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Topolectrical-circuitrealizationoftopologicalcornermodes
Imhof,Stefan;Berger,Christian;Bayer,Florian;Brehm,Johannes;Molenkamp,LaurensW;
Kiessling,Tobias;Schindler,Frank;Lee,ChingHua;Greiter,Martin;Neupert,Titus;Thomale,
Ronny
Abstract:Quantizedelectricquadrupoleinsulatorshaverecentlybeenproposedasnovelquantumstates
ofmatterintwospatialdimensions.Gappedotherwise,theycanfeaturezero-dimensionaltopological
cornermid-gapstatesprotectedbythebulkspectralgap,reectionsymmetriesandaspectralsymmetry.
Hereweintroduceatopolectricalcircuitdesignforrealizingsuchcornermodesexperimentallyandreport
measurementsinwhichthemodesappearastopologicalboundaryresonancesinthecornerimpedance
proleofthecircuit.Whereasthequantizedbulkquadrupolemomentofan electroniccrystaldoes
nothaveadirectanalogueintheclassicaltopolectrical-circuitframework,thecornermodesinheritthe
identicalformfromthequantumcase.Duetotheexibilityandtunabilityofelectricalcircuits,they
areanidealplatformforstudyingthereectionsymmetry-protectedcharacterofcornermodesindetail.
Ourworkthereforeestablishesaninstancewheretopolectricalcircuitryisemployedtobridgethegap
betweenquantumtheoreticalmodellingandtheexperimentalrealizationoftopologicalbandstructures.
DOI:https://doi.org/10.1038/s41567-018-0246-1
PostedattheZurichOpenRepositoryandArchive,UniversityofZurich
ZORAURL:https://doi.org/10.5167/uzh-158542
JournalArticle
AcceptedVersion
Originallypublishedat:
Imhof,Stefan;Berger,Christian;Bayer,Florian;Brehm,Johannes;Molenkamp,LaurensW;Kiessling,
Tobias; Schindler,Frank; Lee,ChingHua; Greiter,Martin;Neupert,Titus; Thomale,Ronny(2018).
Topolectrical-circuitrealizationoftopologicalcornermodes.NaturePhysics,14(9):925-929.
DOI:https://doi.org/10.1038/s41567-018-0246-1
Topolectrical circuit realization of topological corner modes
Stefan Imhof,
1
Christian Berger,
1
Florian Bayer,
1
Johannes Brehm,
1
Laurens Molenkamp,
1
Tobias Kiessling,
1
Frank Schindler,
2
Ching Hua Lee,
3
Martin Greiter,
4
Titus Neupert,
2
and Ronny Thomale
4
1
Experimentelle Physik 3, Physikalisches Institut, University of W
¨
urzburg, Am Hubland, D-97074 W
¨
urzburg, Germany
2
Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
3
Institute of High Performance Computing, A*STAR, Singapore, 138632
4
Institute for Theoretical Physics and Astrophysics, University of W
¨
urzburg, Am Hubland, D-97074 W
¨
urzburg, Germany
(Dated: August 15, 2017)
Quantized electric quadrupole insulators have been recently proposed as unprecedented quantum states of
matter in two spatial dimensions. Gapped otherwise, they may feature zero-dimensional topological corner
midgap states protected by the bulk spectral gap, reflection symmetries, and a spectral symmetry. We develop
and measure a topolectrical circuit design for such corner modes which manifest themselves as topological
boundary resonances in the corner impedance profile of the circuit. While the quantized bulk quadrupole mo-
ment of the electronic crystal does not have a direct analogue in the classical topolectrical circuit framework,
the corner modes inherit the identical form from the quantum case and, due to the accessibility and tunability of
electrical circuits, lend themselves to a detailed study of their reflection symmetry-protected character. Our work
therefore establishes an instance where topolectrical circuitry is employed to bridge the gap between quantum
theoretical modeling and the experimental realization of topological band structures.
Introduction — The Berry phase provides a powerful lan-
guage to describe the topological character of band structures
and single-particle systems
1,2
. Manifestly, it allows to treat
fermionic and bosonic quantum systems on the same footing.
Furthermore, the Berry phase concept is not tied to Hilbert
space, but applies to the connectivity of any given coordinate
space, and as such accounts for classical degrees of freedom
as well
3
. It is thus intuitive that, with the discovery of various
topological quantum states of matter such as quantum Hall
4
and quantum spin Hall effect
5
, classical systems with simi-
lar phenomenology could also be identified. This was initi-
ated in the context of photonics
6
, and subsequently transferred
to other fields such as mechanics
7
, acoustics
8
, and electron-
ics
9
. Even though spectra and eigenstates of the single parti-
cle problem, including edge modes, might look similar or even
identical, it is the fundamental degrees of freedom which pose
the central distinction between quantum systems and their de-
signed classical analogues. First, quantization phenomena de-
riving from topological invariants usually necessitate the non-
commutativity of phase space and as such are often reserved
to quantum systems. Second, internal symmetries pivotal to
the protection of a topological phase might not carry over to
classical systems as the degrees of freedom are changed. For
instance, this applies to time-reversal symmetry T as the pro-
tecting symmetry of the quantum spin Hall effect, where the
half integer spin of electrons implies Kramer’s degeneracy due
to T
2
= −1 in the quantum case, while it does not in the clas-
sical case T
2
= 1. Whereas the classical counterpropagat-
ing edge modes might still be detectable, there is no particu-
lar topological protection left, rendering the classical system
much more vulnerable to perturbations
9,10
.
From this perspective, at least two directions appear as
most promising to develop classical topological band struc-
ture models that are universally stable beyond fine-tuning.
The first is the realization of classical analogues to topolog-
ical semimetals
11–16
, where the extensive edge mode degen-
eracy suggests unambiguous persistent spectral edge features
also in the presence of small perturbations. The second is
to focus on topologically insulating quantum electronic states
where either no protecting symmetries are needed such as for
the quantum Hall effect
6
, or where the protecting symmetries
obey the same algebraic relations in the classical and quantum
mechanical case.
Electric quadrupole insulators
17
fall in the latter category.
While the quantum case is most suitably constructed from the
viewpoint of quantized multipole moments of an electronic
crystal, the complementary protecting symmetry perspective
is most intuitive for the classical system design. The sym-
metry group that protects the quantization of the quadrupole
moment includes two non-commuting reflection symmetries
M
x
and M
y
as well as a C
4
rotation symmetry. In partic-
ular, they obey M
2
x,y
= 1, and as such directly carry over
to the classical degrees of freedom. In analogy to the rela-
tion between the quantization of bulk dipole moment (which
is quantized to half-integer values by inversion symmetry) and
the appearance of protected end states in the topological Su-
Schrieffer-Heeger model, an additional spectral symmetry, the
chiral symmetry, is needed to pin the topological boundary
modes in the middle of the bulk energy gap. All these sym-
metries are realized in the microscopic model given in Ref. 17.
Hence, the only task is to implement the hopping model given
by a four site unit cell and real, but sign-changing hybridiza-
tion elements. Due to recent progress in implementing waveg-
uide elements that invert the sign of hybridization
18
, the com-
plexity of this model could recently be captured by a photonic
cavity lattice structure
19
. We turn to topolectrical circuits to
realize the quadrupole insulators in a classical environment.
Linear circuit theory and topology — We consider non-
dissipative linear electric circuits, i.e., circuits made of ca-
pacitors and inductors. Labeling the nodes of a circuit by
a = 1, 2, ···, the response of the circuit at frequency ω is
given by Kirchhoff’s law
I
a
(ω) =
X
b=1,2,···
J
ab
(ω) V
b
(ω) (1)
that relates the voltages V
a
to the currents I
a
via the grounded
arXiv:1708.03647v1 [cond-mat.mes-hall] 11 Aug 2017
2
FIG. 1. Electrical circuit exhibiting a topological corner state with nodes of the circuit indicated by black dots. a) Unit cell of the circuit.
Blue and black circuit elements correspond to weak and strong bonds in a tight-binding or mechanical analogue of the circuit. Red circuit
elements connect to the ground. All capacitor-inductor pairs have the same resonance frequency ω
0
= 1/
√
L
1
C
1
= 1/
√
L
2
C
2
= 1/
p
L
g
1
C
g
1
.
b) Layout of the full circuit which has been realized experimentally. The corners (i) and (iii) are invariant under the mirror symmetry that
leaves the dashed green line invariant. They are compatible with the bulk unit cell choices (I) and (II), respectively, which correspond to an
interchange of strong and weak bonds. As a consequence we expect a topological bound state at corner (i) but not at corner (iii). c) Unit cell
of the experimentally realized circuit.
circuit Laplacian
J
ab
(ω) = iω C
ab
−
i
ω
W
ab
. (2)
Here, the off-diagonal components of the matrix C contain
the capacity C
ab
between nodes a 6= b, while its diagonal
component is given by the total node capacitance
C
aa
= −C
a0
−
X
b=1,2,···
C
ab
(3)
including the capacitance C
a0
between node a and the ground.
Similarly, the off-diagonal components of the matrix W con-
tain the inverse inductivity W
ab
= L
−1
ab
between nodes a 6= b,
while its diagonal components are given by the total node in-
ductivity
W
aa
= −L
−1
a0
−
X
b=1,2,···
L
−1
ab
(4)
including the inductivity L
a0
between node a and the ground.
At fixed frequency ω, J
ab
(ω) determines the linear re-
sponse of the circuit in that the impedance Z
ab
between two
nodes a and b is given by
Z
ab
(ω) = G
aa
(ω) + G
bb
(ω) − G
ab
(ω) − G
ba
(ω), (5)
where G(ω) = J
−1
(ω) is the circuit Green’s function. The
impedance is thus dominated by the smallest eigenvalues
j
n
(ω) of J(ω) at this given frequency, provided that the sites
a and b are in the support of the corresponding eigenfunctions.
In turn, frequencies ω for which an exact zero eigenvalue
j
n
(ω) = 0 exists correspond to eigenmodes of the circuit.
They are determined by the equations of motion satisfied by
the electric potential φ
a
(t) at node a
X
b=1,2,···
C
ab
d
2
dt
2
φ
b
(t) +
X
b=1,2,···
W
ab
φ
b
(t) = 0. (6)
The spectrum ω
2
of eigenmodes of the circuit is thus given by
the spectrum of the dynamical matrix
D = C
−1/2
W C
−1/2
, (7)
with matrix multiplication implied.
We now explain why topological properties can be defined
for the matrices J(ω) and D that describe the physics of the
circuit. In order to define topological properties of a physi-
cal system, the notions of locality and adiabaticity (enabled
by spectral gaps) are of central importance. Locality naturally
arises when we consider circuits in which the nodes a are ar-
ranged in a (in the case at hand two-dimensional) lattice. This
3
also allows to define spatial symmetry transformations. Adia-
baticity in turn follows from the spectral continuity of J(ω) as
a function of ω, that is, if a specific frequency ω
0
lies in a gap
in the spectrum of D, the spectrum of J(ω
0
) also has a gap
around zero eigenvalues. Furthermore, a spectrally isolated
eigenvalue (which may be a topological bound state) of D at
frequency ω
0
is in correspondence with a spectrally isolated
zero mode of J(ω
0
).
Due to these relations between J(ω) and D, protected
boundary modes of a circuit can arise from the topological
properties of either matrix. In this work, we choose to build a
two-dimensional circuit for which the topology of J(ω
0
) at a
specific frequency ω
0
protects corner modes. The topological
protection of spectrally isolated zero modes always requires a
spectral (chiral or particle-hole) symmetry that relates eigen-
values of equal magnitude and opposite sign. Spectrally and
locally isolated eigenstates of this symmetry, if present, are
protected in that they are pinned to the eigenvalue zero. As an
eigenstate of J(ω), such a state naturally dominates the linear
repose of the circuit.
Circuit with corner states —To realize a quadrupole in-
sulator with topologically protected corner states, the system
should have two anticommuting mirror symmetries, as well as
a
ˆ
C
4
rotation symmetry in the bulk. The fundamental mirror
symmetries in classical systems commute. To build a classi-
cal analogue of a electric quadrupole insulator, we thus devise
a circuit that has an emergent pair of anticommuting mirror
symmetries
ˆ
M
x
and
ˆ
M
y
for modes near a specific frequency
ω
0
. This means that J(ω
0
) commutes exactly with
ˆ
M
x
and
ˆ
M
y
and the eigenspaces of D are approximately invariant un-
der
ˆ
M
x
and
ˆ
M
y
for frequencies near ω
0
.
We first discuss the bulk properties of a periodically re-
peating circuit unit cell, depicted in Fig. 1, before consider-
ing boundary modes. The circuit unit cell contains four sites
denoted by pairs (i, j) ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}. We
use two pairs of capacitors and inductors (C
1
,L
1
) and (C
2
,L
2
)
which have the same resonance frequency ω
0
= 1/
√
L
1
C
1
=
1/
√
L
2
C
2
to couple these sites. The latter equality is au-
tomatically satisfied if we set C
2
= λC
1
, L
2
= L
1
/λ
for some real positive parameter λ. Sites 1 and 4 are con-
nected to the ground via an LC circuit with C
g
1
= 2C
1
and
L
g
1
= L
1
/2 such that it has the same resonance frequency ω
0
.
Sites 2 and 3 are connected to the ground via an inductivity
L
g
2
= L
1
/[2(1 + λ)]. In this setup, the circuit is parametrized
by the parameters ω
0
and λ.
We now describe the circuit with periodic boundary con-
ditions in momentum space. The Fourier components of the
matrix J
λ
(ω), denoted by
˜
J
λ
(ω, k), are 4 × 4 matrices that
satisfy
M
x
˜
J
λ
(ω
0
, k
x
, k
y
)M
−1
x
=
˜
J
λ
(ω
0
, −k
x
, k
y
),
M
y
˜
J
λ
(ω
0
, k
x
, k
y
)M
−1
y
=
˜
J
λ
(ω
0
, k
x
, −k
y
),
C
4
˜
J
λ
(ω
0
, k
x
, k
y
)C
−1
4
=
˜
J
λ
(ω
0
, k
y
, −k
x
),
(8)
where M
x
= σ
1
τ
3
, M
y
= σ
1
τ
1
, and 2C
4
= (σ
1
+ iσ
2
)τ
0
+
(σ
1
− iσ
2
)(iτ
2
) are the representations of the symmetries sat-
isfying M
x
M
y
= −M
y
M
x
and C
4
M
x
C
−1
4
= M
y
. Here, σ
µ
and τ
µ
, µ = 0, 1, 2, 3 are the 2 × 2 identity matrix and the
three Pauli matrices acting on the i and j sublattice index, re-
spectively. Note that the circuit is then also invariant under
the combined symmetries
ˆ
M
x¯y
= C
4
M
x
and
ˆ
M
xy
= C
4
M
y
that map (x, y) → (−y, −x) and (x, y) → (y, x) , respec-
tively. In addition,
˜
J
λ
(ω
0
, k) has a chiral symmetry C = σ
3
τ
0
,
which by C
˜
J
λ
(ω
0
, k)C
−1
= −
˜
J
λ
(ω
0
, k) implies a spectral
symmetry. Up to an overall factor of i, the circuit Laplacian
˜
J
λ
(ω
0
, k) takes exactly the same form as the Bloch Hamilto-
nian matrix of the quadrupole insulator introduced in Ref. 17
(see Methods section). For λ 6= 1 the spectrum of
˜
J(ω
0
, k) is
gapped, and the gapless point λ = 1 corresponds to a topolog-
ical phase transition between a quadrupole circuit for λ > 1
and a trivial circuit for λ < 1.
We now turn to a circuit with open boundary conditions to
realize topologically protected corner modes. In general, two
criteria must be met to realize a topological bulk-boundary
correspondence. First, the symmetries which protect the topo-
logical character may not be broken by the boundary. Second,
the system termination must be compatible with the choice
of bulk unit cell for which a topological invariant has been
defined, i.e., the boundary should not cut through unit cells.
We demonstrate all of these properties on a single circuit by
choosing different boundary terminations as follows.
In order for the open system to obey the chiral symmetry C,
the diagonal elements of J(ω) need to vanish at ω
0
. This holds
for all bulk sites by the construction of the model. Imposing
this symmetry also for edge and corner sites in an open ge-
ometry fixes the circuit elements (capacitor and or inductor)
that connect each site to the ground. (See the supplemental
material for the specific grounding that was used for the open
circuit.)
With this condition imposed on the boundary sites, we ter-
minate the upper left edge of the circuit in a way compatible
with the choice of bulk unit cell denoted as (I) in Fig. 1 c). The
lower left circuit termination is chosen to be compatible with
the unit cell denoted as (II) in Fig. 1 c). This edge termination
preserves the mirror symmetry
ˆ
M
x¯y
= C
4
M
x
and breaks all
other spatial symmetries mentioned above. Topological cor-
ner modes could thus potentially be protected at the upper left
and the lower right corner, which are invariant under
ˆ
M
x¯y
, but
not at the other two corners. However, the bulk circuit Lapla-
cians which correspond to the two choices of unit cell (I) and
(II) satisfy
˜
J
(II)
λ
(ω
0
, k) = λ
˜
J
(I)
1/λ
(ω
0
, k) for an appropriate la-
beling of unit cell sites. Recalling that the topological phase
transition occurs at λ = 1, this implies that when
˜
J
(I)
(ω
0
, k)
is in a topological phase,
˜
J
(II)
(ω
0
, k) is trivial and vice versa.
As a result, our choice of boundary termination renders one
corner topological (the upper left one for λ > 1) and the op-
posite corner trivial.
We thus expect that for λ > 1 and at eigenfrequency ω
0
the
circuit depicted in Fig. 1 c) supports a localized topological
corner state at the upper left corner, and none at the lower right
corner. We further note that the corner mode should be an
exact eigenstate of the
ˆ
M
x¯y
symmetry. We will now present
impedance measurements that support this expectation.
Experimental results — For the experimental realization
4
FIG. 2. Comparison of experimental and theoretical results for the circuit spectrum and corner mode. (a) Theoretical spectrum of the circuit
Laplacian J(ω) as a function of the driving frequency. All frequency scales are normalized to the resonance frequency ω
0
. An isolated mode
crossing the gap, which corresponds to a zero energy eigenvalue of J(ω) at ω = ω
0
is clearly visible. It corresponds to the topological corner
mode. The calculation includes a random disorder of 1% for all capacitors and 2% for all inductors. (b) Theoretical weight distribution of the
eigenstate of J(ω
0
) that corresponds to the corner mode, where only the circuit nodes near the corner are shown. (c) Comparison between the
experimental corner mode impedance at ω = ω
0
, measured between nearest neighbor nodes along the horizontal and vertical edges, and along
the diagonal, and the theoretically computed weight of the corner mode eigenstate. Both decay with the decay constant λ = 3.3 set by the
ratio of alternating capacitors/inductors. (d) Frequency scan (normalized with respect to ω
0
) of the impedance between two nearest-neighbor
sites at the corner, at the edge, and in the bulk. Both the experimental and theoretical curves show the corner state resonance isolated in the
gap of bulk and edge states.
of topological corner modes a circuit board with 4.5 × 4.5
unit cells was designed. The line spacing on the board was
chosen large enough such that spurious inductive coupling be-
tween the circuit elements was below our measurement reso-
lution. All impedance measurements were performed with a
HP 4194A Impedance/Gain-Phase Analyzer in a full differen-
tial configuration. In order to achieve a clearly resolvable cor-
ner state resonance on the superimposed resistive background
of the bulk states (i. e., the combined impedance contribution
of our RLC circuit), which is of the order of a few hundreds
of milli-ohm, the values of the circuit elements where chosen
for the resonance frequency to be in the MHz-range. The ratio
λ between the capacitors/inductors was set to 3.3.
Figure 2 compares the experimental data with the theoreti-
cal predictions, finding excellent agreement between the two.
It demonstrates the existence of a spectrally and spatially lo-
calized topological corner state. In Fig. 2 a) the frequency-
dependent spectrum of the circuit Laplacian shows the iso-
lated corner mode and illustrates the connection between a
(bulk and edge) spectral gap of J(ω) at fixed frequency ω and
a gap in the spectrum of the dynamical matrix D, which corre-
sponds to a range of frequencies without zero modes of J(ω).
In Fig. 2 b) and c) the corner mode at ω = ω
0
is mapped
out with single-site resolution. The exponential decay of the
measured impedance is in excellent correspondence with the
theoretical expectation. The experimental demonstration that
the corner mode is indeed a spectrally isolated is contained in
Fig. 2 d).
Physical interpretation of corner modes — Along the x
and y directions, the circuit corresponds to a collection of
connected pairs of linear circuits with alternating capacitors
and inductors, respectively. With the appropriate boundary
conditions discussed previously, electric charge on the capac-
itors forms “dimerized”, isolated oscillators as described in
Ref. 16 and 20. Note that the capacitances alternate between
C
1
and C
2
with C
1
< C
2
. Therefore, by virtue of being an
eigenmode of the circuit Laplacian in terms of potential and
current profile where every second node exhibits no current
and accordingly no potential difference
16
, a fixed amount of
charge Q between each pair of capacitors give rise to a po-
tential difference V
1
> V
2
, since Q = V
1
C
1
= V
2
C
2
. With
appropriate boundary conditions, we can thus infer the exis-
