††thanks: Contributed equally to this work.††thanks: Corresponding author:
<EMAIL_ADDRESS>Corresponding author<EMAIL_ADDRESS>
# The simulation of non-Hermitian disordered system in linear circuits

Luhong Su  Beijing National Laboratory for Condensed Matter Physics, Institute
of Physics, Chinese Academy of Sciences, Beijing 100190, China  School of
Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,
China  Hui Jiang  Department of Physics, National University of Singapore,
Singapore 117542  Zhan Wang  Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Shu Chen  Beijing National Laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  CAS Center for Excellence in Topological Quantum Computation
and School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  The Yangtze River Delta Physics Research Center,
Liyang, Jiangsu 213300, China  Dongning Zheng  Beijing National Laboratory for
Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  CAS Center for Excellence in Topological Quantum
Computation and School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Songshan Lake Materials Laboratory, Dongguan,
Guangdong 523808, China

###### Abstract

Non-Hermitian skin effect (NHSE) is a novel phenomenon appearing in non-
Hermitian systems. Here, we report the experimental observation of NHSE.
Different from the previous non-reciprocal circuit implementation scheme using
logic components, we construct our one-dimensional (1D) circuits using linear
components only. Besides, we achieve the non-reciprocity by proportionally
varying the parameter value of the components. By measuring the voltage
response of each site, the information of eigenstates can be mapped out. The
results show that the voltage response is always larger on one end of the
circuit no matter on which end voltage driving is applied, indicating clearly
the presence of the NHSE. Furthermore, we also simulate the interplay of NHSE
and Anderson localization (AL) when additional disorder is introduced. Upon
increasing the disorder strength, we observe the transformation from the skin
effect phase to the localized phase. In the regime of skin effect phase, the
eigenstates are all localized at one edge while eigenstates are affected by
the voltage supply input in localized phase. Our findings unveil a possible
new route for simulation of topological phenomena in non-Hermitian systems.

##  I Introduction

While the Hamiltonian of a closed quantum system is always Hermitian, it has
been demonstrated that some open quantum systems, optic systems and electric
circuits can be effectively modeled by non-Hermitian HamiltoniansAshida  _et
al._ (2020). Non-Hermitian systems have been unveiled to exhibit some unique
features, such as non-Hermitian skin effect (NHSE), which is characterized by
the accumulation of a majority of eigenstates on the boundariesLee (2016); Yao
and Wang (2018). To date, the NHSE has been extensively investigated from
various theoretical aspectsLee  _et al._ (2019); Sounas and Alù (2017); Huang
_et al._ (2016); Yang  _et al._ (2020); Longhi (2019); Borgnia  _et al._
(2020); Gong  _et al._ (2018); Yuce (2020); Zhang  _et al._ (2021); Tang  _et
al._ (2021); Li  _et al._ (2020a); Liu  _et al._ (2020a); Yuce (2021); Lee
_et al._ (2020); Prodan and Schulz-Baldes (2016).

The theoretical progress has further promoted experimental studies on the
properties of non-Hermitian systemsLi  _et al._ (2019); Wang  _et al._ (2019);
Wu  _et al._ (2019); Wang  _et al._ (2021); Naghiloo  _et al._ (2019), which
have been performed in optical fieldZhu  _et al._ (2020); Sebastian Weidemann
and Szameit (2020); Xiao  _et al._ (2020), micro-resonatorQi  _et al._
(2020),cold atomLi  _et al._ (2020b), robotic metamaterialsGhatak  _et al._
(2020); Martin Brandenbourger and Coulais (2019) and electric circuitsYakir
Hadad and Alù (2018). The non-Hermitian system with NHSE is sensitive to
boundary conditionsGuo  _et al._ (2021); Li  _et al._ (2020a). When the
boundary condition of the system changes from the periodic boundary to open
boundary, the energy spectrum of the system changes drastically, which
reflects the boundary sensitivity of the skin effect. Due to the design
flexibility and simplicity, electric circuits have become a powerful platform
for studying topological effects in recent yearsNingyuan  _et al._ (2015);
Zhang  _et al._ (2019); Albert  _et al._ (2015); Stefan Imhof and Thomale
(2018); Lu  _et al._ (2019); Helbig  _et al._ (2019); Ezawa (2019a). Some
topological phenomena have been demonstratedNingyuan  _et al._ (2015), such as
the boundary states with topological protection in the one-dimensional SSH
modelZhang  _et al._ (2019); Albert  _et al._ (2015); Stefan Imhof and Thomale
(2018); Lu  _et al._ (2019); Helbig  _et al._ (2019), the generalized
Brillouin zone and the generalized bulk boundary correspondenceLiu  _et al._
(2021a); T. Helbig and Thomale (2020), the high-order topological state in
high-dimensional lattice circuitsStefan Imhof and Thomale (2018); Zangeneh-
Nejad and Fleury (2019). Majority of the circuit design realizes non-
Hermiticity relying on using active devicesEzawa (2019b); T. Helbig and
Thomale (2020); Liu  _et al._ (2021a); Zhang and Franz (2020); Xu  _et al._
(2021); Liu  _et al._ (2020b); Stegmaier  _et al._ (2021). The simplest model
exhibiting the NHSE is the Hatano-Nelson (HN) modelHatano and Nelson (1996,
1997),which can be realized by generating non-reciprocal hopping. However,
this has not been demonstrated experimentally in RLC linear electric circuit
without any logic devices.
Using the platform of RLC linear electric circuit, we can realize the non-
Hermitian system with non-reciprocal hopping and study the topological phase
transition induced by the competition of non-reciprocal hopping and disorder.
NHSE is characterized by the emergence of bulk eigenstates localized in the
boundary due to the regular unidirectional gain or loss during the
transmission of electronic waves between lattice points. In contrast, if the
electron wave is perturbed randomly, the wave function of the electron will be
confined to a certain range and no longer propagate, which is known as the
Anderson localization (AL)Anderson (1958),111Although the Anderson
localization is originally referred as the localization induced by the random
disorder, in some references people also called the localization phenomena
induced by quasiperiodic potentials as Anderson localization. Here we call
both localization induced by random and quasiperiodic potentials as Anderson
localization.. The similarities and differences between the two phenomena have
received attentions. An interesting issue is how the wave function behaves if
the particle is subjected to both regular unidirectional gain and random
perturbation? And what will the wave function be? Recently the interplay of
NHSE and AL has been studied theoretically by considering the non-reciprocal
quasiperiodic modelJiang  _et al._ (2019),which stimulates intensive
theoretical interest on the study of localization phenomena in non-reciprocal
systemsLiu  _et al._ (2021b); Longhi (2021); Claes and Hughes (2021). However,
experimental study on this interesting issue is still lacking. In this paper,
we address the above issue in the RLC linear electric circuit with non-
reciprocal hopping.

Figure 1: Reciprocity breaking in a non-Hermitian linear RLC circuit. (a)
Schematic diagram of non-reciprocal one-band hopping model, where $J_{R(L)}$
is the right(left)-hopping amplitude. (b) Experimental implementation of a
non-reciprocal circuit model based on Fig. 1(a). The asymmetrical couplings
are achieved by ratio of two adjacent inductors $L_{n}=Lg^{-n}$. The $\gamma$
term is achieved by the resistors. The brown dotted line represents the on
site potential, which divides the samples into one-band non-Hermitian system
Fig. 1(c) and the interplay system Fig. 1(d). The numbers label the circuit
components: (1)surface mounted device (SMD) inductor; (2)SMD capacitor
($C_{n}$$\approx$$Cg^{n}$); (3)SMD resistor ($R_{n}=Rg^{-n}$), (4)additional
SMD inductor in interplay system: $\Delta_{n}=L_{n}/l_{n}$, left-skin phase:
$J_{L}=g=2.2$, $\Delta_{n}$ random select in $[7\times 10^{-5},0.33]$,
localized phase: $J_{L}=g=2.2$, $\Delta_{n}=2\Delta\text{cos}(2\pi\beta n)$,
$\beta$=$(\sqrt{5}-1)/2$.

##  II Non-Reciprocity circuit

In this work, we choose the HN model to exhibit the NHSE experimentally. The
model is illustrated schematically in Fig. 1(a), where $J_{R(L)}$ is the right
(left)-hopping amplitude and $J_{R}\neq J_{L}$ . The Hamilton of the model can
be written as:

$H=\sum_{n}J_{R}|n+1\rangle\langle n|+J_{L}|n\rangle\langle n+1|.$ (1)

To simulate this model, we construct a chain circuit that consists of 10 unit-
cells using linear components (such as capacitors and inductors) only.
Obviously, non-reciprocity cannot be achieved by simply using the same
repeating unit. Instead, we build the circuit with capacitance and inductance
of each unit vary proportionally at a constant ratio $g$, as suggested in Ref.
Jiang  _et al._ (2019). According to Kirchhoff‘s law, for the $n$th site, the
voltage of adjacent unit-cells can be written as: (The details of the
derivation are given in Appendix A.)

$V_{n-1}+gV_{n+1}=(\omega^{2}/\omega_{0}^{2}+1+g)V_{n}.$ (2)

Eq. (2) can be described by the Hamiltonian Eq. (1) with $J_{R}=1$ and
$g=J_{L}/J_{R}$. Hence, the behavior in HN model can be understood by
investigating the voltage response of our system. In Fig. 1(b), we show two
adjacent unit-cells of circuit. In each unit-cell, the brown inductor in
parallel is used for the interplay system in section IV, and the resistors are
used to stabilize the circuit. Capacitive and inductive elements achieve
static equation $\mathcal{HV}=E\mathcal{V}$, where $\mathcal{H}$ is the matrix
representation of the non-reciprocal Hamiltonian under the open boundary
condition (OBC) and $\mathcal{V}$=${(...,V_{n},...)}^{T}$, as defined in Eq.
(1). The eigenvalue $E=\omega^{2}/\omega_{0}^{2}+1+g$ and the parameter
$\omega_{0}=1/\sqrt{L_{n}C_{n}}=1/\sqrt{LC}$ donates resonance frequency of
single unit-cell. Therefore, we can obtain the information of eigenstates by
measuring the voltage response at each site. However, the measurement of the
static system requires a strong self-sustained energy gain, that is, after a
short time of source feed, the voltage response can still be measured at each
site respectively. But in the experiment the self-oscillating dissipation of
the static system is strong, and the voltage decays rapidly.
Remarkably, dynamic measurements are more widely available in circuit
experiments, so we introduce dynamic response. The information of the static
equation can be obtained through the dynamic evolution equation when circuit
is applied an alternating current (a.c.) voltage driving. The inhomogeneous
equation with dimensionless parameters reads:

$\frac{d^{2}}{d\tau^{2}}\mathcal{V}(\tau)-(\mathcal{H}-1-g)\mathcal{V}(\tau)=\mathcal{V}_{e},$
(3)

where $\tau=\omega_{0}t$ and $\mathcal{V}_{e}$ donates the position of the
external source $V(f)$, with $\omega=2\pi f$. For example, if the a.c. voltage
$V(f)$ is imposed on the left side of the circuit, we have
$\mathcal{V}_{e}=(V(f),0,...,0)^{T}$. The solution of Eq. (3) corresponds to
the eigenstates of the $\mathcal{H}$. Thus, the distribution of the voltage
response characterizes the eigenstates of the static equation.

In fact, Eq. (3) takes into account both the initial state and the first
derivative of the initial state. While it is generally difficult to measure
directly, we introduce the damping term into Eq. (3). We eventually arrive at
a evolution equation:

$\frac{d^{2}}{d\tau^{2}}\mathcal{V}(\tau)+\gamma\frac{d}{d\tau}\mathcal{V}(\tau)-(\mathcal{H}-1-g)\mathcal{V}(\tau)=\mathcal{V}_{e},$
(4)

where $\gamma$=$(1/R)\sqrt{L/C}$, with large values of resistance which is 20
times of the series inductance at the resonance frequency in our circuits, so
that the system can stabilize quickly. The solution of Eq. (4) is :

$\mathcal{V}(\tau)=e^{-\gamma\tau/2}\mathcal{V}_{0}(\tau)+\sum_{n}\mathcal{V}_{n}\mathcal{W}_{n}^{T}\mathcal{V}_{e}({a_{n}}\text{cos}\tilde{\Omega}t+b_{n}\text{sin}\tilde{\Omega}t).$
(5)

Here, $\mathcal{V}_{n}$ and $\mathcal{W}_{n}^{T}$ are $n$th right and left
eigenvectors of $\mathcal{H}$. The $\tilde{\Omega}$ means the frequency which
is normalized to $\omega_{0}$. $\mathcal{V}_{0}(\tau)$ is related to initial
condition, and
$a_{n}(\tilde{\Omega},\gamma),\,\,b_{n}(\tilde{\Omega},\gamma)\in[-1,\,1]$.
When $\gamma\ll 1$, the system is resonant with a large value of $a_{n}$ and
vanishing $b_{n}$, so the voltage response distribution of the system actually
reflects the overlap of $\mathcal{W}_{n}^{T}$ and $\mathcal{V}_{e}$.
By introducing resistance, the initial state can be ignored and the properties
of the static system of the HN model can be obtained. The device parameters we
use cover three orders of magnitude, which is completely different from the
single repeating unit in traditional electric topology circuits reported in
the literatureNingyuan  _et al._ (2015); Zhang  _et al._ (2019); Albert  _et
al._ (2015); Stefan Imhof and Thomale (2018); Lu  _et al._ (2019); Helbig  _et
al._ (2019); Ezawa (2019a).
We fabricated a number of circuit boards with different $g$ values. The total
circuit configuration is shown in Fig. 2(b) where the dashed and dotted lines
represent the left-side and right-side driving, respectively. The circuit
elements of the unit-cell are specified in Fig. 1(b), while a physical unit
cell board cutout is presented in Figs. 1(c) and 1(d).

Figure 2: (a) Schematic diagram of the measured transmission curve,
$V_{i}^{+/-}$represents input/output a.c. voltage in right or left end. (b) A
voltage source of a.c. driving $f_{n}$ is imposed on the left or the right
side of the circuit. We measure voltage response of 10 sites respectively. (c)
Transmission curves of Hermitian extended state. The solid line is the
imaginary part of $S_{12}$ while the dotted line is $S_{21}$ which show
reciprocity, $S_{12}=S_{21}$, where $S$ is the scattering matrix. The point
with zero imaginary part is the resonance point, which is the eigenfrequency
we choose. (d) Voltage response of each site in Hermitian extended state for
input voltage a.c. frequency $f\,=\,433.0$ kHz. (e) Voltage response of each
site in left-skin state for $f\,=\,260.8$ kHz. The voltage response all
localized on the left boundary, and the right side voltage response is much
higher than left side one. (f) Comparison chart of voltage response of
$g\,=\,1$ vs. $g\,=\,2.2$ in left side voltage. We use logarithmic coordinates
for ordinates. $V_{L(R)}$ represent left (right) side Voltage in the Fig. 2
and Fig. 3.

##  III Dynamic measurement

In this section, we give a more detailed description about the dynamic
measurements and show the results. During the experiment, we apply a.c.
voltage $V(f_{n})$ to the left and to the right side respectively. Here,
$f_{n}$ is the resonance frequency of the system. By measuring the voltage
response of each site, we can get the wave function distribution corresponding
to the eigenenergy $E_{n}$. Firstly, we find out the resonance frequency by
measuring the transmission coefficient of the system. The total system is
treated as a two - ports device, through the equation:

$\left(\begin{array}[]{c}V_{1}^{-}\\\
V_{2}^{-}\end{array}\right)=S\left(\begin{array}[]{c}V_{1}^{+}\\\
V_{2}^{+}\end{array}\right)=\left[\begin{array}[]{ccc}S_{11}&S_{12}\\\
S_{21}&S_{22}\end{array}\right]\left(\begin{array}[]{c}V_{1}^{+}\\\
V_{2}^{+}\end{array}\right)\ ,$ (6)

where $V_{i}^{+/-}$ are the voltages of the incoming and outgoing signals at
ports i, port 1 (2) is the left (right) - side of the circuit in Fig. 2(a),
where $S$ is the transmission coefficient matrix. The a.c. voltages are
measured by a lock-in amplifier (Zurich Instruments UHF) working in the
frequency scanning mode. The physical circuits for determining resonant
frequency is shown in Fig. 2(a). We applied a voltage source drive at the port
1 (2), and measure the transmission response at each frequency at the port 2
(1). The Lorentz reciprocity theorem requires that the scattering matrix
satisfies the symmetry condition $S^{T}=S$, so the signal transmission between
two ports is the same for both propagation directions. For Hermitian system,
$S_{12}=S_{21}$, so the left and right transfer functions are exactly the
same. For the one-band non-reciprocal system, $S_{12}\not=S_{21}$ results in
the different transmission coefficients on the left and right and a port-
dependent frequency shift. The transmission curve of Hermitian system is shown
in Fig. 2(c), its imaginary part is plotted as a function of frequency $f$.
In electric circuits, the impedance of inductors and capacitors are
$Z_{L}=i2\pi fL$ and $Z_{C}=-i/(2\pi fC)$ respectively, showing that inductors
lead the phase while capacitors lag the phase. Therefore, in the transmission
curve, the frequency of the point with zero imaginary part is the resonance
frequency $f_{n}$. The Hermitian transmission curves are coincided completely,
which are drastically different from the non-reciprocal transmission curves.
For non-reciprocal systems, the voltage responses of $S_{21}$ and $S_{12}$ are
very small due to the skin effect. Therefore, in the actual measurement, we
use the resonance frequency $f_{n}$ calculated theoretically.
As shown in Fig. 2(b), we apply a.c. voltage $V(f_{n})$ and measure the
voltage response of each site, where $V_{L(R)}$ is equal to port 1 (2) in Fig.
2(a). The results of the Hermitian system are shown in Fig. 2(d). Clearly, the
curves of the voltage response obtained from Hermitian circuits look the same
up to a left-right reflection, being consistent with the theoretical
calculation that Hermitian systems exhibit extended states under OBCs. For
Hermitian system, the wave functions are all extended states. Furthermore, the
overlap of each extended state is similar, which meets
$\sum_{n}\mathcal{V}_{n}\times\mathcal{W}_{n}^{T}=1$, so that the voltage
response reflects the information of the external source. The results in Fig.
2(d) shows that the voltage profile decreases with distance to the feed unit-
cells. This is due to the presence of the resistors introduced for stabilizing
dynamic measurements. Nevertheless, we see the voltage does not drop to very
small value even for the $10$th site, indicating the extended nature of the
system.
In Fig. 2(e), we show the result with $g=2.2$. Remarkably, our circuit
produces a dominant voltage signal at the left edge whether the voltage feed
is imposed on left or right side. When driving from left side, $V_{1}$ is much
larger than $V_{10}$, and $V_{1}/V_{10}\approx 150$. The voltage response
rapidly decays to the right with the change of the position of the site, which
fully reflects the skin effect that the eigenstates of the system are
accumulated to the left boundary. When driving from right side, as shown in
Eq. (5), the overlap of $W_{n}^{T}$ and $V_{e}$ is very large which induces
higher voltage response. Nevertheless, we can still see a trend of decreasing
voltage from the left end to the right end, being consistent with the left
skin effect.

Figure 3: (a) Phase diagram. The phase boundaries are determined by hopping
amplitude $\lvert J_{R}\rvert$($\lvert J_{L}\rvert$) and disorder strength
$\Delta$. Pentagon and diamond makers represent the parameters used in Fig.
2(c) and (d). The circle, hexagon and square markers correspond to localized
phase, Hermitian phase and left-skin phase respectively and indicate the
parameters used for (c)-(e). (b) Comparison of hopping coupling strength and
disorder strength. (c) Voltage response of each site in left skin phase for
$f\,=\,825.3$ kHz, $J_{L}>J_{R}$ while disorder strength $\Delta\approx 0.25$.
(d) Voltage response of each site in Hermitian case for $f\,=\,571.8$ kHz,
$J_{L}=J_{R}$ and $\Delta_{n}$ = 1. (e) Voltage response of each site in
localized phase for $f\,=\,1.82$ MHz, $J_{L}>J_{R}$ while $\Delta\approx 10$.

So far, we have observed the NSHE. Meanwhile, for $J_{L}>J_{R}$ ($g=2.2$) the
corresponding system under periodic boundary condition is topologically
nontrivial, which is characterized by the winding number $w=-1$. Whereas for
$J_{L}=J_{R}\,(g=1)$, the corresponding system is topologically trivial with
$w=0$Gong  _et al._ (2018). The NSHE is characterized by the accumulation of
eigenstates at the boundary. By comparing the skin states of the non-Hermitian
system with the extended eigenstates of the Hermitian system, we observe that
the topologically different phases display quite different behaviors under the
OBC. Remarkably, in our circuits, topological phase transitions can be
realized by tuning the value of $g$ even for a single-band lattice. In
contrast, for Hermitian systems, at least two band systems (such as the SSH
model) are required to observe topological phenomena. Furthermore, in a
classical circuit, when a source is added, the voltage response gradually
decreases along the loop, which is quite different from the NHSE in the non-
reciprocal circuit.

##  IV The Interplay of NHSE and AL

The NHSE can be achieved through a one-band non-reciprocal system, which is
based on a simple hopping between the inductors and the capacitors. But the
circuit platform by introducing tunable on-site potentials can simulate more
complex systems. The circuit system can simulate the interplay of NHSE and AL
when each site is connected with an inductor to form $\Delta_{n}$ to act as a
disorder term. When $\Delta_{n}=2\Delta\cos(2\pi\beta n)$, the model becomes
the non-reciprocal Aubry-Andr$\acute{e}$ (AA) model where $\Delta$ is the
amplitude of the quasiperiodic potential and $\beta$ takes the value of the
golden ratio ($\sqrt{5}-1)/2$. We arrive at interplay system:

$H=\sum_{n}J_{R}|n+1\rangle\langle n|+J_{L}|n\rangle\langle
n+1|+\Delta_{n}|n\rangle\langle n|.$ (7)

By adjusting the value of disorder strength $\Delta_{n}$, topological phase
transition from skin phase to localized phase can be realized. Under the OBC,
the non-reciprocal interplay model has three phases due to the competition
between the AL and the NHSE. As shown in Fig. 3(a) and (b), corresponding to
dominant term, there are three phases including left-skin phase, localized
phase and right-skin phase. When the $\Delta_{n}$ is small, the system is
dominated by skin effect. Due to $g>1$, system is in the left skin phase.
Regardless of whether the source drive is added from the left or right, the
voltage response is always localized on the left, as shown in Fig. 3(c). When
the source is added from the left, the voltage response decays in turn,
$V_{1}/V_{10}\approx 100$, similar to the non-Hermitian left skin state.
However, when the source is added from the right, the left skin effect is
further weakened, because small disorder is added to the system which compete
with the skin effect. In this case, the hopping coupling strength is still
much greater than the disorder strength, so the voltage response is all
localized in the left boundary. It is worth noting that
$\Delta_{n}=L_{n}/l_{n}$ means that the selection of $l_{n}$ should be as
large as possible for the same $L_{n}$. As a reference, we design the
Hermitian system of $g=1$, $\Delta_{n}$=1. In the Hermitian system, the
transmission curves of the left and right ends are basically the same, and the
voltage response of each site fluctuates less, as shown in the Fig. 3(d).

When disorder strength $\Delta_{n}$ is further increased, the competition
between localization and skin effect is strengthened, which can lead to a
transition from skin phase to localized phase. When the average strength
multiple of the localization is about 10 times of the hopping amplitudes, the
skin phase is completely destroyed and the system is in the localized phase.
When the source drive is added from the left end, the voltage response
decreases rapidly from left to right $V_{1}/V_{10}\approx 10^{4}$. When the
source drive is added from the right end, the voltage response decays from
right to left, $V_{10}/V_{1}\approx 30$, as shown in Fig. 3(e). In comparison
with the left skin phase, non-reciprocal hopping still plays a role, but it is
completely overwhelmed by the disorder strength.

Because the circuit of Printed Circuit Board (PCB) has inductance, when the
cosine periodic potential is small, the inductance has a great influence on
the circuit, resulting in the total system is still a localized phase.
Therefore, for the parameter design of the left skin phase, the maximum
$l_{n}$ should be selected as large as possible. The three distinct phases are
located at different locations in the phase diagram. For the left skin phase,
although the skin effect is affected by the AL, the skin effect is still
dominant. Phase transition occurs as the disorder becomes larger and dominant.
For the localized phase, since the wave function is localized, it is
insensitive to boundaries. Unlike the left skin phase, the voltage response is
localized at the source of the added voltage, and the disorder results in
attenuation as it gradually moves away from the source of the added voltage.
The measurement process is similar to the non-reciprocal one-band model, for
which the dynamic evolution is used to obtain the characteristics of the
static system.

##  V Conclusion

In this paper, the NHSE is observed on a one-dimensional non-reciprocal one-
band system with open boundary realized by linear electric circuits.
Furthermore, this platform also simulates competition between disorder and
skin effect, which leads to three different topological phases. When the
disorder strength is small, the left skin phase is obtained. The voltage
response is localized on the left whether the source is added from the left or
the right side. When the the disorder strength is dominant, a dramatic change
occurs, and the voltage response is localized on the side of the source. The
skin effect is suppressed by the disorder term, but a great difference between
the left and right end sources can still be observed. At this time, the system
changes into the localized phase. We have successfully observed the
characteristics of different phase regions.

Using linear circuits to simulate non-Hermitian phenomena is not only
convenient, but also easy to operate. The measurement system is relatively
easy to control, and the limitations of the environment are small. We have
built the circuit with capacitance and inductance of each unit vary
proportionally at a constant ratio $g$ to realize non-reciprocity. We predict
that it may provide a potential scheme for further research of critical skin
effect in the future.

###### Acknowledgements.

This work is supported by the State Key Development Program for Basic Research
of China (Grant No. 2017YFA0304300), the Key-Area Research and Development
Program of Guangdong Province, China (Grant No. 2020B0303030001), the National
Natural Science Foundation of China (Grant No. T2121001, No. 12174436 and No.
11974413) and the Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB28000000).

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##  Appendix A Non-reciprocal circuits design

For the $n$th site in Fig. 4, due to Kirchhoff’s law, the incoming current is
equal to the outgoing current

$i_{n}=i_{n+1}+i_{c}+i_{l}$ (8)

Taking time derivative to the above equation, then we have:

$\frac{di_{n}}{dt}=\frac{di_{n+1}}{dt}+\frac{di_{c}}{dt}+\frac{di_{l}}{dt}.$
(9)

With voltage - current characteristics of capacitance and inductance that

$C\frac{dV}{dt}=i\,\,and\,\,L\frac{di}{dt}=-V,$ (10)

the Eq. (9) is hence replaced by

$V_{n-1}+\frac{L_{n}}{L_{n+1}}V_{n+1}-\frac{L_{n}}{l_{n}}V_{n}=(L_{n}C_{n}\omega^{2}+1+\frac{L_{n}}{L_{n+1}})V_{n}$
(11)

Here, we take $g={L_{n}}/{L_{n+1}}$ to represent the non-reciprocal hopping
amplitude, $\Delta_{n}={L_{n}}/{l_{n}}$ is random on-site potential energy,
the LC oscillator is characterized by its resonance frequency $\omega$, and
$\omega_{0}^{2}=1/L_{n}C_{n}=1/LC$ denotes natural frequency. So, Kirchhoff’s
law can be rewritten asJiang  _et al._ (2019):

$V_{n-1}+gV_{n+1}-\Delta_{n}V_{n}=(\omega^{2}/\omega_{0}^{2}+1+g)V_{n}.$ (12)

Consider all the voltages, written in matrix form $H|V\rangle=E_{n}|V\rangle$,
so the $nth$ eigenvalue $E_{n}=\omega_{n}^{2}/\omega_{0}^{2}+1+g$. The form of
the Eq. (12) is equivalent to the one-dimensional non-reciprocal competitive
system constructed by us :

$H=\sum_{n}|n+1\rangle\langle n|+g|n\rangle\langle
n+1|+\Delta_{n}|n\rangle\langle n|\ .$ (13)

Since each site is connected with an inductor to form $\Delta_{n}$ to act as
an Anderson disorder term. When $\Delta_{n}=2\Delta\cos(2\pi\beta n)$, the
model becomes a non-reciprocal AA model where $\Delta$ is the amplitude of the
quasiperiodic potential and $\beta$ takes the value of the golden ratio
$(\sqrt{5}-1)/2$. For lack of on-site potential ($\Delta_{n}=0$), Eq. (13)
translates to

$H=\sum_{n}|n+1\rangle\langle n|+g|n\rangle\langle n+1|.$ (14)

which denotes the HN model. Different skin states are formed according to
$g>1\,(J_{L}>J_{R})$ or $g<1\,(J_{L}<J_{R})$.

Figure 4: Circuit diagram of two adjacent unit-cells.

##  Appendix B Experimental implementation

A circuit consisting of ten unit cells was realized on a PCB. The concise
values for the circuit components are detailed in Tabs. 1, 2. The parameter
error of the components is within 5%. In the implementation of the NHSE, we
employ the chip capacitors, inductors and resistors. To reduce the influence
of the inductors by the external magnetic field, we add a shielding cover to
each inductor. We have made the same three circuit boards in each table.
To perform the spectral measurements, a constant a.c. voltage is fed into the
beginning/last site of the board, while lock-in amplifiers are used to measure
the each site’s voltage response of the circuit separately. We use the
transmission characteristics to get the resonance frequency. Three resonance
frequencies are selected for each circuit board for measurement.

##  Appendix C Winding number

Topological winding number is defined as Gong  _et al._ (2018):

$\omega\equiv\int_{-\pi}^{\pi}\frac{dk}{2\pi i}\partial_{k}ln(detH(k))$ (15)

Let $E_{n}(k)$(n=1,2,…,N) be the eigenenergy of H(k), where N is the total
number of bands. Then, Eq. (15) can be rewritten as:

$\omega=\sum_{n=1}^{N}\int_{-\pi}^{\pi}\frac{dk}{2\pi}\partial_{k}argE_{n}(k),$
(16)

where arg$E_{n}(k)$ is the argument of the complex energy $E_{n}(k)$.
Therefore, for Hermitian Hamiltonians, the real energy implies
Arg$E_{n}(k)$=0, $\pi$.
We consider the HN model with asymmetric couplings as Eq. (7). By making
Fourier transformation to moment space, so we obtain the Bloch Hamiltonian as:

$H(k)=J_{R}e^{-ik}+J_{L}e^{ik},$ (17)

whose winding number is evaluated to give

$\omega=\left\\{\begin{aligned} 1&,&\left|J_{R}\right|<\left|J_{L}\right|,\\\
-1&,&\left|J_{R}\right|>\left|J_{L}\right|.\end{aligned}\right.$ (18)

There is a topological phase transition as the winding number changes from 1
to -1. We note that $\omega=0$ when $\left|J_{R}\right|=\left|J_{L}\right|$.

Table 1: Parameters of Fig. 2 | Hermitian system (Fig. 2(d)) | left-skin state (Fig. 2(e))
---|---|---
subscript | $L\,(\mu H)$ | $C\,(nF)$ | $R\,(\Omega)$ | $L\,(\mu H)$ | $C\,(nF)$ | $R\,(\Omega)$
1 | 15 | 10 | 100 | 100 | 4.7 | 470
2 | 15 | 10 | 100 | 47 | 10 | 220
3 | 15 | 10 | 100 | 22 | 22 | 100
4 | 15 | 10 | 100 | 10 | 47 | 47
5 | 15 | 10 | 100 | 4.7 | 100 | 22
6 | 15 | 10 | 100 | 2.2 | 220 | 10
7 | 15 | 10 | 100 | 1 | 470 | 4.7
8 | 15 | 10 | 100 | 0.47 | 1000 | 2.2
9 | 15 | 10 | 100 | 0.22 | 2200 | 1
10 | 15 | 10 | 100 | 0.1 | 4700 | 0.47
11 | 15 |  |  | 0.047 |  |
Table 2: Parameters of Fig. 3 | left-skin phase (Fig. 3(c)) | Hermitian system (Fig. 3(d)) | localized phase (Fig. 3(e))
---|---|---|---
subscript | $L\,(\mu H)$ | $C\,(nF)$ | $l\,(\mu H)$ | $R\,(\Omega)$ | $L\,(\mu H)$ | $C\,(nF)$ | $l\,(\mu H)$ | $R\,(\Omega)$ | $L\,(\mu H)$ | $C\,(nF)$ | $l\,(\mu H)$ | $R\,(\Omega)$
1 | 33 | 1 | 100 | 1000 | 15 | 10 | 15 | 220 | 33 | 1 | 22 | 1000
2 | 15 | 2.2 | 220 | 470 | 15 | 10 | 15 | 220 | 15 | 2.2 | 2.2 | 470
3 | 6.8 | 4.7 | 150 | 220 | 15 | 10 | 15 | 220 | 6.8 | 4.7 | 0.68 | 220
4 | 3.3 | 10 | 330 | 100 | 15 | 10 | 15 | 220 | 3.3 | 10 | 33 | 100
5 | 1.5 | 22 | 330 | 47 | 15 | 10 | 15 | 220 | 1.5 | 22 | 0.15 | 47
6 | 0.68 | 47 | 470 | 22 | 15 | 10 | 15 | 220 | 0.68 | 47 | 0.15 | 22
7 | 0.33 | 100 | 220 | 10 | 15 | 10 | 15 | 220 | 0.33 | 100 | 0.1 | 10
8 | 0.15 | 220 | 100 | 4.7 | 15 | 10 | 15 | 220 | 0.15 | 220 | 0.012 | 4.7
9 | 0.068 | 470 | 150 | 2.2 | 15 | 10 | 15 | 220 | 0.068 | 470 | 0.15 | 2.2
10 | 0.033 | 1000 | 470 | 1 | 15 | 10 | 15 | 220 | 0.033 | 1000 | 0.0039 | 1
11 | 0.015 |  |  |  | 15 |  |  |  | 0.015 |  |  |