq. (34)). As such, j 0 and j L can be realized by the configurations illustrated in Fig. 6d, e. give rise to a non-Hermitian Laplacian Eq. (33). b Asymmetric couplings of the simplest form (Eq. (34)) can be realized with a parallel configuration containing one INIC and two capacitors. c An impurity bond consisting of tunable equivalently rescaled asymmetric couplings (Eq. (36)) can be realized with a variable inductor l connected in series with J NN . d, e Explicit example realizations of j 0 , j L grounding components needed to make the grounding terms of the impurity nodes equivalent to the others'. Changing the AC frequency ω leads to a uniform shift in Laplacian eigenvalues through ðiωl gr Þ À1 .
All in all, our circuit Laplacian takes the form whose realization is illustrated in Fig. 5 of the main text.

Data availability
The data that support the plots within this paper and other findings of this study are available from any of the authors upon reasonable request.

Code availability
Though not essential to the central conclusions of this work, computer codes used to generate Figs