Square-root higher-order Weyl semimetals

Lingling Song, Huanhuan Yang, Yunshan Cao, and Peng Yan∗

School of Electronic Science and Engineering and State Key Laboratory of
Electronic Thin Films and Integrated Devices, University of Electronic Science
and Technology of China, Chengdu 610054, China.

The mathematical foundation of quantum mechanics is built on linear algebra,
while the application of nonlinear operators can lead to outstanding
discoveries under some circumstances. In this Letter, we propose a model of
square-root higher-order Weyl semimetal (SHOWS) by inheriting features from
its parent Hamiltonians. It is found that the SHOWS hosts both “Fermi-arc”
surface and hinge states that connect the projection of the Weyl points. We
theoretically construct and experimentally observe the exotic SHOWS state in
three-dimensional (3D) stacked electric circuits with honeycomb-kagome
hybridizations and double-helix interlayer couplings. Our results open the
door for realizing the square-root topology in 3D solid-state platforms.

Nearly all the operators encountered in quantum mechanics are linear (or
antilinear) operators, such as the rotation, translation, parity, time
reversal, etc, which allows us to construct the mathematical basis of quantum
mechanics formulated on linear algebra. Square-root operator is one of the few
exceptions. Historically, Paul Dirac derived the Dirac equation through a
square-root operation on the Klein-Gordon (KG) equation to describe all
spin-$\frac{1}{2}$ massive particles that inherent the Lorentz-covariance of
the parent KG equation [1, 2, 3]. The approach has inspired Arkinstall _et
al._ [4] to propose the concept of square-root topological insulator (TI) by
taking the nontrivial square-root of a tight-binding Hamiltonian in periodic
lattices. The most appealing feature of square-root TI is that it inherits the
nontrivial nature of Bloch wave function from its parent Hamiltonian. The
square-root TI was subsequently observed in a photonic cage [5]. Recently, the
square-root operation has been applied to higher-order topological insulators
(HOTIs) that allow topologically robust edge states with codimension larger
than one [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Besides the gapped solution,
e.g., the electron-positron pair, the Dirac equation allows another crucial
gapless or massless solution called Weyl fermion [16] that plays an important
role in quantum field theory and the Standard Model. Although not yet observed
among elementary particles, Weyl fermions are shown to exist as collective
excitations in Weyl semimetals [17, 18, 19]. It is thus intriguing to ask if
the square-root operation can apply to semimetals [20] or higher-order
semimetals [21, 22, 23, 24, 25], and particularly how to realize these exotic
states in experiments.

In this article, we propose a tight-binding (TB) model of the square-root
higher-order Weyl semimetal (SHOWS) by a vertical stacking of two-dimensional
(2D) square-root HOTIs with interlayer couplings in a double-helix fashion. It
is found that the SHOWS hosts both 2D surface arc states and one-dimensional
(1D) hinge states with the topological feature being fully characterized by
the quantized bulk polarization. We construct the TB model in honeycomb-kagome
(HK) hybridized inductor-capacitor (_LC_) circuit networks. By performing both
the impedance and voltage measurements in the stacked HK circuit, we identify
the fingerprint of the SHOWS by directly observing the Weyl points, the
“Fermi-arc” surface states, and the hinge states. It is revealed that both the
surface states and the hinge states ideally connect the projections of the
Weyl points, consistent with theoretical calculations.

Fig. 1:  a Illustration of an infinite three-dimensional (3D) stacked HK TB
model. The unit cell including five nodes is represented by the dashed black
rhombus. The intralayer hoppings are $t_{a}$ (green) and $t_{b}$ (blue) in the
$x-y$ plane, whereas the interlayer double-helix hopping is $t_{z}$ (brown). b
The equivalence between the squared Hamiltonian of the HK circuit and its
parents. c The bulk dispersion along the $k_{z}$ direction with
$(k_{x},k_{y})=(4\pi/3,0)$. The dashed blue line indicates the position of the
degenerate points. d The first Brillouin zone and the distribution of the Weyl
points. e Bulk polarization $p_{1}$ as a function of $k_{z}$. For TB
calculations in c and e, we set $t_{a}=0.5$, $t_{b}=1$, and $t_{z}=0.5$.

Model
Figure 1a shows the lattice structure of the proposed model, the square of
which can be viewed as the direct sum of a stacked honeycomb and a breathing
kagome lattices (see Fig. 1b and the analysis in Supplementary Information
Sec. I [26]). The tight-binding Hamiltonian is given by

$\displaystyle\mathcal{H}=$ $\displaystyle t_{a}\sum_{\langle
m,n\rangle}\left(a_{m}^{\dagger}c_{n}+a_{m}^{\dagger}d_{n}+a_{m}^{\dagger}e_{n}\right)$
(1) $\displaystyle+t_{b}\sum_{\langle
m,n\rangle}\left(b_{m}^{\dagger}c_{n}+b_{m}^{\dagger}d_{n}+b_{m}^{\dagger}e_{n}\right)$
$\displaystyle+t_{z}\sum_{\langle\langle
m,n\rangle\rangle}\left(b_{m}^{\dagger}c_{n}+b_{m}^{\dagger}d_{n}+b_{m}^{\dagger}e_{n}\right)+{\rm
H.c.},$

where $a^{\dagger}$ ($a$), $b^{\dagger}$ ($b$), $c^{\dagger}$ ($c$),
$d^{\dagger}$ ($d$), and $e^{\dagger}$ ($e$) are the creation (annihilation)
operators on the site 1-5, respectively, $\langle m,n\rangle$ and
$\langle\langle m,n\rangle\rangle$ label the nearest-neighbor and next-
nearest-neighbor coupling, respectively, and $t_{a}$, $t_{b}$, and $t_{z}$ are
the hopping parameters. Without loss of generality, we assume all hopping
paramaters are positive. In momentum space, the Hamiltonian can be expressed
as

$\mathcal{H}=\left(\begin{matrix}O_{2,2}&\Phi_{\bf k}^{\dagger}\\\ \Phi_{\bf
k}&O_{3,3}\\\ \end{matrix}\right),$ (2)

where $O_{2,2}$ and $O_{3,3}$ are the $2\times 2$ and $3\times 3$ zero matrix,
respectively, and $\Phi_{\bf k}$ is the $3\times 2$ matrix

$\Phi_{\bf
k}=\left(\begin{matrix}t_{a}&t_{b}+2t_{z}\text{cos}(\mathbf{k}\cdot{\mathbf{a}_{3}})\\\
t_{a}&[t_{b}+2t_{z}\text{cos}(\mathbf{k}\cdot{\mathbf{a}_{3}})]e^{-i\mathbf{k}\cdot{\mathbf{a}_{1}}}\\\
t_{a}&[t_{b}+2t_{z}\text{cos}(\mathbf{k}\cdot{\mathbf{a}_{3}})]e^{-i\mathbf{k}\cdot{\mathbf{a}_{2}}}\\\
\end{matrix}\right).$ (3)

Here $\mathbf{k}=(k_{x},k_{y},k_{z})$ is the wave vector, and
$\textbf{a}_{1}=\frac{1}{2}\hat{x}+\frac{\sqrt{3}}{2}\hat{y}$, $\textbf{
a}_{2}=-\frac{1}{2}\hat{x}+\frac{\sqrt{3}}{2}\hat{y}$ and
$\textbf{a}_{3}=\hat{z}$ are three basic vectors.

By taking the square of the original Hamiltonian (2), we can conveniently
obtain the dispersion relation of $[\mathcal{H}]^{2}$ (see Supplementary
Information Sec. I [26])

$E_{\bf k}=0\ \ \text{and}\ \frac{3}{2}\left[t_{a}^{2}+t_{b}^{\prime
2}\pm\sqrt{(t_{a}^{2}-t_{b}^{\prime 2})^{2}+4t_{a}^{2}t_{b}^{\prime
2}|\Delta({\bf k})|^{2}}\right],$ (4)

with $t_{b}^{\prime}=t_{b}+2t_{z}\text{cos}(k_{z})$ and $\Delta({\bf
k})=(1+e^{i\mathbf{k}\cdot{\mathbf{a}_{1}}}+e^{i\mathbf{k}\cdot{\mathbf{a}_{2}}})/3$.
The band structure of the original Hamiltonian is thus given by
$\varepsilon_{\bf k}=\pm\sqrt{E_{\bf k}}$. It is found that the band structure
closes at the twofold degenerate points $K_{\pm}=(4\pi/3,0,\pm k_{zw})$, as
shown in Fig. 1c, with $k_{zw}=\text{arccos}[(t_{a}-t_{b})/(2t_{z})]$ when
$|t_{a}-t_{b}|<2t_{z}$. It is straightforward to verify that their time-
reversal counterparts are $G^{{}^{\prime}}_{\pm}=(-4\pi/3,0,\pm k_{zw})$, and
their equivalence points locate at $G_{\pm}$, $G^{{}^{\prime\prime}}_{\pm}$,
$K^{{}^{\prime}}_{\pm}$, and $K^{{}^{\prime\prime}}_{\pm}$,

Fig. 2: a Top view of a stacked 10-layer HK circuit with 2860 nodes. b
Admittances for $C_{A}=C_{B}/2=0.5$ nF, $L_{A}=30$ $\mu$H, $L_{B}=7.5$ $\mu$H,
$L_{C}=18$ $\mu$H, and $L_{G}=21.829$ $\mu$H. The red, blue, black symbols
represent the hinge, surface, bulk states, respectively. Spatial distribution
of the wave-function of the normalized hinge ($j_{n}=3.534\times 10^{-7}$
$\Omega^{-1}$) state c, surface ($j_{n}=-0.0003762$ $\Omega^{-1}$) state d,
and bulk ($j_{n}=0.001042$ $\Omega^{-1}$) state e.

as shown in Fig. 1d. By evaluating the topological charge $C_{\rm FS}$, we
find that the hollow and solid circles plotted in Fig. 1d denote the Weyl
points with opposite topological charges, i.e., $+1$ and $-1$, respectively
(see Supplementary Information Sec. II and Fig. S1 [26]). In addition, we
derive the low-energy effective Hamiltonian near the degeneracy points, and
obtain a linear crossing in the vicinity of the Weyl points (see Supplementary
Information Sec. II [26]). The computation of Berry curvatures are plotted in
Figs. S1c and S1d, which indeed demonstrates that the Weyl points manifest as
singularities (source and drain), a close analog to the magnetic monopole in
momentum space.

For a system with the rotational symmetry (it is $C_{3}$ in our model), the
bulk polarization is the appropriate invariant to characterize the topological
features. For the $n$th band, the bulk polarization as a function of $k_{z}$
is written as

$2\pi p_{n}(k_{z})=\text{arg}\theta_{n}(\mathbf{k})\ (\text{mod}~{}2\pi),$ (5)

where $\mathbf{k}=(4\pi/3,0,\pm k_{z})$, and $\theta_{n}({\bf
k})=u^{{\dagger}}_{n}({\bf k})U_{{\bf k}}u_{n}({\bf k})$ with $u_{n}({\bf k})$
the $n$th eigenvector and the $U$-matrix

$U_{\bf k}=\left(\begin{matrix}1&0&0&0&0\\\
0&e^{-i\mathbf{k}\cdot{\mathbf{a}_{2}}}&0&0&0\\\ 0&0&0&0&1\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ \end{matrix}\right).$ (6)

Here we are particularly interested in the 1st (or 5th) band, because the Weyl
points only appear in the intersecting between the first and second energy
bands (or between the fourth and fifth energy bands). As shown in Fig. 1e,
$p_{1}$ takes $1/3$ for $|k_{z}|<|k_{zw}|$, and 0 for $|k_{z}|>|k_{zw}|$. The
topological phase transition occurs at $k_{z}=\pm k_{zw}$. A non-vanishing
$p_{1}$ indicates the very presence of the higher-order topologial edge
states. The present model unambiguously demonstrates the bulk-hinge
correspondence and manifests itself as an ideal SHOWS (see Fig. S2c in
Supplementary Information Sec. III [26]). It is noted that a pair of Weyl
points emerge with opposite wave vectors (see Fig. 1c) because of the
inversion-symmetry breaking in our model. It is worth mentioning that the
present model also allows a 3D square-root HOTI phase (see Figs. S2d-S2f in
Supplementary Information Sec. III [26]). In what follows, we construct the
tight-binding SHOWS model in 3D stacked HK _LC_ circuits.
Circuit realization of SHOWS
We consider a stacked 10-layer HK circuit with $\mathcal{N}=2860$ nodes, as
depicted in Fig. 2a. The circuit dynamics at frequency $\omega$ obeys
Kirchhoff’s law $I_{a}(\omega)=\sum_{b}J_{ab}(\omega)V_{b}(\omega)$, with
$I_{a}$ the external current flowing into node $a$, $V_{b}$ the voltage of
node $b$, and $J_{ab}$($\omega$) being the circuit Laplacian

$J(\omega)=\left(\begin{array}[]{ccccccc}J_{0B}&-J_{B}&0&0&0&0&\ldots\\\
-J_{B}&J_{0C}&-J_{A}&0&0&0&\ldots\\\ 0&-J_{A}&J_{0A}&-J_{A}&-J_{A}&0&\ldots\\\
0&0&-J_{A}&J_{0C}&0&-J_{B}&\ldots\\\ 0&0&-J_{A}&0&J_{0C}&0&\ldots\\\
0&0&0&-J_{B}&0&J_{0B}&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{array}\right)_{\mathcal{N}\times\mathcal{N}},$ (7)

with $J_{0A}=3i\omega C_{A}+1/(i\omega L_{A})$, $J_{0B}=3i\omega
C_{B}+1/(i\omega L_{B})$, $J_{0C}=i\omega(C_{A}+C_{B})+1/(i\omega L_{C})$,
$J_{A}=i\omega C_{A}$, and $J_{B}=i\omega(C_{B}+2C_{Z})$. Under the resonance
condition
$\omega_{0}=1/\sqrt{3C_{A}L_{A}}=1/\sqrt{(3C_{B}+6C_{Z})L_{B}}$=$1/\sqrt{(C_{A}+C_{B}+2C_{Z})L_{C}}$,
the circuit Laplacian (7) exactly recovers the tight-binding Hamiltonian by
the following one-to-one correspondence:
$-\omega_{0}C_{A}\leftrightarrow{t_{a}}$,
$-\omega_{0}C_{B}\leftrightarrow{t_{b}}$, and
$-\omega_{0}C_{Z}\leftrightarrow{t_{z}}$. To explore the square-root
topological semimetal phase, we set $C_{A}=C_{B}/2=0.5$ nF, $C_{Z}=0.5$ nF and
$L_{A}=30$ $\mu$H, $L_{B}=7.5$ $\mu$H, and $L_{C}=18$ $\mu$H in the following
calculations, if not stated otherwise.

Fig. 3: a Theoretical impedance versus the driving frequency in a disordered
circuit. b Measured impedance as a function of the frequency. Calculated c and
measured d impedance distribution of hinge state over the system. Hinge-state
voltage distribution in theory e and experiment f.

To facilitate the detection of the hinge states through a direct two-point
impedance measurement [28], we connect a grounded inductor $L_{G}=22$ $\mu$H
to all nodes to move the hinge modes to the zero admittance without modifying
their wave functions [6]. By measuring the impedance, one can precisely
characterize the wave function of the zero-energy hinge states in the circuit
[28, 6]. Figure 2b exhibits the corresponding admittance spectrum, where the
red, blue, and black dots represent the hinge, surface, and bulk states,
respectively. It is obvious to see the three-fold degeneracy of the in-gap
hinge states. The spatial distributions of each mode are plotted in Figs.
2c-2e, from which one can straightforwardly distinguish them.

Fig. 4: a The projected admittances along the $k_{z}$ direction. b Hinge state
dispersion. The red dots and the colour map in a and b represent the theory
and experimental hinge spectrum, respectively. c The numerical “Fermi arc” of
the surface states at $j_{n}=0.004082$ $\Omega^{-1}$, which connects the
projections of the Weyl points (the hollow and solid dots). d “Fermi arc” of
the surface states at 860 kHz. The colour map and the white circles represent
the experimental and theoretical results, respectively.

The photograph of 3D _LC_ electric circuits fabricated on a printed circuit
board is displayed in Fig. 5 in Methods. We choose electric elements
$C_{A}=C_{B}/2=0.5$ nF, $C_{Z}=0.5$ nF and $L_{A}=33$ $\mu$H, $L_{B}=7.5$
$\mu$H, $L_{C}=18$ $\mu$H and $L_{G}=22$ $\mu$H, the same as those for
theoretical computations above, but with a practical $2\%$ tolerance. The
resonant frequency is then $f_{c}=1/(2\pi\sqrt{3C_{A}L_{A}})=755$ kHz. We
first measure the impedance between three representative nodes and the ground
as a function of the exciting frequency with the impedance analyzer (Keysight
E4990A). Experimental results are shown in Fig. 3b, which well agree with
theoretical calculations plotted in Fig. 3a. Here we select the 1432th,
1564th, and 1711th nodes to characterize the properties of the hinge, surface,
and bulk states, respectively. We then measure the spatial distribution of the
impedance and voltage over the circuit (see Figs. 3d and 3f), which compare
reasonably well with the theoretical results plotted in Figs. 3c and 3e.

To characterize the hinge states more carefully, we project the dispersion to
the $k_{z}$ axis, as shown by the color map in Fig. 4a. In numerical
simulations, one can conveniently take different $j_{n}$ and analyze the
spectrum subsequently, but we cannot set the specific value of $j_{n}$ in
circuit experiments. Fortunately, by mapping Kirchhoff’s law to the
Schrödinger equation in circuit [27, 29], we obtain the frequency dispersion
(see Supplementary Information Sec. V [26]) that significantly facilitates our
experimental measurements. Experimentally, we impose a voltage source in the
middle of one hinge of the circuits, and scan the voltage distribution along
the hinge. Specifically, we input a signal $v_{s}(t)$ = 5sin$(\omega t)$ V at
a hinge node with the arbitrary function generator (GW AFG-3022), and then
collect the voltage $v(\omega,z)$ with frequency $f=\omega/(2\pi)$ ranging
from 500 kHz to 1600 kHz by using the oscilloscope (Keysight MSOX3024A). We
perform the Fourier transformation on the $v(\omega,z)$ and obtain the
projected dispersion along the $k_{z}$ direction, shown by the color map in
Fig. 4b. It can be seen that the hinge states connecting two Weyl points at a
resonant frequency around 755 kHz, which perfectly agrees with the simulation
results marked by the solid red circles.

Furthermore, it is known that the “Fermi arc” surface state is an unique
feature of Weyl semimetals. Figure 4c shows the “Fermi arc” surface dispersion
at $j_{n}=0.004082$ $\Omega^{-1}$. Figure 4d shows the “Fermi arc” surface
dispersion at $f=860$ kHz. The colour map represents the measured data and the
white circles denote the simulated equal-admittance contour, whereas the
hollow and solid dots denote the projections of Weyl points with opposite
topological charges $+1$ and $-1$, respectively. Our experiment therefore
unambiguously supports the bulk-hinge correspondence and identifies the
emergence of SHOWS.
Conclusions
To summarize, we proposed a TB model of the SHOWS and constructed it in 3D
double-helix stacked _LC_ circuits. Through the impedance and voltage
measurements, we directly observed both the 1D prismatic states and the 2D
“Fermi arc” surface states connecting the projected Weyl points, the
fingerprint of SHOWS. Comparing with the normal Weyl semimetal [17], the SHOWS
supports robust hinge states, besides the arc surface states. The emergence of
Weyl pairs in SHOWS with both positive and negative energies marks its
difference from the conventional higher-order Weyl semimetals [21, 23, 25].
One of the parent sublattices, i.e., the honeycomb lattice, originally does
not support any hinge states or flat-band states. The square-root operator,
however, makes it inherit these exotic states from the other parent
sublattice. Our results pave the way to realizing the square-root higher-order
topological states, and may inspire the exploration in other solid-state
systems, such as cold atoms, photonic crystals, and elastic lattices.

Methods
PCB image in experiments and circuit Laplacians.

Fig. 5: Side a and top b view of the printed circuit board in experiment.

The circuit dynamics at frequency $\omega$ obeys Kirchhoff’s law
$I_{a}(\omega)=\sum_{b}J_{ab}(\omega)V_{b}(\omega)$, with $I_{a}$ the external
current flowing into node $a$, $V_{b}$ the voltage of node $b$, and
$J_{ab}$($\omega$) being the circuit Laplacian

$J_{ab}(\omega)=i\mathcal{H}_{ab}(\omega)=i\omega\left[-C_{ab}+\delta_{ab}\left(\sum_{n}C_{an}-\frac{1}{\omega^{2}L_{a}}\right)\right],$
(8)

where $C_{ab}$ is the capacitance between $a$ and $b$ nodes, $L_{a}$ is the
grounding inductance of node $a$, and the sum is taken over all nearest-
neighboring nodes. For a finite circuit, the Laplacian of the circuit can be
written as Eq. (7) in the main text. Considering the resonance condition
$\omega=\omega_{0}$, one can obtain all eigenvalues $j_{n}$ (admittances) and
eigenfunctions $\psi_{n}$ ($n=1,2,...,\mathcal{N}$). We set
$C_{A}=C_{B}/2=0.5$ nF, $C_{Z}=0.5$ nF, $L_{A}=30$ $\mu$H, $L_{B}=7.5$ $\mu$H,
and $L_{C}=18$ $\mu$H in the following calculations. In the calculation of
Fig. 2 in the main text, we consider the ideal situation that all inductors
and capacitors have no loss and disorder. Considering the practical loss and
tolerance of capacitors and inductors, we introduced $2\%$ disorder to each
capacitor and inductor in theoretical calculations thereafter. In experiments,
we stacked 10 identical 2D printed circuit boards (PCBs) along the $z$
direction, as shown in Fig. 5a. Figure 5b shows the top view of the PCB with
the inset zooming in the design details of the electrical circuit.

Data availability
The data that support the plots within this paper and other findings of this
study are available from the corresponding author on reasonable request.

References

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Acknowledgement
We acknowledge Z.-X. Li for useful discussions. This work was supported by the
National Natural Science Foundation of China (Grants No. 12074057, No.
11704060, and No. 11604041).

Author contributions
P.Y. conceived the idea and contributed to the project design. L.S. and H.Y.
designed the circuits and performed the measurements. L.S. developed the
theory and wrote the manuscript. All authors discussed the results and revised
the manuscript.

Competing interests
The authors declare no competing interests.

Correspondence and requests for materials should be addressed to P.Y. by
<EMAIL_ADDRESS>