Observation of Impurity-Induced Scale-Free Localization
in a Disordered Non-Hermitian Electrical Circuit

Growing efforts have been invested to intriguing phenomenon of non-Hermitian systems in recent year Ashida et al. (2020); Lee (2016); Leykam et al. (2017); Xu et al. (2017); Gong et al. (2018); El-Ganainy et al. (2018); Yao and Wang (2018); Zhang et al. (2020); Yokomizo and Murakami (2019); Yao et al. (2018); Kunst et al. (2018); Liu et al. (2019); Song et al. (2019); Zhang et al. (2021a); Lee et al. (2019); Kawabata et al. (2019a); Lee et al. (2019); Sun et al. (2021); Ge et al. (2019); Zhou and Lee (2019); Zhao et al. (2019); Kawabata et al. (2019b); Borgnia et al. (2020); Liu et al. (2020); Li et al. (2020); Yokomizo and Murakami (2021); Okuma et al. (2020); Zou et al. (2023); Fan and Liang (2021); Yi and Yang (2020); Liu et al. (2021); Li et al. (2021); Bergholtz et al. (2021); Li et al. (2022); Zhang et al. (2022a); Lin et al. (2023); Ren et al. (2022); Kawabata et al. (2023); Zhang et al. (2023a); Li et al. (2023a); Liu et al. (2023); Cai et al. (2024); Li et al. (2024); Cai et al. (2024); Liu et al. (2023); Li et al. (2024); Hu et al. (2024); Wang et al. (2024); Xie et al. (2024a); Zhang et al. (2024). One of unique features in non-Hermitian systems is the non-Hermitian skin effect (NHSE) Yao and Wang (2018); Zhang et al. (2020); Yokomizo and Murakami (2019); Yao et al. (2018); Kunst et al. (2018); Liu et al. (2019); Song et al. (2019). This effect is characterized by an extreme sensitivity of eigenspectra to boundary conditions, where most of bulk modes become localized at the boundaries under open boundary conditions (OBCs). A lot of exciting non-Hermitian phenomena without their Hermitian counterparts are related to the NHSE, e.g., breakdown of conventional Bloch band theory Yao and Wang (2018), scale-free localization Li et al. (2021), and disorder-free entanglement phase transitions Kawabata et al. (2023). The NHSEs have been experimentally observed in many physical systems and have also shown potential applications in sensors due to the extreme sensitivity to the boundary conditions Koch and Budich (2022); Budich and Bergholtz (2020); McDonald and Clerk (2020) .

For NHSE, the localization length of bulk modes is usually independent of the system’s size under OBCs. Recently, an anomalous skin localization, dubbed scale-free localization, was found and extensively explored in non-Hermitian system Li et al. (2021); Guo et al. (2023); Li et al. (2023b); Wang et al. (2023); Yokomizo and Murakami (2021); Fu and Zhang (2023); Xie et al. (2024b); Molignini et al. (2023). Unlike the conventional NHSE, the localization length of scale-free modes relies on the system size, and the localization direction is not indicated by the bulk. This intriguing localization phenomenon has been largely investigated in various non-Hermitian systems Li et al. (2021); Guo et al. (2023); Li et al. (2023b); Wang et al. (2023); Yokomizo and Murakami (2021); Fu and Zhang (2023); Xie et al. (2024b); Molignini et al. (2023). Recently, the scale-free localization has been experimentally observed in an electrical circuit with a Hermitian lattice subjected to a parity-time-symmetric non-Hermitian defect Xie et al. (2024b). While, the experimental observation of the scale-free localization, resulting from the interplay of nonreciprocal hopping in the bulk and the single impurity, is still lacking.

An electrical circuit has become a powerful platform to realize topological structures even with complicated lattice geometries, e.g., higher-order topological Anderson insulator, novel topological states in hyperbolic lattices, and among others Zhang et al. (2021b, 2023b); Imhof et al. (2018); Zangeneh-Nejad and Fleury (2019); Serra-Garcia et al. (2019); Zhang et al. (2023c, 2022b). Due to the design flexibility, the nonreciprocal hopping can be easily realized by using operational amplifiers arranged as impedance converters through current inversion (INIC) Hofmann et al. (2019). Therefore, the electrical circuits have been utilized to realize novel non-Hermitian phenomena Lee et al. (2018); Helbig et al. (2019); Hofmann et al. (2019); Zhang et al. (2019); Wang et al. (2020); Bao et al. (2019); Lu et al. (2019). In this work, we experimentally designed the non-Hermitian electrical circuit in the presence of the nonreciprocal hopping, a single non-Hermitian impurity and onsite disorder. We measure and observe the scale-free localization in the disordered non-Hermitian chain. Such anomalous scale-free accumulations of eigenstates are controlled by the single non-Hermitian impurity, and their localization direction can be opposite to the bulk hopping direction. Our experiment verifies the existence of the anomalous skin effects induced by the single impurity in the nonreciprocal systems, and the results open the door to further explore the interesting localization phenomena in non-Hermitian electrical circuits.

In order to study the anomalous skin effect due to the interplay of disorder and impurity, we consider the disordered Hatano-Nelson (HN) chain in the presence of a single non-Hermitian impurity, with its Hamiltonian reading Molignini et al. (2023)

where $t\pm\gamma$ indicate the asymmetric hopping strengths, $V_{n}$ is onsite disorder potential, sampled in a random uniform distribution $[-V,~{}V]$, and $v\pm\delta$ are the asymmetric hopping strengths between the first and last sites, severing as a single non-Hermitian impurity [see Fig. 1(a)]. By controlling the impurity’s parameters $v\pm\delta$ in the presence of the disorder, one can observe anomalous skin-localization phenomena Molignini et al. (2023), where the system undergoes Anderson localization and scale-free skin localization, as indicated by the phase diagram in Fig. 1(b). The phase diagram is obtained by calculating the mean center of mass (mcom), which is defined as the amplitude squared of all right eigenvectors $\ket{\psi_{R,n}}$, averaged over many disorder realizations $N_{r}$ Molignini et al. (2023), i.e.,

with

Here, $\left<\cdot\right>_{V}$ indicates disorder averages.

The nonreciprocal hopping typically leads to non-Hermitian skin effects in the clean system. While, the scale-free localization in the presence of a single non-Hermitian impurity is distinct from the non-Hermitian skin effect, where its localization is not dictated by the bulk, and the localization length is proportional to the system size Molignini et al. (2023). Furthermore, the localization at the left or right boundary of the chain is controlled by the impurity hopping strength in the 1D disordered HN chain [see Fig. 1(b)]. Note that such anomalous skin-localization feature is determined by the interplay of bulk hopping strength and the single non-Hermitian impurity, but it is stabilized to a nonmonotonic localization behavior as a function of the hopping terms by the random disorder Molignini et al. (2023), as shown in Fig. 1(b).

In order to experimentally observe the anomalous skin-localization phenomena due to the interplay of a single non-Hermitian impurity and the bulk nonreciprocal hopping, we design non-Hermitian electrical circuits, corresponding to the model in Eq. (II). Figure 1(c) plots the electrical circuit network, where the nonreciprocal hopping between nodes $n$ and $n+1$ is realized by the negative impedance converters through current inversions (INICs) Hofmann et al. (2019). Figure 1(d) shows the experimental circuit board, where the first node and the last node are connected by the external wires acting as the single non-Hermitian impurity. The disorder term $V_{n}$ in Eq. (II) is introduced by the grounded capacitor $C_{sn}$ ($n=1,2,\cdots,N$) and the tolerance of the grounded inductance $L_{0}$ [see Fig. 1(c)]. The model in Eq. (II) is represented by the circuit Laplacian $J(\omega)$ of the circuit Lee et al. (2018). The Laplacian is defined as the grounded-voltage vector $\mathbf{V}$ to the vector $\mathbf{I}$ of input current by $\mathbf{I}(\omega)=J(\omega)\mathbf{V}(\omega)$. As shown in Fig. 1(c), the circuit Laplacian reads (see Appendix A)

with

where $C_{sn}$ signifies a grounded capacitor at the node $n$ within the range $[0,~{}C_{S}]$. By further writing $J$ as $J=i\omega[\mathcal{H}-\varepsilon(\omega)]$, one found that $J$ and $\mathcal{H}$ share the same eigenstates, if we set $\pm C_{2}-C_{1}=t\pm\gamma$, $\pm C_{\delta}-C_{v}=v\pm\delta$, and $C_{sn}-C_{S}/2=V_{n}$. The eigenvalues and eigenstates of $J$ can be obtained by measuring the voltage response at the circuit nodes.

It has shown that the single non-Hermitian impurity can induce a scale-free accumulation of all eigenstates opposite to the bulk hopping direction [see Fig. 1(b)], distinct from the NHSE occurring at open boundaries Molignini et al. (2023). Such scale-free localization phenomenon is simulated using electrical circuit, as shown in Fig. 2. Here, we set $C_{1}=C_{2}$, $C_{v}=C_{\delta}$, and introduced onsite disorder through random variations in the fabricated grounded inductors due to imperfect manufacturing processes.

The voltage distribution at resonance frequency can be used to represent the state distribution of the circuit Laplacian. Figure 2(a,b) plots the frequency-resolved voltage distribution $\absolutevalue{\mathbf{V}_{n}(\omega)}$ excited by alternating current (AC) at the different node $n$ for $C_{1}=9.4$ nF and $C_{v}=47$ nF, and (b) for $C_{1}=22$ nF and $C_{v}=2.2$ nF, corresponding to the skin-mode localized at the left and right sides of the chain, respectively. This indicates that the bulk states localized towards different directions can be controlled by changing the hopping strength within the bulk chain and the the hopping strength at the single-impurity site in spite of the nonreciprocal hopping direction within the bulk.

Figure 1(b) shows the existence of the scale-free localization controlled by the single non-Hermitian impurity in the presence of weak disorder, where the localization length is dependent on the lattice size. In order to demonstrate the scale-free localization, we calculate the normalized spatial distribution $\Phi_{n}$ of the peak voltage at the node $n$, which is defined as

where $\mathbf{V}_{n}^{2}|_{\textrm{peak}}$ denotes the peak voltage at node $n$, which is normalized to the maximum value of peak voltages of all the nodes.

Figure 2(c,d) shows the normalized spatial distribution $\Phi_{n}$ of the peak voltage as a function of the normalized node index $(n-1)/(N-1)$ for the left- and right-localized skin modes at the different lattice size $N$, where the node index is mapped to the range $[0,~{}1]$. The state distributions at the different size are collapsed close to each other, indicating the size-dependent localization length. By exponentially fitting the state distribution, we extract the localization length $\xi$ at different lattice size $N$ [see blue dots in Fig. 2(e,f)]. After linearly fitting these dots, the localization length $\xi$ exhibits the linear dependence on the lattice size $N$. This indicates the existence of the scale-free localization for the skin modes controlled by the single non-Hermitian impurity.

Our main results are the experimental verification of the scale-free localization induced by the single non-Hermitian impurity Molignini et al. (2023) using the electrical circuit. The electrical-circuit network and fabricated experimental circuit board are shown in Fig. 1(c,d). As shown in Fig. 1(c), two nodes within the circuit are interconnected via capacitors $C_{1}$ and INICs, where the INICs have the equivalent capacitance of $\pm C_{2}$ in opposite directions. The first and last nodes are connected through distinct capacitors and INICs, denoted as $C_{v}$ and $\pm C_{\delta}$, which serve as the single non-Hermitian impurity. The parameters of the experimental electrical circuits are the same as ones used in the simulation. In addition to the random variations suffering from imperfect manufacturing processes, disorder is mainly introduced by the grounded capacitors $C_{sn}$. The grounded capacitor $C_{sn}$ of each node is randomly chosen from a diverse set of capacitors with capacitance ranging from $0$ to $C_{S}=10$ nF. The diagonal element of the circuit Laplacian is given by $1/(i\omega L_{0})+i\omega(C_{0}+2C_{1}+C_{S}/2)$, and the circuit’s reference frequency reads $f_{0}=(2\pi)^{-1}(C_{0}+2C_{1}+C_{S}/2)^{-1/2}L_{0}^{-1/2}$, where the capacitance $C_{1}$ is $C_{1}=9.4$ nF for the left-localized states, corresponding to $f_{0}=176$ kHz, and it is $C_{1}=22$ nF for the right-localized states, corresponding to $f_{0}=164.5$ kHz. A chirp signal spanning the frequency band from $0$ kHz to $250$ kHz is used as the excitation. Details of the sample fabrication and experimental measurements are provided in the Appendix B.

The experimentally measured voltages of the admittance under the excitation of chirp signals are shown in Fig. 3. We have experimentally designed two electrical circuits with different parameters of boundary capacitors acting as the single impurity, where the bulk parameters are fixed. To be specific, for the $C_{1}=9.4$nF and $C_{v}=47$nF, we plot the frequency-resolved voltage distribution [see Fig. 3(a)]. The voltage is peaked around the frequency of $172.5$ kHz, which matches well with the simulated result. By extracting the peak voltage at each node, its spatial distribution is shown in Fig. 3(b)], corresponding to the state distribution of non-Hermitian Hamiltonian $\mathcal{H}$ for the specific eigenvalue. This state is localized at the left side, indicating the occurrence of NHSE. While, for $C_{1}=22$nF and $C_{v}=2.2$nF, the voltage is peaked around the frequency of $164$ kHz [see Fig. 3(b)], where we observe the right-side localized state [see Fig. 3(d)]. The experimental results indicate the existence of the anomalous skin-mode localization controlled by the single impurity in spite of the bulk hopping direction.

To verify the scale-free localization property, we measured the site-resolved peak voltages for different sizes, as shown in Fig. 4(a,b). For the left-side skin modes, the parameter of the impurity for all the samples is set as $C_{1}=9.4$nF and $C_{v}=47$nF, and for the right-side skin modes, it is $C_{1}=22$nF and $C_{v}=2.2$nF. Figure 4(a) plots the peak voltages of different samples as a function of normalized node index $(n-1)/(N-1)$, where the node index is normalized to the range $[0,~{}1]$. These left-side skin modes are not collapsed, indicating the absence of scaled localization for $C_{1}=9.4$nF and $C_{v}=47$nF. There is also absence of scaled localization for right-side skin modes with the single-impurity parameters set as $C_{1}=22$nF and $C_{v}=2.2$nF. After performing a linear fit of the localization length at different sizes, we observe scale-free localization behavior. This size-dependence of the localization behavior exhibits a significant deviation from the NHSE, where the localization length remains consistent for different system size $N$, as predicted in Refs. Molignini et al. (2023); Li et al. (2021). This unique phenomena of scale-free eigenstates are usually accompanied by the emergence of complex eigenspectrumLi et al. (2021), which has also been presented in Fig. 4(e,f).

In summary, we have experimentally observed the anomalous non-Hermitian skin effects with skin-mode localization directions controlled by a single non-Hermitian impurity in non-Hermitian disordered electrical circuits. Furthermore, anomalous skin modes are verified to show the scale-free localization induced by the single non-Hermitian impurity by measuring the size-dependent localization length. Our experimental results have proved the theoretical proposal on the scale-free localization induced by the single non-Hermitian impurity. In the future, it would be interesting to investigate scale-free localization in higher dimensions.