\section{Topolectrical circuits design}



We design a circuit that implements both non-reciprocal and non-Hermitian couplings in a periodical network. Topological effects in electrical circuit networks have initially been considered for the Hofstadter model~\cite{PhysRevX.5.021031,PhysRevLett.114.173902}. The Laplacian formalism, with the complex admittance as its observable, has been introduced in~\cite{leecommphys}. It has established the notion of admittance band structures, which are fully accessible experimentally~\cite{helbigprb}. Topolectrical circuits are a newly emerging platform for synthetic topological matter which allows to implement any tight-binding model of any coupling range or dimensionality. As such, topolectric circuits manage to host nodal knots or links as zero admittance eigenvalues in their three-dimensional variant~\cite{Research2018Yu,lee2019topological}. Topological corner states as implementations of higher order topology can likewise be realized~\cite{quadrupolar,PhysRevB.100.045407}. 
Our non-Hermitian non-reciprocal circuit network is described by the model Laplacian introduced in Eq.~(1). An idealized conceptual unit cell of the circuit is shown in Fig.~\ref{fig:circuit_schematic_simplified}a. The unit cell consists of two sublattice nodes, $A$ and $B$. The sublattices are connected in the unit cell by an $LC$-resonating circuit, which represents the Hermitian and reciprocal coupling $\ii \omega\,v(\omega) = \ii \omega C_1 + 1/(\ii \omega L_1) $. The frequency-dependence of its impedance allows us to modulate the intracell coupling and to resolve the fundamentally different parameter regimes of the model on the same circuit board. The $LC$ oscillator is characterized by its resonance frequency $\omega_0^{-1} = \sqrt{L_1 C_1}$, at which any current is blocked from flowing	through the component by an impedance divergence, such that the connectivity $v(\omega_0) = 0$. The coupling $C_2$ in between unit cells acts as the reciprocal intercell connection $\ii\omega \, r = \ii \omega C_2$.
Inevitable resistive losses in the circuit can, to first approximation, be modeled by an effective resistance $R_\text{x}$ connecting the nodes $A$ and $B$ within an unit cell.

The key non-Hermitian component in our circuit is the negative impedance converter with current inversion (INIC), which implements non-reciprocity~\cite{PhysRevLett.122.247702}. Its setup is shown in Fig.~\ref{fig:circuit_schematic_simplified}b. Through an operational amplifier (OpAmp) in a negative feedback configuration, we realize an antisymmetric connection, that implements a positive capacitance $C_3$ in one direction and a negative capacitance with equal magnitude in the opposing direction, leading to the breaking of reciprocity. Solving Kirchhoff's laws for the INIC leads to the reduced two-node Laplacian
\begin{align}
\begin{pmatrix}
I_\text{in} \\
I_\text{out}
\end{pmatrix} = 	\ii \omega C_3 \,	
\underbrace{\begin{pmatrix}
	-1 &  1 \\
	- 1 & 1
	\end{pmatrix}}_{= J_\text{INIC}}
\begin{pmatrix}
V_\text{in} \\
V_\text{out}
\end{pmatrix}.
\end{align}
The in- and output currents are related by $I_\text{in} = - I_\text{out}$, meaning, that both currents flow in opposite directions. For this, the OpAmp must introduce a current source or sink in the system, with currents either flowing inwards to $V_0$ or outwards from $V_0$ in both directions. From the viewpoint of $V_\text{in}$, the INIC acts as a negative capacitance $-C_3$, whereas $V_\text{out}$ experiences the positive capacitance $C_3$. The INIC breaks reciprocity, $J_\text{INIC} \neq J_\text{INIC}^\top$, in an antisymmetric fashion as the off-diagonal elements differ by a minus sign. Its diagonal contribution is proportional to $\sigma_z$. To avoid it, we add a grounding term of $C_0 \equiv 2 C_3$ to the node at $V_\text{in}$ in order to change the diagonal terms to a unit matrix contribution.

The circuit in Fig.~\ref{fig:circuit_schematic_simplified}a is mathematically described by a Laplacian matrix, which in momentum space under PBCs, takes the form
\begin{align}
J_{\omega}(k)  = \ii \omega \Bigg[ 
&\left(C_1 - \frac{1}{\omega^2 L_1}\right) \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}
+ C_2 \begin{pmatrix} 1 & -\ee^{-\ii k} \\ -\ee^{\ii k} & 1 \end{pmatrix}
+ C_3 \begin{pmatrix} -1 & 1 \\ -1 & 1 \end{pmatrix} \nonumber \\
&- \frac{\ii}{\omega R_\text{x}} \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} -\left( \frac{1}{\omega^2 L_0} + \frac{\ii}{\omega R_0}\right)\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + C_0 \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
\Bigg]. \label{eq:SC Laplacian}
\end{align}
Additional to the introduced coupling terms, we include grounding terms $1/(\ii \omega \, L_0)$, $1/R_0$ at all nodes as well as $\ii \omega \, C_0$ at sublattice $A$. By omiiting the resistive intracell connection $R_x$ and representing equation \eqref{eq:SC Laplacian} in the Pauli matrix basis, we arrive at the effective circuit Laplacian stated in equation~(1).


The inductive grounding at each node is used as an $A$-$B$ symmetric grounding, which shifts the imaginary part of the admittance in a frequency-dependent fashion. We damp the circuit globally by grounding all nodes with a resistance $R_0$. The damping is needed to stabilize the circuit and eliminate instabilities (self-amplifying energy gains) in the system, that arise due to the INIC as an active element. The resistance $R_0$ shifts the real part of the admittance eigenvalue spectrum globally by $1/R_0$ but leaves the eigenmodes invariant.


\section{Experimental implementation}

A circuit consisting of ten unit cells was realized on a Printed Circuit Board (PCB). The concise values for the circuit components are detailed in Tab.~\ref{tab:Nominal values}. They are chosen to optimize several aspects of the experimental setup: The Lock-In amplifiers, which are employed for the measurement, operate up to an upper frequency bound of 100\,kHz, the choice of specific parameters facilitates a more stable circuit, in the way that it requires less damping to stabilize it and the non-local response is seen more dominantly if the parameters are adjusted to $r = 2 \gamma$. 


In the implementation of the INIC, we employ the unity-gain stable operational amplifier LT1363.
In principle, the auxiliary components in the positive and negative feedback loop of the OpAmp can be chosen arbitrarily, provided they are equal. For a real OpAmp with its finite gain bandwidth product, limited stability conditions, and non-zero output impedance, however, a careful stability analysis~\cite{SuppMat} is required to guarantee the stable operation of LT1363 with our capacitive load $C_3$. This does not only concern the frequency range at which the experiment was conducted, but also frequencies in the MHz range to avoid undesired gain resonances in this regime. Considering those aspects we chose $R_\text{a} = \text{0.2}\,\Omega$ and $C_\text{a} = \text{0.94}\,$ $\mu$F. 

As the INIC is an active, non-reciprocal circuit element it pumps energy into the system. This can lead to eigenmodes of the dynamical matrix, which are described by a complex energy eigenvalue with negative imaginary part, such that their magnitude increases exponentially with time. Hence, energy accumulates in the system until the OpAmp shows non-linear saturation effects and discontinues to function properly. In order to prevent this self-sustained energy gains, we add resistors $R_0$ from each node to ground, which consume the desired amount of power. The suited value for $R_0$, which differs between OBC and PBC, should be small enough to avoid instabilities but large enough to avoid oversized damping, which localizes the voltage response and likewise decreases the measurement accuracy. A numerical computation of the eigenfrequencies using the circuit Hamiltonian formalism~\cite{SuppMat} requires $R_0 \leq \text{45.0} \,\Omega$ for PBC and $R_0 \leq \text{66.8} \,\Omega$ for OBC to remove all global divergences. The discrepancy between those values roots in $\min(\mathfrak{Re}\,(J_\text{OBC})) \geq \min(\mathfrak{Re}\,(J_\text{PBC}))$ for the PBC and OBC spectrum, where a negative real part of the admittance is prone to cause global instabilities. In the actual experimental implementation, parasitic resistances help to stabilize the system and further increase the necessary $R_0$ to stabilize the system.
For the spectral measurements we chose $R_0 = \text{20}\,\Omega$, while for the non-local voltage response measurements we used $R_0 = \text{120}\,\Omega$. 

In order to further eliminate undesired parasitic effects in the experiment and increase its accuracy, we design the circuit boards such that the unintended electromagnetic coupling between different wires and inductors is minimized. We add additional shielding around the inductors, use two circuit boards with five unit cells each, and place them on a metallic mesh. The shielding of the inductors changes their nominal values specified in Tab.+\ref{tab:Nominal values} and furthermore alters their frequency dependence. Moreover, we move the supply lines for the OpAmps out of the PCB board plane. In order to keep parametric disorder to a minimum and thereby preserve an approximate translational invariance, all circuit components were precharacterized by a BK Precision 894 LCR-meter and sorted into groups with only 1\% tolerance.

To perform the spectral measurements, a constant AC current is fed into one node of the board, while lock-in amplifiers with a high dynamical range are used to measure the voltage response of the circuit. The current is realized by a voltage source connected to the PCB through a shunt resistance of $R_\text{S} = \text{12.0}\,\Omega$. The PBC results were acquired by feeding an input current to sublattice $A$ and measuring the voltage at all the other nodes, then repeating the same procedure for sublattice $B$. The spectrum was then calculated through Fourier transformation, which is possible due to circuit periodicity.  For the OBC spectra, the current was fed at each node individually and the voltages of all the other nodes were measured. The results of these measurements are used to populate the Green’s function, as the inverse of the Laplacian,
\begin{align}
\nvec{V} = \begin{pmatrix}
G_{1,m} \, I_m \\
\vdots \\
G_{N,m} \, I_m
\end{pmatrix} = \begin{pmatrix}
G_{1,1} & \cdots & G_{1,N} \\
\vdots & \ddots & \vdots \\
G_{N,1} & \cdots & G_{N,N}
\end{pmatrix}
\cdot \begin{pmatrix}
0 \\ \vdots \\ I_m \\ \vdots \\ 0
\end{pmatrix} \qquad \Leftrightarrow \qquad G_{n,m} = \frac{V_n}{I_m}
\end{align}	
where $I_m$ is the excitation current at node $m$, and $V_n$ is the measured voltage at node $n$. Using this construction, the Laplacian $J$, as well as its eigenvalues and eigenvectors can be computed numerically.

\section{Results and Analysis}

\subsection{Full-scale experimental fit of the theoretical model.}\label{subsec:Full-scale experimental fit of the theoretical model}


To fit the experimental results theoretically, we extend our description of the circuit to a full-scale fitted model that takes the most prominent imperfections into account. They occur as parasitic resistances in the inductive couplings of the circuit, and can be modeled as a serial resistor $R_{L_i}$ which modifies the inductances according to
$L_i \quad \rightarrow \quad  L_i + \frac{R_{L_i}}{\ii \omega }$,
as depicted in Fig.~\ref{fig:parasitics}. This results in the full-scale fitted model
\begin{align}\label{eq:full_scale_Laplacian}
J_{\omega,\text{fit}}(k) = \ii \omega \, \Bigg[&\left(C_1 + C_2 + \frac{C_0}{2} - \frac{1}{\omega^2 L_0 - \ii \omega R_{L_0} } - \frac{1}{\omega^2 L_1 - \ii \omega R_{L_1} } -\frac{\ii}{\omega R_0}\right) \, \mathbbm{1} \nonumber \\
-&\left(C_1 -\frac{1}{\omega^2 L_1 - \ii \omega R_{L_1} } +C_2 \cos(k)  \right) \, \sigma_x - \Big(C_2 \sin(k) - \ii\, C_3 \Big) \, \sigma_y \nonumber \\
&+\left(\frac{C_0}{2} -C_3\right)\sigma_z\Bigg],
\end{align}
with capacitances given by $C_1 =$ 300\,nF, $C_2 =$ 94\,nF, $C_3 =$ 47\,nF and $C_0 =$ 94\, nF. The serial resistance to ground, which ensures stability, is chosen to be $R_0 = 20 \, \Omega$ for spectral measurements in accordance with the stability condition evaluated in~\cite{SuppMat}. 

In comparison to the inductors, the serial resistances of the capacitors are negligible. The filling material of the selected capacitances for the experimental implementation is chosen such that the capacitance is stable across the measured frequency regime. In contrast, the inductors and their corresponding serial resistances $L_1$, $L_0$, $R_{L_1}$, $R_{L_0}$ vary significantly with the AC driving frequency $f=\omega/2\pi$. We fit the PBC and OBC spectra of $J_{\omega,\text{fit}}(k)$ to the measured eigenvalues in Fig.~\ref{fig:Spectral measurements} (a1-a5), and obtain the fit parameters for $L_1$, $L_0$, $R_{L_1}$, and $R_{L_0}$ (see Tab.~\ref{tab:Fit parameters}).


Incorporating the parasitic resistance $R_{L_1}$ in the intracell connection given by $L_1$, the intracell parametric hopping $v(\omega) = C_1 -\frac{1}{\omega^2 \, L_1}$ is adjusted to
\begin{align}
v(\omega) \rightarrow \left(C_1 - \frac{L_1}{\omega^2 \, L_1^2 + R_{L_1}^2}\right) +\  \ii \  \left(-\frac{1}{\omega^2} \cdot\frac{ \, R_{L_1}}{\omega^2 \, L_1^2 +  \, R_{L_1}^2}\right) \equiv v_R+i v_I,
\end{align}
where $v_R$ and $v_I$ label its real and imaginary part. In a theoretical analysis, we are able to connect the limit of periodic and open boundary conditions for $J_{\omega,\text{fit}}(k)$, which differ by a connection with the boundary coupling of $\ii \omega C_2$. To find the full OBC admittance spectrum $j_\text{OBC}$ from the PBC spectrum $j(k)$, we recast the boundary coupling as a pumping of an effective imaginary flux. As detailed in section~\cite{SuppMat}, the OBC spectrum is reached at the smallest, in general finite $\kappa$, where $J(k + i \kappa)$, as an analytic continuation of momentum to the complex plane, recovers the Bloch state limit of plane waves. 

The magnitude of the Bloch factor $\abs{\alpha}$ for the OBC modes relates to the imaginary momentum $\kappa$ by $\abs{\alpha} = e^{- \kappa}$. For the concrete model, a computation (see~\cite{SuppMat} with $v_R\leftrightarrow v$ and $v_I\leftrightarrow \beta$) yields 
\begin{align}\label{eq:OBC Bloch factor}
\abs{\alpha} = \sqrt{\abs{\frac{v_R+ \ii \, v_I + \gamma}{v_R + \ii \, v_I - \gamma}}} = \sqrt[4]{\frac{(v_R+\gamma)^2+v_I^2}{(v_R-\gamma)^2+v_I^2}}.
\end{align}
The corresponding localization length of the bulk eigenmodes is given by
\begin{align}
\xi = \frac{1}{\kappa} = -\frac{1}{\ln(\abs{\alpha})} = \left[\frac{1}{4} \, \ln(\frac{(v_R-\gamma)^2+v_I^2}{(v_R+\gamma)^2+v_I^2})\right]^{-1},
\end{align}
which translates to the OBC bulk modes through $e^{-x/\xi} = e^{-\kappa x}$, where $x$ labels the circuit unit cells from left to right. The inverse localization length $\kappa$ is plotted in Fig.~\ref{fig:Phase diagram} as a function of $\omega$. The OBC eigenvalues are correspondingly given by all analytically continued PBC eigenvalues for different momenta $k$,
\begin{align}
j_{\text{OBC}} = \bigcup_k \,  j(k + i \kappa),
\end{align}
where $\kappa = \ln\sqrt[4]{[(v_R-\gamma)^2+v_I^2]/[(v_R+\gamma)^2+v_I^2]}$. For $v_R \neq 0$, the nonzero $\kappa$ yields boundary localized modes for OBC, with the intrinsic localization length of $\xi$. 


In Fig.~\ref{fig:Spectral measurements}a, we plot the PBC and OBC admittance spectrum obtained from experimental measurements at differentfrequencies $\omega$. By continuous curves, we show the theoretical band structure obtained from the fitted model including all parasitic effects and circuit fit parameters chosen as detailed in Tab.~\ref{tab:Fit parameters}. For $v_R \neq 0$, we observe an extensive breakdown of bulk-boundary correspondence, as the PBC eigenvalues lying on closed loops coalesce to open arcs for OBC. The corresponding eigenvectors are calculated from the measurement data and displayed in Fig.~\ref{fig:Spectral measurements}b, showcasing the localization of all bulk voltage eigenmodes at either one boundary of the OBC circuit depending on the sign of $v_R(\omega)$. For $v_R <0$ as in Fig.~\ref{fig:Spectral measurements}\,(b1),(b2), all modes are localized at the left edge due to $\kappa > 0 $ ($\xi>0$) as shown in Fig.~\ref{fig:Phase diagram}. The localization length switches its sign as a function of frequency with $v_R >0$ as seen in Fig.~\ref{fig:Phase diagram} and converts the localization of the OBC modes to the right edge in Fig.~\ref{fig:Spectral measurements}\,(b4),(b5).





In our experimental frequency sweep, we cross one transition point, where all modes are delocalized for both PBCs and OBCs as depicted in Fig.~\ref{fig:Spectral measurements}\,(b3). It is found at $v_R= 0$ at 
\begin{align}
\tilde{\omega}_0 = \sqrt{\frac{1}{L_1 \, C_1}- \frac{R_{L_1}}{L_1^2}},
\end{align}
yielding a transition frequency of $\tilde{f}_0 \approx \ $91.3\,kHz. At this point, the localization length $\xi$ for the bulk modes diverges, which is visualized for $\kappa \rightarrow 0$ in Fig.~\ref{fig:Phase diagram}. Bulk-boundary correspondence is restored, leading to a perfect matching of PBC and OBC eigenvalues in Fig.~\ref{fig:Spectral measurements}\,(a3) except for topological SSH eigenvalues. Consequently, the PBC spectrum lies on arcs and follow an extended reciprocity condition, stating that the PBC spectrum is reciprocal around a symmetry point $k_s$ in the Brillouin zone, $j(k_s+k) = j_s(k_s-k)$ (see~\cite{SuppMat}). In the fitted model, the PBC spectrum is symmetric around $k_s = -\arctan(\gamma/v_I) \approx \ $0.4$\pi$ at $\tilde{f}_0$. 


For $v_I \neq 0$, due to the incorporation of parasitic resistances, the exceptional points in the OBC spectrum disappear. The strongest localization of the OBC bulk modes is found at the respective global maximum (left edge) and minimum (right edge) of the imaginary momentum $\kappa$, as depicted in Fig.~\ref{fig:Phase diagram}. Due to the small parametric adjustment of $v$ by the incorporation of $R_{L_1}$ as in $v_I(\omega) \ll v_R(\omega)$ and their respective non-trivial frequency dependencies, the frequencies of maximum localization of the OBC modes are marginally adjusted from $\abs{v} = \abs{\gamma}$ to $f_l \approx \ $84.5\,kHz and $f_r \approx \ $100.0\,kHz for the left and right edge localization respectively. Experimental data close to those frequencies is shown in Fig.~\ref{fig:Spectral measurements}\,(a2),(b2) as well as Fig.~\ref{fig:Spectral measurements}\,(a5),(b5), where we observe a strong localization of OBC modes. 

In Fig.~\ref{fig:Spectral measurements}, we further notice the emergence of Su-Schrieffer-Heeger type topological modes, whose eigenvalues reside between the two bulk admittance bands. The topological modes are localized at the boundary of the sample. To find the regime, where topological states exist, we resort to the full-scale fitted model. In analogy to the argument given in~\cite{SuppMat}, a similarity transformation involving the Bloch factor $\alpha$ acting on the OBC Laplacian of $J_\text{fit}$ yields a fully reciprocal model. It presents an intracell hopping of 
\begin{align}
t_0 = \sqrt{(v_R + \ii \,v_I)^2-\gamma^2}
\end{align}
and an intercell hopping of $r$. Two zero-admittance solutions corresponding to boundary-localized topological modes exist, for $\abs{t_0 } < \abs{r}$, which translates to
\begin{align}
\abs{v_R} < \sqrt{\sqrt{r^4-(2 \gamma v_I)^2}-v_I^2+\gamma^2}.
\end{align}
This leads to a frequency regime of 
\begin{align}
f_L < f < f_U,
\end{align}
where we expect zero-admittance solutions in the OBC spectrum and the corresponding topological modes to exist. The zero eigenvalues are shifted according to the frequency-dependent unit matrix contribution in the fitted model. For the given parameters, the topological transition frequencies are given by $f_L \approx \ $78.8\,kHz and $f_U \approx \ $113.0\,kHz. The computed values agree with the measurement, as topological modes occur in Fig.~\ref{fig:Spectral measurements}\,(a2)-(a5), but not in Fig.~\ref{fig:Spectral measurements}\,(a1). Topological modes hence coexist with the localization of all bulk modes for OBC, \textit{i.e.\ }\/ the non-Hermitian skin effect. Furthermore, the emergence of topological modes is parametrically distorted away from the PBC complex gap-closing point (PBC Exceptional point) due to the non-Hermitian contribution.

We visualize the imaginary flux pumping causing the localization of the OBC modes in an experimental measurement (Fig.~\ref{fig:Spectral measurements}c) supported by a numerical computation (Fig.~\ref{fig:Spectral flow}). The spectral flow of the complex admittance spectrum from PBC to OBC is experimentally achieved through the attenuation of the boundary coupling, that modulates the boundary conditions in Fig.~\ref{fig:Spectral measurements}c.
For PBC the coupling is implemented by the capacitance $C_2$. To realize the imaginary flux pumping, we decrease the capacitance by adding additional capacitors in parallel and, eventually, open the connection completely to achieve OBC. 

In Tab.~\ref{tab:alpha}, we show the theoretically expected magnitude of the OBC Bloch factor $\abs{\alpha(\omega)}$, as computed in equation \eqref{eq:OBC Bloch factor} while using the parameters given in Tab.~\ref{tab:Fit parameters} for the experimentally studied frequencies. The Bloch factor can also be obtained from $\abs{\alpha(\omega)} = e{-\kappa}$ with $\kappa$ being plotted in Fig.~\ref{fig:Phase diagram}. In comparison, we calculate the mean decay of the OBC bulk eigenmodes, as they are obtained from the measurement data in Fig.~\ref{fig:Spectral measurements}b. The experimental values agree with the theoretical data within the scope of the computed standard deviation. 
This confirms the accuracy of the theoretical fitted model. 
Discrepancies might root in additional parasitic effects and couplings in the realistic setting, which are not included in the fitted model as parasitic resistances of the capacitances, point-like resistive defects due to solder joints or capacitive couplings between conducting wires.


\subsection{Non-local voltage response} 

Consider a linear periodic circuit array with periodic boundary conditions and a single current input at a specific circuit node. Then, the voltage response to the current feed must be localized around the input unit cell, \textit{i.e.\ }\/ the voltage profile decreases with distance from this unit cell.
In the limit of a dissipationless circuit, the response can also be delocalized but never increase in a different unit cell. 
Otherwise, the system cannot be stable as the voltage accumulates infinitely in a periodic circuit.
Now consider the same circuit with open boundary conditions and a current input in the bulk of the system. If the circuit exhibited bulk-boundary correspondence, the eigenstates of the system's bulk would only change perturbatively and not effect the principal voltage response in comparison to the PBC case. By implication, a non-local voltage response in a periodic circuit is an inevitable signature of BBC breakdown and localized modes of extensive cardinality.

To compute the voltage response to an external current excitation $\nvec{I}$, we resort to the Greens function $G_J = J^{-1}$ of a system with $N$ unit cells. In its spectral representation, it is given by
\begin{align}
G_J = \sum_{n } \frac{1}{j_n} \cdot \frac{\nvec{V}_n \, \nvec{U}_n^\dagger}{\nvec{U}_n^\dagger \, \nvec{V}_n},
\end{align}
where $\nvec{V}_n$ ($\nvec{U}_n$) are the right (left) eigenvectors of the Laplacian matrix to the eigenvalue $j_n$. The voltage response $\nvec{V}$ is then given by $\nvec{V} = G_J\,\nvec{I}$. The eigenstates, which correspond to the smallest eigenvalues are excited the most.
Assume that one eigenvector, say $\nvec{V}_1$, is dominantly excited by the current profile $\nvec{I} = (0,\,\cdots,\,0,\,I_a,\,0,\,\cdots,\,0)^\top$ with only one nonzero contribution, \textit{i.e.\ }\/ one input current at node $a$. Then, the voltage response behaves as
\begin{align}\label{eq:non-local voltage response approx}
\nvec{V} \sim \frac{1}{j_1} \cdot \frac{U_{1,a}^*}{(\nvec{U}_1^\dagger \, \nvec{V}_1)} \, \nvec{V}_1 ,
\end{align}
where $U_{1,a} \sim \abs{\alpha}^{-a/2}$ assuming $\abs{\alpha}>1$ (right edge localization w.l.o.g.) as the left and right eigenvectors decay with $\abs{\alpha}^{-1}$ into the bulk. The normalization then follows to be $(\nvec{U}_1^\dagger \, \nvec{V}_1) \sim 2 \cdot N \, \abs{\alpha}^{-N}$ in the circuit system with $N$ unit cells. The voltage profile is approximated by
\begin{align}
\nvec{V} \sim \frac{1}{2 N \, j_1} \cdot \abs{\alpha}^{N- \frac{a}{2}} \cdot \nvec{V}_1.
\end{align}
From this formula, we make two observations. First, the voltage response increases with the nodal distance of the current feed to the localized voltage profile at the right edge. We can therefore associate an amplification factor proportional to $\abs{\alpha}$ to each unit cell in the chain. As we move the excitation further away from the mode localization, the magnitude of the response gets exponentially amplified with $\abs{\alpha}$. Second, the normalization factor $(\nvec{U}_1^\dagger \, \nvec{V}_1)$ decreases as we create a system with larger $N$. A feed on the edge opposite to the localized response is therefore also exponentially increased with the number of unit cells $N$. This makes sense, because the number of intermediate amplification cells is increased in the system. In the limit of $N\rightarrow \infty$, the normalization of left and right eigenmodes vanishes. 



Similar to Sec.~\ref{subsec:Full-scale experimental fit of the theoretical model}, we find the mean localization of the OBC bulk eigenmodes, that were obtained from the experimental data to $\abs{\alpha}_\text{nl} = ($2.2 $\pm$ 0.3$)$. It agrees with the theoretically computed value in Tab.~\ref{tab:alpha} within the scope of the standard deviation, but the localization is weakened due to non-linear effects in the operational amplifiers, that become noticeable in the regime of small damping. In Fig.~\ref{fig:non-linear response}a, we show a logarithmic plot of the non-local voltage response with the circuit being excited at node 1 and node 7. It agrees with the expected amplification profile due to the localization of the OBC modes with the Bloch factor of $\abs{\alpha}_\text{nl}$, as shown by a straight line in Fig.~\ref{fig:non-linear response}. Fig.~\ref{fig:non-linear response}b depicts the amplification factor of the circuit with respect to the voltage level at the feed node. The maximum voltage decreases exponentially as a function of the nodal distance to the current feed. The expectation in Fig.~\ref{fig:non-linear response}b using  $\abs{\alpha}_\text{nl}$ as the unit-wise amplification factor agrees with the measurement data. Fig.~\ref{fig:non-linear response} highlights two statements on the non-local response. First, the voltage response retains the decay profile of the localized bulk modes with $\abs{\alpha}_\text{nl}$ and second, $\abs{\alpha}_\text{nl}$ also accurately models the amplification factor as a contribution from each unit cell.

Note, that if we decrease $R_0$ and thereby increase the damping in the system, we need to take the excitation of all modes into account and the approximation formulated in equation \eqref{eq:non-local voltage response approx} breaks down. With larger damping, we introduce interference between all modes in the system leading to a voltage response, that is more and more localized at the point of the current feed.

Fig.~\ref{fig:Phase diagram} displays the localization of OBC modes, where $\kappa = 0$ means full delocalization. We notice, that the maximum of $\abs{\kappa}$ is larger for right edge localized modes ($\kappa <0$) than for left edge localized ones ($\kappa <0$). This asymmetry is caused by the serial resistances, which exhibit a different frequency dependence than capacitive and inductive components. Their effect is therefore weakened for larger frequencies leading to a stronger localized voltage profile at the right edge than at left edge at the respective frequencies.   

{\bf Caption Table 1:} Nominal values for the components used for the circuit implementation. (${}^*$) The potentiometer implementing $R_0$ was adjusted to $20\,\Omega$ for the measurements of the spectra and to $120\,\Omega$ for the measurement of the non-local voltage response.

{\bf Caption Table 2:} Frequency-dependent fitted inductances and their corresponding serial resistances $L_1, L_0, R_{L_1}, R_{L_0}$ at each experimentally studied frequency. Other experimental parameters are chosen to be $R_0 = 20 \, \Omega$, $C_1 =$ 300\,nF, $C_2 =$ 94\,nF, $C_3 =$ 47\,nF and $C_0 =$ 94\, nF.

{\bf Caption Table 3:} Theoretically calculated and experimentally fitted magnitude of the Bloch factor $\abs{\alpha}$ for the experimentally studied frequencies. Experimental values are obtained as a mean value of the exponential decay profile of all eigenmodes, that have been numerically computed from the measurement data. Their error is given by the standard deviation of the mean value while considering all eigenmodes of the system and therefore quantifies the agreement of the localization of all bulk states.

{\bf Caption Figure 4:} \textbf{(a)} Schematic diagram of one unit cell of the skin circuit including all grounding terms. An $LC$ resonating circuit models the frequency dependent intracell coupling $v(\omega)$. Connectivity between unit cells is modeled by a capacitance $C_2$, which is also depicted in Fig.~1 of the main text. \textbf{(b)} Negative impedance converter with current inversion used in the intracell $A-B$ connection based on an operational amplifier to generate the non-Hermitian, non-reciprocal coupling $\gamma$.

{\bf Caption Figure 5:} Schematic adjustment of the inductive coupling as we incorporate parasitic resistances in the circuit model.

{\bf Caption Figure 6:} \textbf{(a1-5)} Complex admittance spectra for frequencies $f_1=70$\,kHz, $f_2=\ $84.2\,kHz, $f_3= \ $91.5\,kHz, $f_4= \ $95.0\,kHz and $f_5=\ $98.5\,kHz with PBCs in red and OBCs in green. Discrete dots denote measurements, joint lines show the parasitics-corrected theoretical values. \textbf{(b1-5)} Magnitude of voltage modes $\abs{\psi}_n$ for OBCs with $n$ indicating the $n$-th eigenvector of the system, computed from experimental data. The topological modes are separated from the bulk modes by a green line. \textbf{(c)} Spectral flow of the admittance spectrum from PBC to OBC for $f= \ $80\,kHz quantified by $\eta$, which is the boundary bond normalized to its PBC value. Black dots are experimental data points, red and green lines indicate the admittance spectrum of the fitted model for PBC and OBC respectively.

{\bf Caption Figure 7:} Top: Variation of the intracell coupling $v(\omega)$ as a function of frequency. The two OBC exceptional points for the ideal model at $v = \pm \gamma$ and the transition point at $v=0$ with delocalized OBC modes are marked by black lines. Bottom: Localization length $\xi$ of the bulk OBC eigenmodes for the ideal effective model as a dashed red curve and for the full-scale fit model as a blue joint line.

{\bf Caption Figure 8:} Spectral flow of the admittance in the complex plane at the frequency $f = \ $80\,kHz for the effective model (left) and the aberration-corrected fit model (right). The flow is quantified by the parameter $\eta$, which describes the attenuation factor of the 1-20 bond. 

{\bf Caption Figure 9:} \textbf{ (a)} Logarithmic plot of the absolute voltage response in a measurement involving an external current feed with AC driving frequency $f = \ $98.5\,kHz at the left edge of the sample at node 1 in red and at node 4 in blue. The voltage increases exponentially to the right side of the circuit due to a unit-wise amplification. The expected voltage profile normalized to the voltage at the feed node is shown as a straight line, where the slope has been numerically calculated as the average decay profile of the OBC bulk eigenmodes.  \textbf{ (b)} Logarithmic plot of the amplification factor of the circuit defined as the ratio of the maximum voltage $V_\text{max}$ to the voltage  $V_\text{feed}$ at the feed node. The voltage response increases with nodal distance to the right edge, which is supplemented by the expected drop in amplification calculated from the mode localization as in \textbf{ (a)}.

\clearpage

\begin{table}
	\centering
	\begin{tabular}{|c|c|c|c|c|c|c|c|}
		\hline 
		$C_1$ [nF] & $C_2$ [nF] & $C_3$ [nF] & $L_1$ [$\mu$H] & $L_0$ [$\mu$H] & $R_0$ [$\Omega$] & $R_\text{a}$ [$\Omega$] & $C_\text{a}$ [nF] \\ 
		\hline 
		300 & 94 & 47 & 10 & 33 & 20, $120^*$ & 0.2 & 940 \\ 
		\hline
	\end{tabular} 
	\caption{} 
	\label{tab:Nominal values}
\end{table}


\begin{table}
	\centering
	\begin{tabular}{|c|c|c|c|c|c|c|}
		\hline 
		Frequency $f$ [kHz] & 70.0 & 80.0 & 84.2 & 91.5 & 95.0 & 98.5 \\ 
		\hline 
		Inductance $L_1$ [$\mu$H] & 10.1 & 10.1 & 10.1 & 10.1 & 10.1 & 10.1 \\ 
		\hline 
		Inductance $L_0$ [$\mu$H] & 33.0 & 32.0 & 31.7 & 30.3 & 30.1 & 30.0 \\ 
		\hline 
		Serial resistance $R_{L_1}$ [m$\Omega$]& 220 & 230 & 250 & 280 & 300 & 320 \\ 
		\hline 
		Serial resistance $R_{L_0}$ [m$\Omega$]	& 660 & 700 & 750 & 780 & 780 & 780 \\ 
		\hline 
	\end{tabular} 
	\caption{} 
	\label{tab:Fit parameters}
\end{table}




\begin{table}
	\centering
	\begin{tabular}{|c|c|c|c|c|c|}
		\hline 
		frequency $f$ [kHz]	& 70.0 & 84.2 & 91.5 & 95.0 & 98.5 \\ 
		\hline 
		Theory (fit): $\abs{\alpha}_\text{theo}$  & 0.80 &  0.42 & 1.02 & 1.60 & 2.53 \\ 
		\hline 
		Experiment:  $\abs{\alpha}_\text{exp}$ & 0.79 $\pm$ 0.03 & 0.42 $\pm$ 0.03 & 1.0 $\pm$ 0.1 &  1.6 $\pm$ 0.2 & 2.7 $\pm$ 0.3 \\ 
		\hline
	\end{tabular} 
	\caption{} 
	\label{tab:alpha}
\end{table}

\clearpage

\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{SC_figure_comp_unit_cell_INIC.pdf}
	\caption{}
	\label{fig:circuit_schematic_simplified}
\end{figure}


\begin{figure}
	\centering
	\includegraphics[width=0.7\linewidth]{SC_Figure_parasitics.pdf}
	\caption{}
	\label{fig:parasitics}
\end{figure}


\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{SC_figure_eigenvalues_appendix_v2.pdf}
	\caption{}
	\label{fig:Spectral measurements}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[width=0.5\linewidth]{SC_Figure_phase_diagram.pdf}
	\caption{}
	\label{fig:Phase diagram}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{SC_figure_flow.pdf}
	\caption{}
	\label{fig:Spectral flow}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[width=\linewidth]{SC_figure_amplification.pdf}
	\caption{}
	\label{fig:non-linear response}
\end{figure}

\end{document}
