††thanks: Z. Wang and X.-T. Zeng contributed equally to this work.††thanks: Z.
Wang and X.-T. Zeng contributed equally to this work.

# Realization of a Hopf insulator in circuit systems

Zhu Wang  Wuhan Institute of Quantum Technology, Wuhan 430206, China  School
of Physics and Technology, Wuhan University, Wuhan 430072, China  Xu-Tao Zeng
School of Physics, Beihang University, Beijing 100191, China  School of
Physics and Technology, Wuhan University, Wuhan 430072, China  Yuanchuan Biao
Wuhan Institute of Quantum Technology, Wuhan 430206, China  School of Physics
and Technology, Wuhan University, Wuhan 430072, China  Zhongbo Yan
<EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of
Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen
University, Guangzhou 510275, China  Rui Yu<EMAIL_ADDRESS>Wuhan Institute
of Quantum Technology, Wuhan 430206, China  School of Physics and Technology,
Wuhan University, Wuhan 430072, China

###### Abstract

Three-dimensional (3D) two-band Hopf insulators are a paradigmatic example of
topological phases beyond the topological classifications based on powerful
methods like $K$-theory and symmetry indicators. Since this class of
topological insulating phases was theoretically proposed in 2008, they have
attracted significant interest owing to their conceptual novelty, connection
to knot theory, and many fascinating physical properties. However, because
their realization requires special forms of long-range spin-orbit coupling
(SOC), they have not been achieved in any 3D system yet. Here we report the
first experimental realization of the long-sought-after Hopf insulator in a 3D
circuit system. To implement the Hopf insulator, we construct basic pseudo-
spin modules and connection modules that can realize $2\times 2$-matrix
elements and then design the circuit network according to a tight-binding Hopf
insulator Hamiltonian constructed by the Hopf map. By simulating the band
structure of the designed circuit network and calculating the Hopf invariant,
we find that the circuit realizes a Hopf insulator with Hopf invariant
equaling $4$. Experimentally, we measure the band structure of a printed
circuit board and find the observed properties of the bulk bands and
topological surface states (TSS) are in good agreement with the theoretical
predictions, verifying the bulk-boundary correspondence of the Hopf insulator.
Our scheme brings the experimental study of Hopf insulators to reality and
opens the door to the implementation of more unexplored topological phases
beyond the known topological classifications.

In 2008, the pioneering tenfold-way classification based on non-spatial
symmetries provided the first systematic understanding of non-interacting
topological phases of matter Schnyder  _et al._ (2008); Kitaev (2009), and
founded the basis for the later discovery of a long list of symmetry-protected
topological phases based on powerful methods such as symmetry indicators Chiu
_et al._ (2016); Po  _et al._ (2017); Bradlyn  _et al._ (2017); Zhang  _et
al._ (2019); Tang  _et al._ (2019); Vergniory  _et al._ (2019). Despite its
systematicity and fundamental significance, the existence of topological
phases beyond the tenfold way classification was soon noticed. Just in the
same year, Moore, Ran, and Wen theoretically showed that a class of 3D two-
band magnetic topological insulators Moore  _et al._ (2008), later dubbed Hopf
insulators as characterized by an integer-valued Hopf invariant Deng  _et al._
(2013); Kennedy (2016), exist outside the tenfold-way periodic table Schnyder
_et al._ (2008); Kitaev (2009). Besides the prominent conceptual significance,
the two-band Hopf insulators have attracted considerable interest both in
theory and experiment due to their many fascinating properties Lapierre  _et
al._ (2021); Alexandradinata  _et al._ (2021). The bulk-boundary
correspondence, a central property of topological phases, is also unique in
Hopf insulators. The uniqueness is manifested through the dependence of TSS on
the surface’s orientation and the support of gapless surface Dirac cones, even
though the time-reversal symmetry is broken. Besides enriching topological
phases, the study of Hopf insulators also substantially advances the
understanding of 2D out-of-equilibrium topological phases. The Hopf invariant
is found to play an important role in the topological characterization of
quenched Chern insulators Wang  _et al._ (2017); Tarnowski  _et al._ (2019),
quenched Euler insulators Ünal  _et al._ (2020); Zhao  _et al._ (2022), and
Floquet Chern insulators Ünal  _et al._ (2019).

Although Hopf insulators have been proposed for more than one decade and the
great importance of their physical realization is well appreciated Moore  _et
al._ (2008); Schuster  _et al._ (2021a, b), to date they have only been
simulated in a single-qubit quantum simulator Yuan  _et al._ (2017) and have
not been implemented in any 3D system yet. The main challenges for
implementing Hopf insulators are the demand of having exactly two bands and a
peculiar pattern of long-range SOC. These requirements rule out the
implementation in many quantum material systems as well as many artificial
systems.

In this paper, we report the first bulk realization of the long-sought-after
Hopf insulators in a 3D circuit. Because of the extremely high level of
connection freedom, circuit networks have been used to realize many novel
states of matter, such as 2D topological insulators Ningyuan  _et al._ (2015);
Albert  _et al._ (2015), 3D topological semimetals Lee  _et al._ (2018); Luo
_et al._ (2018); Lu  _et al._ (2019), and even 4D topological phases Yu  _et
al._ (2020); Ezawa (2019); Zhang  _et al._ (2020); Wang  _et al._ (2020). To
carry out the experiment, we use basic building blocks, which in principle
admit the implementation of any arbitrary two-band model, Hermitian or non-
Hermitian, to design a 3D periodic circuit according to a Hopf insulator model
constructed by the Hopf map. By numerically simulating the band structure and
calculating the Hopf invariant $N_{h}$, we find a Hopf insulator phase with
$N_{h}=4$ exists in a sizable region of the parameter space. By experimentally
measuring the bulk and boundary energy spectra of a printed circuit board
sample, we find the experimental results agree well with the theoretical
predictions and verify the defining bulk-boundary correspondence of the Hopf
insulator.

Model Hamiltonian.—We start with the theoretical model for two-band Hopf
insulators. It is known that any two-band model can be expressed via the Pauli
matrices $\boldsymbol{\sigma}$=$(\sigma_{1}$,$\sigma_{2}$,$\sigma_{3})$ as

$H(\bm{k})=d_{0}(\bm{k})\sigma_{0}+\bm{d}(\bm{k})\cdot\boldsymbol{\sigma},$
(1)

where $\sigma_{0}$ is the $2\times 2$ identity matrix and
$\bm{k}=(k_{x},k_{y},k_{z})$. Focusing on band topology, all essential
information is encoded in the three-component $\bm{d}$-vector. The first term
on the right-hand side is irrelevant and can be neglected. Moore, Ran, and Wen
showed that theoretical models for two-band Hopf insulators can be
systematically constructed when the $\bm{d}$ vector is descended from a
complex spinor via the Hopf map Moore  _et al._ (2008), i.e.,
$\bm{d}(\bm{k})=z(\bm{k})^{\dagger}\bm{\sigma}z(\bm{k})$, where
$z(\boldsymbol{k})=(z_{1}(\boldsymbol{k}),z_{2}(\boldsymbol{k}))^{T}$,
$z_{1}=\eta_{1}(\bm{k})+i\eta_{2}(\bm{k})$,
$z_{2}=\eta_{3}(\bm{k})+i\eta_{4}(\bm{k})$, with $\eta_{1,2,3,4}(\bm{k})$
being real functions of momentum. The map is characterized by the Hopf
invariant defined as Moore  _et al._ (2008)

$N_{h}=-\frac{1}{4\pi^{2}}\int
d^{3}k\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\rho}A_{\nu},$ (2)

where $A_{\mu}=-i\langle u(\bm{k})|\partial_{\mu}|u(\bm{k})\rangle$ with
$\mu,\nu,\rho=\\{k_{x},k_{y},k_{z}\\}$ and $|u\rangle$ being the negative-
energy eigenfunction of $H(\bm{k})$ is the Berry connection. It is worth
noting that $N_{h}$ does not have any gauge ambiguity even though the Berry
connection $A_{\mu}$ is gauge dependent. Such a property allows us to
numerically calculate the Hopf invariant through discretization of the
Brillouin zone Yan  _et al._ (2017a). When $N_{h}$ is nonzero, the resulting
$\bm{d}(\bm{k})\cdot\boldsymbol{\sigma}$ model realizes a two-band Hopf
insulator.

There are infinite choices for $\eta_{1,2,3,4}(\bm{k})$ to achieve a nonzero
$N_{h}$. For the convenience of experimental implementation, in this work we
consider $\eta_{1}(\bm{k})=t_{1}\sin k_{x}$,
$\eta_{2}(\bm{k})=t_{2}\cos(k_{x}+k_{y}+k_{z})$, $\eta_{3}(\bm{k})=t_{3}\sin
k_{y}$, and $\eta_{4}(\bm{k})=t_{4}\sin k_{z}$. Accordingly, we find $N_{h}=4$
and

$\displaystyle d_{1}(\bm{k})$ $\displaystyle=$ $\displaystyle 2t_{1}t_{3}\sin
k_{x}\sin k_{y}+2t_{2}t_{4}\cos k_{d}\sin k_{z},$ $\displaystyle
d_{2}(\bm{k})$ $\displaystyle=$ $\displaystyle 2t_{1}t_{4}\sin k_{x}\sin
k_{z}-2t_{2}t_{3}\cos k_{d}\sin k_{y},$ $\displaystyle d_{3}(\bm{k})$
$\displaystyle=$ $\displaystyle
t_{1}^{2}\sin^{2}k_{x}+t_{2}^{2}\cos^{2}k_{d}-t_{3}^{2}\sin^{2}k_{y}-t_{4}^{2}\sin^{2}k_{z}.\quad$
(3)

Here we have introduced $k_{d}\equiv k_{x}+k_{y}+k_{z}$ to shorten the
notation. Apparently, all three components of the $\bm{d}$ vector involve
long-range hopping processes in real space. What raises a particular challenge
is that the hopping parameters involving different length scales need to be
comparable in magnitude and satisfy a stringent phase pattern.

Figure 1:  (a) A connection module connects pseudo-spin modules $p_{1}$ and
$p_{2}$. The three ports on the connection module’s left (right) side are
connected to the three nodes of the left (right) pseudo-spin module. (b-d) The
list of designed connection modules $m_{\pm(i)\sigma_{1,2,3}}$ that gives
$\pm(i)\sigma_{1,2,3}$ types of tunneling matrices between the pseudo-spin
space, where solid lines indicate capacitors, dashed lines indicate resistors.
In each module, the red solid and dashed lines indicate that their impedances
are half those of the black solid and dashed lines.

Hopf insulator circuit.—In this paper, we overcome the challenge and implement
the Hopf insulator Hamiltonian (3) in circuit networks as follows. We first
create the pseudo-spin space via the module depicted in Fig. 1(a), where three
identical inductors form a triangle with ${\rm C}_{3}$ rotational symmetry.
Accordingly, the pseudo-spin space in Eq.(3) is provided by the twofold
degenerate eigenstates characterized by the 2D representation of the ${\rm
C}_{3}$ group. Connecting pseudo-spin modules with connection modules, in
which the components form a braided network, allows electrical signals to flip
the pseudo-spin, resulting in a SOC-like effect when signals are transmitted
between the pseudo-spin modules. Based on this idea, we design connection
modules to generate couplings of the form $\pm(i)\sigma_{n}$ ($n=1,2,3$) as
shown in Figs. 1(b-d) Wu  _et al._ (2022); sm . As the parameters of the
capacitors, inductors, and resistors are positive real numbers, we incorporate
the negative sign and the imaginary unit $i$ into the network structure of the
connection modules in order to obtain the hopping matrix with complex
coefficients in Eq. (4) below. With these modules, the challenging long-range
SOC in Eq. (3) can be achieved since components in electronic circuits can be
connected between nodes at arbitrary distances by wires, which is in sharp
contrast to condensed solid materials and many artificial materials.

$m$ | 0 | 1 | 2
---|---|---|---
$\delta_{m}$ | $0$ | $2\hat{x}$ | $2\hat{y}$
$\hat{U}_{m}$ |  $\frac{1}{2}$$($$t_{1}^{2}$$+$$t_{2}^{2}$$-$$t_{3}^{2}$$-$$t_{4}^{2}$$)$$\sigma$1 | $-\frac{1}{4}t{}_{1}^{2}\sigma_{1}$ | $\frac{1}{4}t{}_{3}^{2}\sigma_{1}$
$m$ | 3 | 4 | 5
$\delta_{m}$ | $2\hat{z}$ | $\hat{x}-\hat{y}$ | $\hat{x}+\hat{y}$
$\hat{U}_{m}$ | $\frac{1}{4}t{}_{4}^{2}\sigma_{1}$ | $\frac{1}{2}t_{1}t_{3}\sigma_{3}$ | $\frac{1}{2}(it_{2}t_{4}-t_{1}t_{3})\sigma_{3}$
$m$ | 6 | 7 | 8
$\delta_{m}$ | $\hat{x}-\hat{z}$ | $\hat{x}+\hat{z}$ | $\hat{x}+2\hat{y}+\hat{z}$
$\hat{U}_{m}$ | $\frac{1}{2}t_{1}t_{4}\sigma_{2}$ | $-\frac{1}{2}(it_{2}t_{3}+t_{1}t_{4})\sigma_{2}$ | $\frac{i}{2}t_{2}t_{3}\sigma_{2}$
$m$ | 9 | 10 |
$\delta_{m}$ | $\hat{x}+\hat{y}+2\hat{z}$ | $2\hat{x}+2\hat{y}+2\hat{z}$ |
$\hat{U}_{m}$ | $-\frac{i}{2}t_{2}t_{4}\sigma_{3}$ | $\frac{1}{4}t{}_{2}^{2}\sigma_{1}$ |
Table 1: The hopping vectors $\delta_{m}$ and the corresponding SOC operators
$\hat{U}_{m}$ for the Hopf insulator circuit. $\hat{x}$, $\hat{y}$, and
$\hat{z}$ indicate the unit lattice vectors in the x- , y- and z-directions,
respectively.

Performing Fourier transformation on Eq. (3), the real-space tight-binding
Hamiltonian reads

$H_{TB}=\sum_{\bm{l}}\sum_{m=0}^{10}(c_{\bm{l}+\delta_{m},\bm{l}}^{\dagger}\hat{U}_{m}c_{\bm{l}}+h.c.),$
(4)

where $\bm{l}$ indicate lattice sites, $\delta_{m}$ are hopping vectors,
$\hat{U}_{m}$ are SOC operators as given in Table (1). The operators
$\hat{U}_{m}$ can be implemented with the connection modules illustrated in
Figs. 1(b-d). For example, $\hat{U}_{1}$ can be built by $m_{-\sigma_{1}}$
module and $\hat{U}_{5}$ can be constructed by connecting the
$m_{-\sigma_{3}}$ and $m_{i\sigma_{3}}$ modules in parallel. To reduce the
number of operational amplifiers used in the experiment, we have exchanged the
expressions for $d_{1}$ and $d_{3}$. This operation does not change the band
topology since it is equivalent to a redefinition of the spin basis.

Figure 2: (a) The unit cell of the Hopf insulator circuit. The connection
modules are used for 3D connections. The ports of the connection modules
marked with $\delta_{m}$ ($m=$ 1 to 10) indicate that they are connected to
the pseudo-spin module in cell $\bm{l}+\delta_{m}$, while unmarked ones
indicate that they are connected to the pseudo-spin module in cell $\bm{l}$.
(b) The $\mathcal{H}$-module consists of voltage followers, resistors, and
adder-subtractor operational amplifiers. (c) Detailed circuit diagram of the
red ports in Fig. 2(a).

According to Eq. (4), we construct the 3D Hopf insulator circuit network using
pseudo-spin modules and connection modules. The unit cell of the Hopf
insulator circuit is shown in Figs. 2(a-c). It is worth noting that the
resistors in the modules $m_{\pm i\sigma_{2}}$ and $m_{\pm\sigma_{3}}$ induce
energy loss, which makes the Hamiltonian non-Hermitian. To address this issue,
we use the $\mathcal{H}$-module (Fig. 2(b)) to compensate for the energy loss
and restore the hermiticity sm .

Kirchhoff’s equations for the Hopf insulator circuit can be written as

$(h_{1}(\bm{k})\oplus
H_{h}^{circuit}(\bm{k}))\tilde{\bm{v}}=\omega^{-2}(0\oplus
I_{2})\tilde{\bm{v}}$ (5)

where $\oplus$ stands for a direct sum of the constant representation space
and the pseudo-spin space of the ${\rm C}_{3}$ symmetry group,
$\tilde{\bm{v}}=U^{\dagger}{\bm{v}}$, $\bm{v}=({v}_{1},{v}_{2},{v}_{3})^{T}$
are the node voltages in the unit cell, and $U$ is defined in Supplemental
Material sm . $h_{1}(\bm{k})$ is the Hamiltonian in the constant
representation space.
$H_{h}^{circuit}(\bm{k})=\sum_{i=0}^{3}f_{i}(\bm{k})\sigma_{i}$ is the
Hamiltonian in the pseudo-spin space, where

$\displaystyle f_{0}(\bm{k})$ $\displaystyle=$
$\displaystyle\frac{L}{3}\big{(}4C_{1}+2C_{2}+2C_{3}+2C_{4}+12(C_{5}+C_{6})\big{)},$
$\displaystyle f_{1}(\bm{k})$ $\displaystyle=$
$\displaystyle-\frac{4L}{3}\big{(}C_{1}\sin^{2}k_{x}+C_{2}\cos^{2}k_{d}-C_{3}\sin^{2}k_{y}$
$\displaystyle-C_{4}\sin^{2}k_{z}\big{)},$ $\displaystyle f_{2}(\bm{k})$
$\displaystyle=$
$\displaystyle-\frac{4\sqrt{3}L}{3}\big{(}-\frac{1}{R_{1}\omega}\cos k_{d}\sin
k_{y}+C_{5}\sin k_{x}\sin k_{z}\big{)},$ $\displaystyle f_{3}(\bm{k})$
$\displaystyle=$ $\displaystyle-\frac{4\sqrt{3}L}{3}\big{(}C_{6}\cos k_{d}\sin
k_{z}+\frac{1}{R_{2}\omega}\sin k_{x}\sin k_{y}\big{)},$ (6)

and $R_{\alpha}$ ($\alpha=1,2$), $C_{\beta}$ ($\beta=1$ to $6$), $L$ are
parameters of the components in the circuit. By choosing appropriate
parameters, one can separate the eigenfrequencies of $H_{h}^{circuit}(\bm{k})$
well from that of $h_{1}(\bm{k})$. Therefore, we focus on
$H_{h}^{circuit}(\bm{k})$ and examine its topological properties below.

Figure 3:  (a) The calculated band gap as a function of the parameters of $C$
and $R$. The color map represents the ratio of the band gap to the total
frequency band width. (b) The frequency spectrum of $H_{h}^{circuit}(\bm{k})$
at points A, B, C, and D in the phase diagram, where $\Gamma$=(0,0,0), $\rm
Y$=(0,1,0), $\rm M$=(1,1,0), $\rm R$=(1,1,1) are the high symmetry points in
the Brillouin zone in units of $\pi$ with the unit cell lattice parameters set
to 1. The parameters ($R$, $C$) are equal to (0.03 $\rm k\Omega$, 0.03 nF) at
A point, (0.2 $\rm k\Omega$, 0.56 nF) at B point, (1.2 $\rm k\Omega$, 1.2 nF)
at C point, and (2.3 $\rm k\Omega$, 1.8 nF) at D point. The inductance is
fixed as $L$=2.7 ${\rm\mu}$H in all calculations. (c) The calculated frequency
spectrum along $k_{y}$ and $k_{z}$ directions of a 16-layers slab with
x-direction surfaces. The green color refers to bulk states. The red (blue)
color refers to states localized on the x=1 (x=16) layer. (d) The frequency
spectrum of a 16-layers slab with z-direction surfaces. The TSS on the z=1
(red) and z=16 (blue) layers degenerate and overlap in frequency. The
parameters at point B are used to calculate the TSS in Figs. 3(c-d).

For simplicity of discussion, we set the inductance $L$=2.7 $\mu$H,
$R_{\alpha}$=$R$ ($\alpha=1,2$), and $C_{\beta}$=$C$ ($\beta=1$ to $6$). The
phase diagram of the frequency band gap as a function of $R$ and $C$ is shown
in Fig. 3(a), where the band gaps are finite in the yellow and green regions
and tend to zero in the dark blue region as shown in Fig. 3(b). In the gapped
region, we find $N_{h}=4$ as expected, agreeing with the considered Hopf map.
In the regions with vanishingly small gaps, the Hopf invariant does not
converge as it is not well-defined in the presence of band degeneracy.

One defining characteristic of the Hopf insulator is the unique correspondence
between the number of topologically protected TSS and the Hopf invariant.
Measuring this bulk-surface correspondence can faithfully identify the
topological nature of the system. In Figs. 3(c-d), we show the numerically-
calculated band structures for a 16-layers thickness slab structure terminated
in x- and z-direction, respectively. There are four TSS on the x-direction
surface of the system in Fig. 3(c), where the red (blue) color refers to TSS
located on the x=1 (x=16) layer, and the green color refers to bulk states. On
the z-direction termination, the TSS localized on z=1 and z=16 layers are
degenerate in frequency, as shown in Fig. 3(d). Remarkably, on each
z-direction surface, the crossings of the surface frequency spectra labeled by
the same color at time-reversal invariant momenta reveal the existence of
surface Dirac cones even though the Hamiltonian does not have time-reversal
symmetry as already mentioned. Experimentally, these exotic surface Dirac
cones provide a unique signature to identify the Hopf insulator.

In the following, we experimentally verify the topological nature of the Hopf
insulator designed above. According to the circuit diagram in Figs. 2(a-c), we
prepare a printed circuit board with the number of unit cells being
3$\times$20$\times$6 in x-, y-, and z-directions and set periodic boundary
conditions in the x- and y-directions and open boundary conditions in the
z-direction. The circuit structure of the unit cell and the
$\mathcal{H}$-module are shown in Figs. 4(a-b). A global view of the printed
circuit board is shown in Supplemental Material sm .

Figure 4:  (a) The unit cell of the printed circuit board (in the red box) is
fabricated according to Fig. 2(a). (b) The unit cell of the printed circuit
board of the $\mathcal{H}$-module (in the yellow box). (c), (d) and (e) show
the experimentally measured band structure along $k_{y}$ direction with the
signal source connected to a cell at z=1 (bottom surface), z=3 (bulk), and z=6
(top surface), respectively. The black color represents the experimental data,
and the colored dotted lines are the computed band structure of a slab with
six layers in the z-direction, where red (blue) indicates the wave functions
localized on the z=1 (z=6) surface and green indicates the bulk states.

To extract the frequency spectrum of the circuit lattice, we perform
frequency-domain measurements to obtain the voltage vectors
$\bm{v}(\bm{r},f)$=$(v_{1}(\bm{r},f)$, $v_{2}(\bm{r},f)$, $v_{3}(\bm{r},f))$,
where $\bm{r}$=$(\rm{x},\rm{y},\rm{z})$ labels the unit cell, $f$ is the
frequency, and the subscripts indicate the nodes in each unit cell. By
performing Fourier transformation in the x- and y-directions, the voltage
$\bm{v}(\boldsymbol{k}_{\parallel},{\rm z},f)$ can be obtained in the momentum
space, where $\boldsymbol{k}_{\parallel}$=$(k_{x},k_{y})$. The frequency
dispersions shown in Figs. 4(c-e) are obtained by plotting
$\rho(\boldsymbol{k}_{\parallel},{\rm
z},f)$=$|\bm{v}(\boldsymbol{k}_{\parallel},{\rm z},f)|^{2}$, where the peaks
of $\rho$ indicate the resonance frequency of the circuit system.

The frequency dispersions for the excitation signal applied to the z=1 and z=6
layers are depicted in Fig. 4(c) and (e), where the TSS appear in the band gap
and locate around the time-reversal invariant momenta, which is in good
agreement with the results calculated from the model Hamiltonian (color dotted
lines). The TSS disappear for the excitation signal applied to the z=3 layer
because the signal on the middle layer cannot excite the TSS located on the
surface layers, as shown in Fig. 4(d). These experimental results confirm the
predicted properties of the TSS. In Figs. 4(c-e), we have shifted the
experimental data upward by 2.9$\times 10^{4}$ Hz to compare with the
theoretical results. Details of the distribution of the TSS in the z-direction
are provided in the Supplemental Material sm . As a final remark, the TSS in
the circuit lattice can also be detected by other methods, like impedance
measurements Zhang  _et al._ (2021, 2022).

Conclusions and discussions.—In this work, we present a general scheme for the
implementation of long-range SOC in electric circuits, where the spatial
dependence of the SOC can be modulated with a high degree of freedom. Using
this property, we have successfully implemented the long-sought-after Hopf
insulator in the circuit and observed the TSS enforced by bulk-boundary
correspondence. Our general scheme brings the experimental study of Hopf
insulators to reality and paves the way for exploring other exotic topological
phases associated with peculiar SOC. With our established platforms, the
experimental exploration of links and knots with very rich topological
structures becomes accessible Yan  _et al._ (2017b); Bi  _et al._ (2017);
Belopolski  _et al._ (2022). Moreover, the idea behind our scheme can be
applied to the future design of systems with more but fixed bands to implement
the novel topological phases beyond the known topological classifications,
such as Hopf insulators in three-band systems Neupert  _et al._ (2012) and
models constructed by higher-dimensional generalizations of the Hopf map Liu
_et al._ (2017).

Acknowledgments.—R. Y. was supported by the National Key Research and
Development Program of China (No.2017YFA0304700, No.2017YFA0303402), the
National Natural Science Foundation of China (No.11874048 and No.12274328),
and the Beijing National Laboratory for Condensed Matter Physics. Z. Y. was
supported by the National Natural Science Foundation of China (Grant
No.11904417 and No.12174455) and the Natural Science Foundation of Guangdong
Province (Grant No. 2021B1515020026).

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