# ExPECA: An Experimental Platform for Trustworthy Edge Computing Applications

Samie Mostafavi1, Vishnu Narayanan Moothedath2, Stefan Rönngren3, Neelabhro
Roy4,
Gourav Prateek Sharma5, Sangwon Seo6, Manuel Olguín Muñoz7, and James Gross8
https://expeca.proj.kth.se  KTH Royal Institute of Technology, Stockholm,
Sweden
{1ssmos, 2vnmo, 3steron, 4nroy, 5gpsharma, 6sangwon<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

This paper presents ExPECA, an edge computing and wireless communication
research testbed designed to tackle two pressing challenges: comprehensive
end-to-end experimentation, and high levels of experimental reproducibility.
Leveraging Openstack-based Chameleon Infrastructure (CHI) framework for its
proven flexibility and ease of operation, ExPECA is located in a unique,
isolated underground facility, providing a highly controlled setting for
wireless experiments. The testbed is engineered to facilitate integrated
studies of both communication and computation, offering a diverse array of
Software-Defined Radios (SDR) and Commercial Off-The-Shelf (COTS) wireless and
wired links, as well as containerized computational environments. We exemplify
the experimental possibilities of the testbed using OpenRTiST, a latency-
sensitive, bandwidth-intensive application, and analyze its performance.
Lastly, we highlight an array of research domains and experimental setups that
stand to gain from ExPECA’s features, including closed-loop applications and
time-sensitive networking.

###### Index Terms:

Edge computing experimental platform, reproducibility, end-to-end
experimentation, wireless testbed

##  I Introduction

As we move onto the next decade, edge computing applications are poised for
significant growth, fueled by advancements in wireless technologies such as 5G
and Beyond 5G (B5G), as well as breakthroughs in compute-intensive tasks like
model predictive control, multimedia processing, and machine learning
algorithms. In order to achieve trustworthiness and dependability in these
interdisciplinary, distributed systems, involving computer science and
wireless research, it is imperative to develop sophisticated scientific tools.
On one hand, research in areas like wireless networking methods necessitates
close interaction with radio hardware and a high degree of adaptability, as
facilitated by Software Defined Radios (SDRs). Conversely, the computational
aspect requires performance workloads to be adaptable to fluctuating network
conditions, necessitating the creation and efficient repeatable testing of
innovative computing algorithms. End-to-end experimentation is essential for
obtaining a comprehensive understanding of the system as a whole and for
devising innovative approaches to overcome existing bottlenecks. Additionally,
the ability to efficiently replicate these experiments is key to allowing
other researchers to easily validate findings.

In this realm of research, we find a diverse landscape of testbeds that have
emerged in recent years, each distinguished by its size, applicability, and
feature set. Notable ones include the  1. COSMOS  2. POWDER  3. ARA  4. Drexel
Grid SDR  testbeds. COSMOS (Cloud enhanced Open Software defined MObile
wireless testbed for city-Scale deployment) is a testbed spanning over 1
square mile, equipped with an array of resources, including Software-Defined
Radios (SDRs), millimeter-wave (mmWave) equipment, optical fibers, and various
computational nodes for core functionality and application processing [1, 2].
COSMOS integrates with cloud infrastructure, with a primary focus on advancing
cloud computing research. This testbed includes a diverse set of rooftop,
intermediate, and mobile nodes, all centrally controlled.

POWDER (Platform for Open Wireless Data-driven Experimental Research) serves
as an even larger testbed, offering a platform for exploring wireless and
mobile networks with a high level of programmability that extends down to the
waveform level [3]. Covering an area of 15 $\mathrm{km}^{2}$, POWDER comprises
fixed, programmable radio nodes constructed using off-the-shelf SDR
technology. The platform seamlessly integrates with cloud and edge-like
compute capabilities, allowing mobile nodes to tap into the compute resources.

The Drexel Grid SDR testbed features SDRs that connect over-the-air (OTA),
channel emulator, or alternatively a combination of the two, to enable
realistic and reproducible experimentation. Even though the testbed provides
freedom in terms of experiment development, it is intended primarily for SDR-
related research, and does not integrate any core, cloud or edge components.

ARA (The Agricultural and ruRAl communities) represents a wireless research
platform, spanning a rural expanse with a diameter of 60 km in Iowa [4]. ARA’s
mission is to investigate wireless platforms and technologies within the real-
world context of agricultural settings. The testbed incorporates a diverse
array of wireless platforms, implemented through SDRs and programmable
Commercial Off-The-Shelf (COTS) radios along with automated vehicles and
various cameras and other sensors. ARA’s testbed software builds upon the
Chameleon Infrastructure (CHI) software framework from the Chameleon testbed
project [5]. Chameleon primarily serves as a bare-metal reconfigurable
research testbed designed for edge-to-cloud experimentation, adapting the
cloud paradigm for computer science research. Since its public unveiling in
2015, Chameleon has accommodated over 6,000 users across more than 800
projects, gaining valuable experience and refining methods for automating and
managing research infrastructure sites. This accumulated knowledge has been
encapsulated in the CHI-in-a-Box [6] testbed software distribution, which aims
to simplify and optimize the deployment and operation of future associated or
independent Chameleon sites.

Existing wireless research-focused platforms like COSMOS, POWDER, and Drexel
Grid employ their own specialized software solutions, which are not in line
with prevailing cloud-native standards. These testbeds also utilize
virtualization technology based on Virtual Machines (VMs), rather than opting
for lighter and more edge-compatible solutions such as containers. Similar to
the approach taken by the ARA testbed, we found that CHI-in-a-Box presents an
excellent opportunity to develop experimental platforms at minimal cost, while
offering significant flexibility in both cloud and edge-native technologies.
However, when it comes to reproducibility in wireless edge-computing research,
existing large-scale testbeds have limitations that we aim to address by
introducing our testbed, ExPECA. By ‘reproducibility’, we mean the capacity
for researchers to easily replicate experiments and validate results.
Achieving this is particularly challenging for two main reasons: one, edge
computing experiments involve a range of heterogeneous components like servers
and radios, all interacting within a single experiment; and two, the existence
of wireless links introduces variables such as interference and dynamic
propagation conditions that must be controlled for consistent results. Owing
to its unique design choices, ExPECA provides a highly reproducible
environment for end-to-end wireless and edge computing experimentation, making
it one of the first platforms to effectively address this issue. In the
following section, we delve into these design intricacies and their
consequential impacts.

##  II Testbed Design and Architecture

Figure 1: Conceptual architecture of ExPECA for end-to-end edge computing
experiments.

The ExPECA testbed is designed to address two primary challenges pervasive in
edge computing research: 1) end-to-end experimentation, 2) ensuring
repeatability and reproducibility. To enable end-to-end edge computing
experiments that encompass both communication and computation aspects, ExPECA
is engineered with the following primary goals:

  1. 1.

Provision communication components by utilizing a diverse array of wireless
and wired links.

  2. 2.

Executing and overseeing computational elements within containerized
environments running on the testbed infrastructure.

These objectives are realized through an architectural framework depicted in
Figure 1. Central to this architecture is a versatile network fabric that
ensures IP connectivity among different experimental components, which may
include servers located in the cloud or at the edge, as well as wireless
terminals.

Distinctive design choices have been made to realize these objectives,
distinguishing ExPECA from contemporary solutions. We here describe these
choices and their impacts.

###  II-A Location and Physical Environment

Figure 2: A photograph of the underground site (at KTH R1) of the testbed.

The testbed is located in KTH R1 hall, an experimental facility 25 meters
below ground. This isolated location offers a controlled environment with
minimal radio interference, making it ideal for running reproducible wireless
experiments. With dimensions of 12 meters in width, 24 meters in length, and
12 meters in height, the facility is spacious enough to accommodate a variety
of remotely controlled devices, such as drones and cars, providing researchers
with considerable freedom in designing their experiments.

###  II-B Control and Management Software

We adopted CHI-in-a-Box as a baseline testbed implementation with extensions
and contributions from our team. CHI-in-a-Box is a packaging of the CHameleon
Infrastructure (CHI) software framework which is built primarily on top of the
mainstream open-source OpenStack platform. CHI has mature support for edge-to-
cloud provisioning of compute, storage, and network fabrics, and provides end-
to-end experiment control and user services. Specifically, we leverage the
CHI@Edge flavor of the framework, a novel version that supports running
containerized workloads on compute resources. In the following, we describe
the essential elements of CHI. Users have the flexibility to allocate
resources either on-the-fly or through advance reservations, facilitated by
customized versions of OpenStack’s Blazar and Doni services. These services
have been tailored by the ExPECA team to include unique features such as radio
and Kubernetes node reservations. The scope of allocatable resources
encompasses worker nodes, radios, networks, and IP addresses. Once resources
are secured, the integrated Kubernetes orchestration system enables the
deployment of containerized workloads via OpenStack Zun. Users have the option
to utilize pre-configured Docker images supplied by the ExPECA team (as in the
case of SDR RAN) or to upload their own container images. Notably, the ExPECA
team has extensively modified Zun to support functionalities like network
interface and block storage attachments to K8S containers, features not
originally supported in CHI@Edge. Configuration of complex experiment
topologies such as distributed networking experiments is supported through
CHI’s programmable interface and our adopted flavor of the python-chi Python
library. In addition to these, OpenStack Neutron plays a pivotal role in
managing the network architecture. Neutron provides a robust set of networking
capabilities, including virtual networking, that allows for the creation of
isolated networks, subnets, and routers. We connect all testbed components,
including radios, to managed Layer 2 switches controlled by Neutron.

###  II-C Radio Nodes

One of the standout features of ExPECA is the inclusion of wireless terminals
as reservable resources. These radios are mounted at diverse locations in R1,
at three different levels, as shown in Figure 3. They are categorized into two
types:

  1. 1.

SDR Nodes: These can serve as the air interface for any wireless protocol
implemented for USRP E320 SDRs, including OpenAirInterface (OAI) 5G, OAI LTE,
SRS LTE, Mangocomm IEEE802.11b/g, or any GNU radio-based wireless protocol.

  2. 2.

COTS Radios: Includes an Ericsson private 5G system with radio dots and
Advantech routers serving as 4G/5G COTS UE nodes.

We chose all radios, specifically SDRs, to be equipped with IP-networked
interfaces. This allowed us to enroll them as network segments in Openstack
Blazar and Doni with minimal extension efforts. The users can easily integrate
them into any workload container or SDR host container, through the testbed
software. For example, researchers can seamlessly transition from a high-
quality 5G link between locations sdr-01 and sdr-06 to a poor-quality channel
between sdr-09 and sdr-07, enabling them to empirically assess the impact on
application performance. Alternatively, they can switch to a WiFi link between
sdr-01 and sdr-06 for further experimentation.

(a) 2m high

(b) 5m high

(c) 10m high

Figure 3: Low, mid, and high-level radio nodes locations at R1 on three
different levels as of September 2023.

###  II-D Compute Nodes

ExPECA offers compute resources by allowing users run containerized
applications with the most recent version of Openstack Zun that uses K8S
(Kubernetes) as the orchestration layer. K8S is designed to handle a large
number of containers across a variety of environments, making it well-suited
for complex, large-scale deployments. Also, it offers a rich set of APIs and a
pluggable architecture, allowing for greater customization and extensibility.
This was particularly useful for our team when developing new desired features
in Zun. These new features include 1) network attachment, 2) block storage
attachment, 3) resource management i.e. specifying the number of cores and
memory size for the container. Users are able to run their containerized
applications on ExPECA’s Intel x86 servers, enrolled in the K8S cluster, which
we refer to as ’workers’, not only to deploy and orchestrate application
workloads, but also to deploy SDR communication stacks or apply radio node
configurations. With the advent of Open Radio Access Network (O-RAN) and SDRs,
all components of the wireless network stack can run on general-purpose
processors. They can thus be containerized and distributed across multiple
hosts as long as the containers are connected to the SDRs. We provide
reference container images for SDR 5G, SDR LTE, and WiFi provisioning. Lastly,
it is important to note that all worker nodes achieve sub-$\mu$s
synchronization by exchanging PTP messages with a Grandmaster (GM) clock that
itself uses GPS as a reference. Tight time synchronization is essential for
users who require the same timing reference to timestamp packets across
various locations within the testbed.

##  III Experimental Testbed Validation

This section aims to substantiate two key assertions about ExPECA: its
efficiency in facilitating end-to-end experiments and its ability to ensure
experimental reproducibility111Experiments available as Python Jupyter
notebooks at https://github.com/KTH-EXPECA/examples.

To validate the first claim, we employ OpenRTiST [7], an open-source
application that is computationally demanding, bandwidth-intensive, and
latency-sensitive. OpenRTiST uses machine learning filtering to overlay
various artistic styles onto video frames, creating an augmented reality
experience. In this experiment, the OpenRTiST client connects to the server
either at the edge, specifically the testbed servers, or in the cloud via a
server located at Carnegie Mellon University (CMU), in Pittsburgh,
Pennsylvania, through a COTS 5G system and the adv-03 node. In Figure 4, we
depict the corresponding critical performance indicators such as frames per
second (FPS) and round-trip time (RTT) measured using the testbed. In Figure
4(a), we plot the Cumulative Distribution Function (CDF) of FPS, and in Figure
4(a) we plot the Complementary CDF (CCDF) of RTT. As we move from edge to
cloud, RTT significantly increases which results in lower refresh rates and
less responsive video stream. The performance is benchmarked against different
video frame sizes to analyze network and compute capacity.

(a) CDF of FTT.

(b) CCDF of RTT.

Figure 4: Performance analysis of OpenRTiST on COTS 5G system and adv-03 node
using the ExPECA testbed.

For the second claim, we concentrate on wireless network latency, a metric
inherently challenging to model due to its stochastic nature at the packet
level. While latency can be influenced by factors like transmit power and
channel interference, it also depends on wireless protocol settings such as
modulation schemes, bandwidth, and antenna configurations. ExPECA’s isolated
location minimizes variability in channel quality, mitigating potential
interference. In Figure 5, we present latency measurements across different
radio links over several days, demonstrating the reproducibility level in the
case of COTS 5G wireless links.

Figure 5: Twelve rounds of independent end-to-end latency measurements over
the 5G COTS system between dot-02 and adv-03 node, each based on 160k samples.

The close similarity between these CCDFs confirms the reproducibility of
wireless latency measurements achieved through ExPECA.

##  IV Supported Experimentation

Figure 6: Illustration: Testbed experiment scenarios.

This section delves into the features of the testbed that users can customize
to meet their unique experimental goals. These features can be broadly
categorized into two types: (1) configurations related to workload and (2)
configurations related to the network.

In terms of workload-related configurations, ExPECA offers researchers a range
of options to tailor to their experiments. Users can select from various
geographical locations for computational workloads using the cloud interface
and by reserving public IPs. It is possible to integrate GPUs for tasks
requiring accelerated processing, or choose a CPU with a certain number of
cores, and allocate a specific amount of RAM to benchmark workload
performance. The platform supports experiments involving both static (fixed-
location) and mobile (varying-location, e.g., drone with a wireless
connectivity dongle) wireless end nodes.

For network-related configurations, ExPECA provides a diverse array of network
types, such as SDR 5G, SDR LTE, SDR WiFi, and COTS 5G, among others.
Researchers can also customize network topologies and control interference
levels, thanks to the testbed’s isolated location. The platform supports a
wide range of wireless protocols compatible with USRP e320 and allows users to
configure channel conditions. Additionally, the use of Software Defined Radios
(SDRs) offers further flexibility, including control over transmit power and
the ability to implement a variety of wireless protocols.

ExPECA offers the opportunity to explore various experimentation scenarios
involving edge computing, wireless communication, and machine learning
research. Here we explain various scenarios, each shedding light on some of
the interesting aspects of wireless communication and/or edge computing
offering unique insights into the dynamic landscape of edge computing. In
Figure 6, we provide an illustration that helps a reader to visualize these
scenarios.

Below, we provide a few of such use cases in various research areas. While
some of these aim at direct measurements for data-driven modeling and
prediction, others target initial observations for forthcoming theoretical
work or the verification and validation of existing theoretical models.

###  IV-A Optimization of Closed-Loop Applications

With the booming interest and applications surrounding cyber-physical systems,
it is imperative to study Networked Control Systems (NCSs) [8]. NCSs form
crucial control systems where the control loops are closed through a
communication network. Due to the safety-critical nature of the applications
these NCSs encompass, such as robots, UAVs etc., there is a need to benchmark
and understand the bottlenecks/boundaries associated with these applications
[9]. Namely, the latency and reliability boundaries of robotic control are
crucial to understand before they can be deployed in real-world scenarios,
where their potential failure might prove expensive [10].

Another such feedback system where latency is of importance are human-in-the-
loop applications such as augmented reality, wearable cognitive assistants
(WCA) [11, 9], and ambient safety. In the context of criticality, we have
systems of automated fault detection, in which case the acoustic [12] or
motion amplified visual [13, 14] data is processed for vibration analysis to
potentially trigger some maintenance, safety or emergency procedures [12]. The
ExPECA testbed provides developers with the opportunities to experiment with
workloads corresponding to these types of applications in a containerized
manner with various real network architectures. These experiments could either
be for data collection or for verification of existing theoretical models.

###  IV-B Time-sensitive Wireless Networking

Supporting the above-mentioned CPS and HITL applications requires the network
to offer a certain level of end-to-end network latency, often combined with
extremely high reliability, in order to operate correctly and safely. This has
spurred various efforts to integrate wireless networking technologies (e.g.,
5G URLLC) with wired technologies (e.g., TSN) as shown in Figure 7. However,
an efficient wired-wireless integration realization to support time-critical
applications necessitates accurate system characterization. In particular,
latency characterization of wireless segments is crucial as they are subject
to various stochastic and temporal influences, unlike wired links where
latencies are immune to external influences. Furthermore, accurate predictions
of wireless latency are also envisioned to be leveraged in 6G networks to
support deterministic communications [15].

Figure 7: An illustration for the integration of a wireless domain with TSN
domains.

Lately, several data-driven methods for latency prediction have been proposed
[16, 17]. Training and validating these approaches require a vast amount of
datasets comprising latency measurements alongside network parameters (e.g.,
SINR and MCS). Precise time-synchronization of all nodes of the testbed
ensures accurate latency measurements can be collected between any two nodes
of the testbed. These experiments can employ diverse wireless communication
technologies, including COTS 5G, software-defined 5G and IEEE 802.11g Wi-Fi
integrated in ExPECA. Furthermore, the containerized workflow within ExPECA
enables users to perform automated, reproducible and repetitive latency
measurement experiments.

###  IV-C Hierarchical Learning and Inference

Another set of experiments researchers can use the ExPECA testbed for is the
verification of theoretical results corresponding to Hierarchical Learning
(HI) [18, 19]. In these works, authors develop the idea of Hierarchical
Inference, where a compute constrained edge device chooses to perform local
inference or classification of an image sample, or offloading via a wireless
channel to get help of a high power edge server for classification. A small-
sized ML model with limited accuracy is assumed to have been deployed at the
edge device and offloading to the server incurs cost. In essence, these
policies aim to make intelligent decisions regarding task offloading,
prioritizing complex tasks or tasks where accuracy is paramount. Conversely,
they process simpler tasks or tasks where reduced latency is of utmost
importance locally. It is to be noted that in this specific work on
Hierarchical Inference this decision is made in an online manner. Using the
ExPECA testbed, researchers can plan to implement the various algorithms
proposed and verify if the real-world results match simulations and models.

###  IV-D Federated Learning Validation

Federated learning (FL) is emerging as one of the key approaches for training
machine learning models [20]. FL aims to improve over centralized learning in
terms of privacy, total training time, computation overhead, etc. However, FL
can cause large communication overhead because the machine learning model
should be repeatedly exchanged between the central server and the
participating nodes during the training process. In order to analyze and
optimize communication overhead, many works have proposed models for FL
systems operating in wireless environments [21, 22]. Validating these
theoretical models in real systems is required to reveal otherwise unavailable
insights. The ExPECA testbed supports experimentation involving both
centralized learning (CL) and federated learning to evaluate them in terms of
communication/computation overhead and accuracy with various wireless
conditions. For example in case of communication overhead, CL and FL will
consume communication resources when exchanging data (i.e., using uplink to
send input data in CL and uplink/downlink to send machine learning model in
FL). The ExPECA testbed is able to measure the communication resource usage,
and training time (communication/computation time), and capture wireless
parameters at that time. Thanks to improved reproducibility and repeatability,
the ExPECA testbed supplies stable validation for machine learning
architectures.

##  V Acknowledgements

This research has been partially funded by (1) the VINNOVA Competence Center
for Trustworthy Edge Computing Systems and Applications (TECoSA) at KTH Royal
Institute of Technology; and (2) the Swedish Foundation for Strategic Research
(SSF), through grant number ITM17–0246.

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# Valley Hall Effect and Kink States in Topolectrical Circuits

S M Rafi-Ul-Islam<EMAIL_ADDRESS>Department of Electrical and Computer
Engineering, National University of Singapore, Singapore 117583, Republic of
Singapore  Zhuo Bin Siu<EMAIL_ADDRESS>Department of Electrical and
Computer Engineering, National University of Singapore, Singapore 117583,
Republic of Singapore  Haydar Sahin<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, National University of Singapore,
Singapore 117583, Republic of Singapore  Institute of High Performance
Computing, A*STAR, Singapore 138632, Republic of Singapore  Mansoor B.A. Jalil
<EMAIL_ADDRESS>Department of Electrical and Computer Engineering, National
University of Singapore, Singapore 117583, Republic of Singapore

###### Abstract

We investigate the emergence of topological valley Hall and kink states in a
two-dimensional topolectrical (TE) model as a result of broken chiral and
reflection symmetries. The TE system consists of two segments hosting distinct
topological states with opposite signs of the valley Hall index, and separated
by a heterojunction. In the practical circuit, the valley Hall index can be
flipped between the two segments by modulating the onsite potential on the
sublattice nodes of the respective segments. The presence of resistive
coupling, which introduces non-Hermiticity in the system, subsequently leads
to the emergence of gapped and gapless valley and kink states in the
admittance spectra. These topological modes can be detected electrically by
the impedance readouts of the system which can be correlated to its admittance
spectra. Finally, we confirm the robustness of the valley Hall and kink states
via realistic LTspice simulation taking into account the tolerance windows and
parasitic effects inherent in circuit components. Our study demonstrates the
applicability of TE circuit networks as a platform to realize and tune valley-
dependent and kink topological phenomena.

##  I Introduction

The study of topological phases has emerged as a frontrunner topic in
condensed matter physics owing to the unconventional properties of such phases
[1, 2, 3], which possess a non-trivial band topology. The valley is meanwhile
a new degree of freedom that can be found in lattice models with various
symmetries [4, 5, 6]. This additional valley freedom is useful in various
technologically significant applications such as robust electronic transport
[7, 8], energy propagation [9], and information processing [10, 11]. As a
result, the valley degree of freedom has given rise to a completely new branch
of technology named “valleytronics” [12, 13, 14] that has revolutionized many
existing fields such as photonics [15], metamaterials [16], condensed matter
[17], acoustics [18], and next-generation quantum computing [19, 20, 21].
Breaking the inversion symmetry in a lattice model results in a valley-
dependent Hall conductivity and a quantum valley Hall effect [22, 23]. The
inversion symmetry can be broken by inducing alternating mass terms in the
lattice Hamiltonian [24]. A domain wall-type interface is created when two
lattice segments with opposite valley responses are joined together. Robust
valley kink states appear at the interfaces of such heterojunctions [25, 26].
These novel valley kink states are useful in many promising phenomena such as
Klein tunneling [27, 28, 29], anti-Klein tunneling [30], spin-valley locking
[31], and quantum memory [32].

More recently, topological boundary states have been realized in many
Hermitian [33, 34] and non-Hermitian systems [35, 36, 37, 38, 39]. These
topologically nontrivial boundary states are characterized by the topological
index (i.e., Chern number [40] and Hall conductivity[41]) of their gapped bulk
energy bands, and exhibit gapped and gapless states on their boundaries
depending on the model parameters. These topological boundary states are
protected by symmetries such as time reversal [42] or spatial inversion
symmetry [43], and are robust against local perturbations and disorders. These
novel topological boundary phases and valley Hall states have recently been
demonstrated in different platforms such as photonics systems [15, 20, 44],
metamaterials [21, 45, 46, 47], and quantum wells [48]. Although boundary
states and valley kink modes promise to bring dramatic changes to existing
technologies, it is difficult to realize and observe multiple topological
valley and boundary phases in the same lattice model because of difficulties
in the dynamical modulation of the system parameters (e.g., fixed lattice
constants and weak spin-orbit coupling). Additionally, all these platforms
involve experimentally complex sample preparation, which is very vulnerable to
perturbations and impurities.

Lattice arrays comprising electrical components such as inductors and
capacitors known as topolectrical (TE) circuits [49, 50, 51, 52, 30, 53, 54]
have become the frontier experimental testbed in the quest for alternative
platforms to study different topological states. Compared to other platforms,
TE circuits offer better tunability of system parameters such as the
interaction strength and phases. Recently, many exotic and novel features such
as edge states [55, 50], corner states [56], quantum spin Hall states [57,
58], chiral magnetic effects [59, 60], topological photonic states [61], and
nodal ring states [62, 63, 64, 65] have been proposed in electrical circuit
networks [29, 50, 52, 53]. The topological states depend on the connectivity
between the electrical components rather than their relative locations in real
space. The TE circuit models also provide better flexibility in varying the
system parameters coupled with the convenient and accurate readout of system
characteristics.

In this paper, we design and propose a general framework to realize various
topological valley phases and kink states based on the electrical responses in
a two-dimensional TE circuit model. We have explained in detail the
fundamental relationship between the admittance band structure and the
impedance profiles through the circuit Green’s function in our previous works
[49, 51, 29]. By tuning the onsite interaction strength on the different
sublattice nodes, we obtain a transition from gapless to gapped edge states in
the admittance spectra. The valley-dependent Hall conductivity is calculated
using the Kubo formula and verified through the impedance spectrum. We study
the valley kink states that result from cascading two TE segments with
opposite signs of the valley Hall responses together. The opposite signs of
the valley Hall responses can be realized by reversing the sign of the onsite
capacitance, and hence the Laplacian mass term, on both sides of
heterojunction. Both gapless and gapped kink states can be obtained by varying
the relative strengths of the onsite capacitance to the resistive coupling
strength. The topological kink states are localized at the interface of the
heterojunction and can be distinguished by their terminal impedances. Since
such TE circuit models can be implemented in typical breadboards or printed
circuit boards with basic electrical components, our model not only opens new
experimental possibilities and directions for the realization of various
topological valley phases, but also helps in the design of multifunctional
valleytronic devices.

##  II Topoelectrical Valley Circuit Model

Consider the TE circuit model consisting of basic electrical components such
as inductors and capacitors in Fig. 1. An AC current of angular frequency
$\omega$ flows through the circuit, which comprises two different types of
sublattice nodes labeled as the $A$ and $B$ type nodes respectively indicated
as the red and orange circles in the figure. Along the $x$ direction, the two
adjacent sublattice nodes within the same unit cell are connected by an
inductor with an admittance equivalent to that of a capacitance of $-C_{1}$
(i.e., an inductance of $1/(\omega^{2}C_{1})$ ). The adjacent nodes in
neighboring unit cells are connected by a capacitance $C_{1}$ (see Fig. 1a).
Along the $y$ direction, adjacent nodes on different sublattices are connected
by a capacitance $C_{y}$ while nodes on the same sublattice are connected
through alternative combinations of positive and negative resistive elements
$R_{v}=\frac{1}{i\omega r}$, where $r$ is a resistance. Note that the $A-A$
and $B-B$ couplings along the $y$ direction have a relative $\pi$ phase
difference in their resistive couplings within the same unit cell. Such $\pi$
phases in the resistive elements can be obtained using negative resistance
converters (NRCs) (see Fig. 1b), which break the reflection symmetry along the
$y$ direction. Each $A$ and $B$ node is connected to the ground via a common
capacitor $C$ and inductor $L$. Each $A$ and $B$ node is further grounded by
another onsite potential capacitor $C_{g}$ and inductor $-C_{g}$,
respectively. The common grounding inductor ($L$) is used to adjust the offset
of the admittance dispersion to a common value for all nodes [35, 39, 66].

Figure 1: (a) Schematic of the valley TE lattice model that hosts valley-
dependent topological phases. Here, the red and orange circles represent the
$A$ and $B$ sublattice nodes, respectively. The unit cell is delineated by the
dotted box. The two alternating sublattices are connected by a capacitor
$C_{1}$ and inductor of $-C_{1}$ (i.e., an inductance of
$1/(\omega^{2}C_{1})$) along the $x$ direction for the intracell and intercell
connections respectively . Along the $y$ direction, neighboring nodes on
opposite sublattices are connected by a capacitor $C_{y}$ and nodes on the
same sublattice connected through alternative combinations of resistive
elements $\pm R_{v}$. (Note that there is a $\pi$ phase difference in the
resistive coupling between the $A-A$ and $B-B$ connections along the $y$
direction within the same unit cell.) (b) We can make use of the negative
resistance converter to realize the $\pi$ phase difference between the $A-A$
and $B-B$ connections along the $y$ direction. The combination of resistors
$R_{1}$ and $R_{v}$ along with an ideal operational amplifier with supply
voltages $V-$ and $V+$ acts as a negative resistance converter with current
inversion. (c) Grounding mechanism of our valley TE circuit. All nodes are
connected to ground by a common capacitor ($C$) and inductor ($L$).
Furthermore, each $A$ and $B$ node is coupled to the ground with the same
magnitude but opposite signs of the coupling strength by an onsite capacitor
and an inductor, respectively.

##  III Laplacian Phases

The TE circuit can be described by its Laplacian, which is analogous to the
tight binding Hamiltonian in quantum physics [49, 50, 29, 67]. The Laplacian
at the resonant frequency of $\omega_{r}=1/\sqrt{2C_{y}L}$ multiplied by
$i\omega_{r}$, which we shall refer to as the normalized Laplacian for short
subsequently, is given by

$\displaystyle L_{TE}(k_{x},k_{y})=$
$\displaystyle\left(-C_{1}+C_{1}\cos(k_{x})+2C_{y}\cos(k_{y})\right)\sigma_{x}$
(1)
$\displaystyle+C_{1}\sin(k_{x})\sigma_{y}+\left(C_{g}+2R_{v}\sin(k_{y})\right)\sigma_{z},$

where $\sigma=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrices
denoting the A/B sublattice degree of freedom.

The circuit hosts both topologically trivial and non-trivial phases depending
on the relative magnitudes of the circuit parameters $C_{g}$, $R_{v}$, $C_{1}$
and $C_{y}$. We first investigate the transition points between the
topologically trivial and non-trivial phases in the
$(C_{g},R_{v},C_{1},C_{y})$ parameter space at which the eigenvalue spectrum
of the Laplacian in Eq. (1) becomes gapless. This happens when the
coefficients of all the Pauli matrices in Eq. (1) are simultaneously zero for
some real $\vec{k}$ in the Brillouin zone. For the coefficient of $\sigma_{z}$
to be zero, we require $k_{y}=\pm\sin^{-1}(C_{g}/R_{v})$ which has a real
solution when $|R_{v}|\geq|C_{g}|$, while the coefficient of $\sigma_{y}$ is
zero when $k_{x}=0,\pi$. Substituting $k_{x}=\pi$ and
$k_{y}=\cos^{-1}(C_{g}/R_{V})$ into the coefficeint of $\sigma_{x}$ in Eq.
(1), we have, $C_{1}(1-\cos(k_{x}))+2C_{y}\cos(k_{y})=2(-C_{1}\pm
C_{y}\sqrt{1-(C_{g}/R_{v})^{2}})$. This is zero when
$(C_{1}/C_{y})^{2}+(C_{g}/R_{v})^{2}=1$. Thus, the phase transition points
occur at $(C_{1}/C_{y})^{2}+(C_{g}/R_{v})^{2}=1$, and at $|C_{g}|=|R_{v}|$.

Fig. 2a shows the surfaces on which these phase transition points lie in the
$(C_{g},R_{v},C_{1})$ space at $C_{y}=1.5\ \mu\mathrm{F}$. To determine which
side of the phase transition surfaces correspond to the topologically non-
trivial phases, we numerically calculated the Chern numbers of the normalized
Laplacian using the Fukui algorithm [68] that provides a numerical means of
evaluating the Chern number on a discretized lattice. Fig. 2b shows the Chern
numbers at $C_{y}=1.5\ \mu\mathrm{F}$ and $C_{g}=0.1\ \mu\mathrm{F}$. The
topologically non-trivial phases with finite Chern numbers occur at the
intersections of $(C_{1}/C_{y})^{2}<1-(C_{g}/R_{v})^{2}$ and
$|R_{v}|>|C_{g}|$.

Figure 2:  (a) The phase transition surfaces in $(C_{g},R_{v},C_{1})$ space at
a fixed $C_{y}=0.5\ \mu\mathrm{F}$. The green surfaces outlined by the dotted
lines denote the $(C_{1}/C_{y})^{2}+(C_{g}/R_{v})^{2}=1$ surfaces, and the
blue planes the $|R_{v}|=|C_{g}|$ planes. The unfilled orange box denotes the
$C_{g}=0.1\ \mu\mathrm{F}$ plane shown in panel (b). (b) The Chern numbers as
functions of $C_{1}$ and $R_{v}$ at $C_{g}=0.1\ \mu\mathrm{F}$ and $C_{y}=0.5\
\mu\mathrm{F}$. The gray areas denote the topologically trivial phase where
the Chern number is zero. The dark blue lines denote the $|R_{v}|=|C_{g}|$
lines, and the green lines are the projections of
$(C_{1}/C_{y})^{2}+(C_{g}/R_{v})^{2}=1$ onto the $C_{g}=0.1\ \mu\mathrm{F}$
plane. The points (i) to (iv) denote the values of $R_{v}$ and $C_{g}$ at
which the dispersion relations in the finite $x$-width and finite $y$-width
nanoribbon geometries are plotted in panel (c). The
$(C_{g},C_{y},C_{1},R_{v})$ values of these points are (i) (0.1, 0.5, 0.6,
0.025) $\mu\mathrm{F}$, (ii) (0.1, 0.5, 0.8, 0.3) $\mu\mathrm{F}$, (iii) (0.1,
0.5, 0.1, 0.025) $\mu\mathrm{F}$, and (iv) (0.1, 0.5, 0.1, 0.5)
$\mu\mathrm{F}$. (c) Admittance band dispersions of the TE model at the four
$(C_{1},R_{v})$ points denoted in panel (b) in the nanoribbon geometry with
(left) infinite length along the $x$ direction and 10 unit cells along the $y$
direction, and (right) infinite length along the $y$ direction and 10 unit
cells along the $x$ direction. The nanoribbon geometries are schematically
illustrated by the schematics at the top of the figure where the thick black
borders at the edges denote open boundary conditions at the edges while the
borderless edges extend to infinity. The thick lines in the admittance plots
denote the edge states. Note that those edge states associated with (i), (ii)
and (iv) are trivial because they do not cross the band-gap, while that of
(iii) are non-trivial and band-gap crossing.

The latter condition can also be obtained analytically by considering the
linear response of the valley Hall conductivity. Expanding Eq. (1) around
$\vec{k}_{0}=(0,\eta\frac{\pi}{2})$, $\eta=\pm 1$, we obtain

$L^{\eta}_{\mathrm{DP}}(\vec{q})=2\eta
C_{y}q_{y}\sigma_{x}+C_{1}q_{x}\sigma_{y}+(C_{g}+\eta R_{v})\sigma_{z},$ (2)

where $\vec{q}\equiv\vec{k}-\vec{k}_{0}$. One may thus identify the two
$\vec{k}_{0}$ points as the Dirac points (DPs) of massive Dirac fermion
Hamiltonians Eq. (2) associated with the two valleys, so that $\eta$ takes the
meaning of a valley index where $\eta=+1(-1)$ denotes the $K$ and $K^{\prime}$
valley index, respectively. The low-admittance equation reads

$\epsilon^{\eta}_{\pm}=\pm\sqrt{(2C_{y}q_{y})^{2}+(C_{1}q_{x})^{2}+(C_{g}+\eta
R_{v})^{2}},$ (3)

where $\pm$ denotes the particle- and hole-like bands respectively. The
analogous DC Hall conductivity $\sigma_{xy}$ at each valley can be calculated
through the standard Kubo formula [69] as

$\sigma^{\eta}_{xy}=\int\frac{\mathrm{d}q^{2}}{\pi^{2}}\frac{1}{(\epsilon^{\eta}_{+}-\epsilon^{\eta}_{-})^{2}}\mathrm{Im}(M^{\eta+}_{x}(\vec{q})M^{\eta-}_{y}(\vec{q}))$
(4)

where
$M^{\eta\pm}_{x}(\vec{q})=\langle{+},\vec{q},\eta|(\partial_{q_{x}}L^{\eta}_{\mathrm{DP}})|-,k,\eta\rangle$
and
$M_{y}^{\eta\mp}(\vec{q})=\langle{+},\vec{q},\eta|(\partial_{q_{y}}L^{\eta}_{\mathrm{DP}})|-,\vec{q},\eta\rangle$.
Here, $|\pm,\vec{q},\eta\rangle$ is the right eigenvector of Eq. 2, and and
$\langle{\pm},\vec{q},\eta|$ its Hermitian conjugate. The valley-dependent DC
Hall conductivity can be evaluated as

$\sigma_{xy}^{\eta}=-\frac{1}{8\pi^{2}}\mathrm{Sgn}(C_{g}+\eta R_{v}).$ (5)

From Eq. 5, one can see that the Hall conductivity contributions for the two
valleys are unequal. We define the total quantum valley Hall conductivity
($\sigma_{xy}^{\textrm{valley}}\equiv\sigma_{xy}^{\eta=1}-\sigma_{xy}^{\eta=-1}$).
We obtain zero and finite $\sigma_{xy}^{\textrm{valley}}$ for
$|C_{g}/R_{v}|>1$ and $|C_{g}/R_{v}|<1$ respectively. When $|R_{v}|<|C_{g}|$,
the two valleys have the same Hall conductivities. The quantum valley Hall
conductivity is hence zero, and there are no topologically non-trivial edge
states which cross the bulk bandgap. We find, however, that topologically
trivial edge states still exist when boundaries are introduced along certain
directions but these do not cross the bandgap.

Figure 3: Simulated admittance and impedance profiles of the valley TE
circuit. The simulation is performed via the electric circuit simulation
software LTspice. The top rows show the admittance dispersion relations of
$x$-confined nanoribbons with 10 unit cells along the $x$ direction at the
resonant frequency $f=25.250\ \mathrm{Hz}$. For a realistic simulation, the
components are selected from the LTspice component database i.e., $C_{1}=3\
\mu\mathrm{F}$ ($2\times$ Murata GRM033R60G155ME14D), and $C_{y}=1.5\
\mu\mathrm{F}$ (Murata GRM033R60G155ME14D) for (a) edge states with $C_{g}=0\
\mu\mathrm{F}$ and $R_{v}=0$, (b) the non-trivial topological phase
$C_{g}=0.56\ \mu\mathrm{F}$ (KEMET C1206C564K3RACTU) and $R_{v}=2$, and (c)
the trivial valley phase with $C_{g}=2.2\ \mu\mathrm{F}$ (KEMET
C0603C225K9PAC) and $R_{v}=0.5$. For the common grounding inductors, $L=10\
\mu\mathrm{H}$ (Würth Elektronik 744042100) is used. To realize the negative
resistive element $R_{v}$, we performed the simulation with the high precision
operational amplifier LT1056. The bottom rows show the respective spectra of
the impedances measured between the two edges of the nanoribbons.

Fig. 2c shows the dispersion relations at representative points on the
$R_{v}-C_{1}$ plane for nanoribbon geometries of the TE circuit in which the
circuits have infinite length along the $x$ direction and 10 unit cells in the
$y$ direction (left), and in which the circuits have infinite length along the
$y$ direction and 10 unit cells in the $x$ direction (right). (The admittance
dispersion is the TE analogue of the energy dispersion in quantum mechanical
systems.) In general, the dispersion relations for the finite $x$ and finite
$y$ nanoribbons appear markedly different from one another although their
topological character remains unchanged when the finite direction is
exchanged. Points (i) and (iii) on Fig. 2b exemplify the $|R_{v}|<|C_{g}|$
scenario, with $|C_{1}|>|C_{y}|$ at point i and $|C_{1}|<|C_{y}|$ at point
(iii). At both of these points, no edge states exist when the nanoribbon
confinement direction is along the $y$ direction. However, edge states emerge
when the nanoribbon confinement is along the $x$ direction. The edge states
are more prominent at $|k_{y}|<\pi/2$ in the case of $|C_{1}|>|C_{y}|$
corresponding to point (i) whereas the edge states confined to a narrow range
of $k_{y}$ in the vicinity of $|k_{y}|=\pm\pi/2$ in the case of
$|C_{1}|<|C_{y}|$ corresponding to point (iii). Point (iv) corresponds to the
scenario of $|R_{v}|>|C_{g}|$, $(C_{y}/C_{1})^{2}>1-(C_{g}/R_{v})^{2}$
scenario. In contrast to points (i) and (iii) discussed previously, edge
states appear on point (iv) only when the nanoribbon confinement direction is
in the $x$ direction, but not in the $y$ direction. Note that the edge states
corresponding to (i), (iii) and (iv) do not cross the bandgap, thus indicating
their trivial character. Finally, point (ii) exemplifies the topologically
non-trivial phase which exists when $|R_{v}|>|C_{g}|$,
$(C_{y}/C_{1})^{2}<1-(C_{g}/R_{v})^{2}$. Topologically protected edge states
which cross the bulk gap are present for both nanoribbon confinement
directions. In the remainder of this paper, we will focus on the phases
exemplified by points (i) (topologically trivial) and (ii) (topologically non-
trivial) for the case of $|C_{1}|>|C_{y}|$, and without loss of generality set
$C_{1}=2C_{y}$.

To further investigate the effect of $C_{g}$ and $R_{v}$ on the TE model when
$|C_{1}|>|C_{y}|$, we simulated the circuit in LTspice with realistic device
parameters and plot the admittance and impedance spectra as functions of
wavevector $k_{y}$ for open boundary conditions along the $x$ direction (see
Fig. 3a-c). The impedance between any two lattice sites $p$ and $q$ in the TE
network model is given by

$\displaystyle Z_{pq}=$
$\displaystyle(L_{TE})_{pp}^{-1}-(L_{TE})_{pq}^{-1}+(L_{TE})_{qq}^{-1}-(L_{TE})_{qp}^{-1}$
(6) $\displaystyle=$
$\displaystyle\sum_{k=1}^{N}\frac{{|\phi_{kp}-\phi_{kq}|}^{2}}{\epsilon_{k}},$

where $\phi_{ij}$ is value of the $i^{th}$ eigenvector at the $j^{th}$ lattice
point and $\epsilon_{i}$ is the $i^{th}$ non-singular eigenenergy of the
Laplacian matrix. Note that LTspice simulations took into account tolerance
windows and parasitic effects. Even after including such
effects/imperfections, the topological behaviour remains largely unaffected.

In the upper plot in Fig. 3a, we find well defined edge states for zero
$C_{g}$ and $R_{v}$. Moreover, the whole admittance spectra is symmetric about
the zero admittance line. This is due to the chiral symmetry of the Laplacian
$\mathcal{C}L_{TE}(\vec{k})\mathcal{C}^{-1}=-L_{TE}(\vec{k})$ where
$\mathcal{C}=\sigma_{z}$ is the chiral inversion operator. However, the
boundary modes evolution would be different for finite mass term i.e.
($C_{g}+\eta R_{v}$) in Eq. (2). When $|C_{g}|<|R_{v}|$, the gapless edge
modes emerge (see Fig. 3b), where the zero-energy edge states split into two
tilted boundary states that intersect each other. However when the resistive
coupling becomes stronger than the onsite capacitor (i.e., $|C_{g}|>|R_{v}|$
), gapped boundary modes appear, as shown in Fig. 3c. Interestingly, a finite
mass term breaks the symmetry of admittance dispersion about the admittance
$E=0$ line (Fig. 3b,c). The impedance spectra for Fig. 3a–c are shown in the
lower plots of the corresponding panel. The nearly zero-admittance edge states
in Fig. 3a are marked by the very large impedance for $|k_{y}|\lesssim\pi/2$ ,
which agrees with the inversely proportional relation between the eigenvalue
(admittance) and the impedance in Eq. 6. However, the impedance falls
significantly in presence of a mass term in the circuit Laplacian and only
discrete impedance peaks are found when the admittance gap between the two
bands reaches close to zero (see Fig. 3b, c). The gapless and gapped edge
states are indicated by the comparatively large and small impedance peaks
respectively (compare the sharp peak in Fig. 3b with the broader lower peak of
Fig. 3c). The close correspondence between the admittance spectra and
impedance readouts obtained from the LTspice simulation demonstrates the
experimental realization of valley-dependent features and their electrical
characterization under realistic conditions.

##  IV Valley kink states

Figure 4:  Schematic of a TE circuit heterojunction that exhibits kink states.
(a) Valley kink circuit. (b) Schematic of the wiring between each node and the
ground in the two regions. Note that the onsite potential parameter (i.e.,
$C_{g}$) changes sign between the two regions. This guarantees that the Hall
conductivity of each of the valleys switches sign across the heterojunction
interface. All the other parameters are the same in both regions.

In the previous section, we studied the relation between the valley-dependent
topological phases and the emergence of edge states. In this section, we study
the evolution of the boundary states evolution in a heterojunction between two
TE segments in which the Hall conductivities of each valley have opposite
signs in the two segments. More explicitly, we realize a topolectrical valley
kink state at the domain wall-like interface between two TE circuit arrays
with opposite signs of the quantum valley Hall conductivities as shown in Fig.
4. Such heterojunctions can be realized by simply reversing the sign of the
onsite capacitance $C_{g}$ between the two array segments for a fixed
$|R_{v}|<|C_{g}|$. The TE lattice in the left and right sides of the Interface
have the same model parameters except for the onsite potential $C_{g}$, which
is positive and negative on the left and right sides respectively. Such a sign
change in one of the mass parameter terms(i.e., $C_{g}$) across the
heterojuction will induce valley-dependent kink states localized at the
interface. Here we focus on heterojunctions where the interface is parallel to
the $y$ direction.

Figure 5: The plots on the left show the admittance dispersions for TE
circuits with $C_{1}=3\ \mu\mathrm{F}$, $R_{v}=0.75\ \mu\mathrm{F}$,
$C_{y}=1.5\ \mu\mathrm{F}$ for (a) a 10 unit cells-wide $C_{g}=0.5\
\mu\mathrm{F}$ nanoribbon and (b) a 10 unit cells-wide $C_{g}=-0.5\
\mu\mathrm{F}$ nanoribbon. The thicker lines in panel (c) show the dispersion
relation of the heterojunction consisting of the nanoribbon in (a) on the left
and the nanoribbon in (b) on the right connected together along the $y$
direction with the dispersion relations of the isolated left and right
nanoribbons superimposed as thinner lines. The thicker dark lines denote the
valley kink states. The plots on the right indicate the voltage amplitudes of
the bands labelled i to xii in the dispersion relations at $k_{y}=0.8$.

Fig. 5 shows the admittance dispersion of the left (Fig. 5a) and right (Fig.
5b) halves of the heterojunction (Fig. 5c) in isolation from each other, and
that of the entire heterojunction when both halves of the heterojunction are
in the topologically non-trivial phase. In the particular heterojunction under
consideration,, the TE circuit array to the left of the heterojunction has a
positive $C_{g}$ while the right half has a negative $C_{g}$ of the same
magnitude. One way of ascertaining the topological character of a particular
state is to plot its spatial distribution. Hence, the square of the voltage
amplitudes summed over the A and B nodes in each unit cell are plotted for
some of the bands as a function of the $x$ coordinate across the transverse
width of the TE. The square of the voltage amplitude constitutes the TE
analogues of the quantum mechanical probabilty densities
$\psi_{x}^{\dagger}\psi_{x}$ for states described by the wavefunction spinor
$\psi_{x}$, and shall be loosely referred to as ‘probability densities’ for
brevity henceforth. The probability densities in panels (a) and (b) show that
the bands crossing the admittance bandgap (ii, iii, vi, and vii) consist of
edge states exhibiting edge localization. Furthermore, when the edge states
have positive (negative) $k_{y}$ slopes, i.e., ii and vi (iii and vii), the
corresponding states are localized on the left (right) edges of the
nanoribbon. The other bands labelled as i, iv, v, and viii consist of bulk
states where the highest probability densities occur near the center of the
nanoribbon and away from the edges. Comparing Fig. 5a and 5b, we find that the
$k_{y}$ dispersion of nanoribbons with opposite signs of $C_{g}$ are
reflections of each other about the $k_{y}$ axis. This reflection symmetry can
be understood from the form of the Laplacian Eq. (1): the Laplacian is
invariant upon the simultaneous replacement of $C_{g}\rightarrow-C_{g}$ and a
reflection about the $x$ axis, which brings $k_{x}\rightarrow-k_{x}$ and
$\sigma_{y,z}\rightarrow-\sigma_{y,z}$.

When the two halves of the heterojunction are connected together in the
heterojunction, the resultant dispersion relation of the bulk bands is roughly
given by the superposition of the bulk band dispersions in the isolated halves
of the heterojunction, as shown in Fig. 5c. There is a slight increase in the
energy separation between corresponding pairs of bulk bands from each of the
isolated halves in the heterojunction due to the band anticrossing. Note that
the edge states localized away from the heterojunction interface in either
half of the TE heterojunction circuit are not significantly perturbed when the
heterojunction is formed. For instance, the dispersion and spatial
distribution of band xi (x) is virtually identical to that of band vii (ii)
which is localized at the right (left) edge of the isolated half circuits. In
contrast, the edge states of the isolated halves localized at the edges
adjacent to the heterojunction interface (iii and viii) have disappeared in
the heterojunction. They are replaced by bands ix and xii that emerge in the
TE heterojunction circuit which do not have corresponding counterparts in the
isolated halves of the heterojunctions circuit. These new bands correspond to
the valley kink states localized at the heterojunction interface as depicted
in the spatial variation of $|V|^{2}$ in Fig. 5(c).

Figure 6: (a) and (b) show the band dispersions of 10 unit cells-wide TE
nanoribbons with $C_{1}=3\ \mu\mathrm{F}$, $R_{v}=0.75\ \mu\mathrm{F}$,
$C_{y}=1.5\ \mu\mathrm{F}$ and (a) $C_{g}=3\ \mu\mathrm{F}$ and (b) $C_{g}=-3\
\mu\mathrm{F}$. (c) shows the dispersion relation of the heterojunction formed
when the $C_{g}=3\ \mu\mathrm{F}$ nanoribbon is on the left and the $C_{g}=-3\
\mu\mathrm{F}$ nanoribbon on the right, and (d) that of the heterojunction
when the $C_{g}=-3\ \mu\mathrm{F}$ nanoribbon is on the left and the $C_{g}=3\
\mu\mathrm{F}$ nanoribbon on the right. The valley kink states are indicated
as the darker lines in panels (c) and (d).

Surprisingly, valley kink states can also be formed in heterojunctions formed
from joining together two topologically trivial nanoribbons segments, i.e.,
each with $|C_{1}|>|C_{y}|$, as shown in Fig. 6. Panels (a) and (b) show the
dispersion relations for the isolated nanoribbons with positive and negative
values of $C_{g}$, respectively. The particle-like (hole-like) edge states
with positive (negative) $k_{y}$ slopes, i.e. i and iii (ii and iv) are
localized at the left (right) edges of the isolated nanoribbons. The
dispersion relations and the energies of the kink states in the heterojunction
depend on the signs of $C_{g}$ in the two halves of the heterojunction. When
the positive (negative) $C_{g}$ nanoribbon is on the left (right) half of the
heterojunction, the particle-like edge bands of both the left and right
isolated halves (i.e. bands (i) and (iii)) which are localized away from the
heterojunction interface are still be preserved in the heterojunction circuit,
while the resultant valley kink states are hole-like (i.e. bands (ii) and
(iv)), as shown in panel Fig. 6(c). Conversely, when the negative (positive)
$C_{g}$ nanoribbon is on the left (right) half of the heterojunction, the
hole-like edge states of the isolated halves are preserved in the
heterojunction circuit while the resultant valley kink states would be
particle-like.

Figure 7: (a) and (b) show the band dispersions (thin lines) and the terminal
impedance between the leftmost and rightmost nodes of the gapless
heterojunction in Fig. 5c and the gapped heterojunction in 6c respectively.

Finally, the heterojunction circuit with the presence of valley kink states
can be characterized by their unique impedance signatures, as shown in Fig. 7.
Unlike the isolated nanoribbons which exhibit only a single impedance peak as
a function of $k_{y}$ across the Brillouin zone, the impedance dispersion of
the heterojunction circuit has a $k_{y}$-reflection symmetry which leads to a
pair of impedance peaks. The impedance peaks in the impedance dispersion of
the gapped heterojunction circuit are located near the points the hole- and
particle-like bands have their minimum energy separation (see Fig. 7b),
similar to the dispersion of isolated nanoribbon in Fig. 3c. In contrast, the
impedance peaks of the gapless heterojunction circuit are displaced from
$k_{y}=0$ where the hole- and particle-like bands meet. Furthermore, the peak
impedance of the gapless heterojunction is much higher than its gapped
heterojunction counterpart.

##  V Conclusion

In summary, we proposed a highly tunable TE platform that exhibits topological
valley Hall states and valley kink modes. The circuit can be switched between
different topological valley Hall states simply by varying the sign of the
onsite capacitance. Moreover, gapped and gapless boundary states emerge in the
admittance spectra with proper tuning of the mass parameters. We also realize
valley kink states by connecting two TE circuits with opposite signs of Hall
conductivites together. We analytically derived the boundary conditions
between trivial and no-trivial states, characterized by their Chern number.
The gapped and gapless states can be switched from one to the other by tuning
the grounding capacitance. The topological boundary modes or valley kink
states are localized in the interface of the kink circuit. There are
significant differences in the impedance dispersion for the different boundary
modes, leading to measurable and distinguishable circuit responses for
different topological valley and kink states in the uniform and heterojunction
TE circuits, respectively. In summary, our work based on the TE circuit model
provides an accessible testbed to realize various topological valley and kink
phases and allows the efficient modulation and switching between different
topological states for valleytronic applications.

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