Topolectrical circuits—Recent experimental advances and developments Open Access

The use of electrical circuit metamaterial arrays to simulate condensed matter phenomena has emerged as a popular approach alongside other metamaterial approaches,¹ such as mechanical,^2–4 acoustical,^5–10 and photonic^11–13 media. Numerous studies, some of which are reviewed in this article, have demonstrated that a wide range of condensed matter phenomena can be experimentally realized through the design of appropriate electrical circuits. The use of electrical circuit arrays to study the resistance or conductance between the farthest points of a lattice continuum has a long history in condensed matter circles^14–19 and has been studied through various approaches, such as lattice Green’s function,^20,21 transfer matrix method,²² recursion–transform approach,²³ and method of images.²⁴ However, the formal connection between the Hamiltonian formalism and the circuit Laplacian was established less than a decade ago. Early work on periodic circuit lattices primarily addressed point-to-point resistance or impedance problems, lacking a comprehensive methodological framework that connect with condensed matter models.

One of the first studies that introduced a systematic approach to this was the paper by Lee et al. in 2018,²⁵ which introduced a systematic approach that formalized the concept of topolectrical (TE) circuits. It is inspired by two earlier studies by Ningyuan et al.²⁶ and Albert et al.²⁷ that provided explicit constructions of circuit lattices exhibiting topological properties.^28–38 Following these foundational studies, there has been a rapid proliferation of follow-up studies, both through circuit simulations^39–44 and experimental implementations^45–53 (many more to be reviewed later).

Early studies primarily focused on demonstrating topological phenomena,^54–60 including the realization of edge,^52,61,62 higher-order corner,^63–67 surface⁶⁸ and hinge states,⁶⁹ and semi-metalic phases of Weyl^51,70–72 or Dirac semi-metals,^73,74 in multi-dimensional lattices or heterogeneous systems.^75,76 Soon after, these efforts expanded to encompass non-Hermitian^77–80 and non-linear^81–83 systems, as well as simulators of non-Abelian operators^84,85 and quantum gates^86,87 and various other applications.^88–94 At this stage, it is clear that implementing various physical phenomena in electrical circuits has become an engineering task, provided the theoretical framework is validated through the theoretical circuit analysis and circuit simulations.

As such, the focus has recently shifted to the direct device applications of physical models realized by electrical circuits, moving beyond their role as proof-of-concept demonstrations, and the integration of active analog signal processing devices. For the former focus, numerous studies have already proposed sensing applications based on electrical circuits.^95–97 Many of these circuit designs hold promise for practical use, as they can be manufactured using integrated circuit (IC) technologies, enabling their incorporation into everyday technological devices.^98,99 For instance, a sensitive boundary-dependent response can be achieved using simple RLC components, thanks to the non-local impedance response of passive electrical circuit components.¹⁰⁰ The latter focus involves the manipulation of signals using devices such as analog multipliers, operational amplifiers (op-amps), and transistors. While op-amps are commonly used to realize non-reciprocal couplings in impedance conversion configurations,^40,46 their potential extends far beyond these applications. Circuit configurations involving multiple op-amps and analog multipliers enable to realize numerous phenomena such as Floquet engineering of lattice models,^101,102 time-varying couplings,¹⁰³ or tunable interaction strength.^104,105 One illustrative example is the realization of complex coupling terms using analog multipliers, where four multipliers are configured appropriately to achieve the desired effect.¹⁰⁶ 

In the following, we highlight some of the draws of using TE circuits.

• Abundant resources: since electrical circuits are ubiquitous and well-established, a wealth of resources, ranging from educational to research materials, is readily available in the literature, along with a variety of simulation tools, such as LTspice,¹⁰⁷ PSpice,¹⁰⁸ Multisim,¹⁰⁹ or Modelithics Library.¹¹⁰ The widespread availability of circuit components ensures that the necessary elements are easily accessible. Moreover, these resources are cost-effective, making them suitable for diverse implementations.

• Linearity and independence from physical embedding: linear circuits can be analyzed using nodal analysis, where each connection at a node represents an additional degree of freedom, analogous to spatial dimensions in physical systems. The wave interactions in these circuits are governed by the principle of linear superposition, allowing the simultaneous synthesis of multiple voltage or current waveforms at a node.^111–113 Unlike most other metamaterial arrays where propagation direction is constrained by physical geometry, circuits provide the flexibility to emulate systems with higher dimensionalities.^50,114 This capability enables the synthetic realization of effective dimensions that transcend physical constraints, offering a unique advantage over many other platforms.

• Straightforward local adjustments and tunability: localized modifications, such as defect engineering,¹¹⁵ are straightforward and adaptable. The interconnectness of electrical lattices ensures accessibility to each node, enabling precise local tuning or perturbation.^116–118 This offers a significant advantage over less modular setups, where specific localized perturbations are not always possible due to fabrication limitations or intrinsic material properties.

• High reliability and precision: well-designed and maintained electrical circuits provide high reliability, even under challenging environmental conditions.¹¹⁹ They are resilient and do not require special operating environments, such as low temperatures or special electromagnetic shielding,¹²⁰ making them versatile for various applications.^121,122

• Scalability and accessibility: scalability is a major challenge in many metamaterial implementations. Electrical circuits offer high scalability compared to other platforms, which often rely on specialized materials or complex manufacturing techniques,^123,124 making scaling up and system modifications more challenging. In addition, impurities in electrical circuits are typically localized, meaning they do not lead to cumulative effects that cause significant overall systems.

• Ease of implementing non-Hermiticity: electrical circuits are intrinsically suitable for the realization of non-Hermitian models, which require non-reciprocal couplings or controlled loss¹²⁵ and gain.¹²⁶ Active components, such as op-amps, allow the direct implementation of highly linear non-reciprocal non-Hermiticity,⁷⁷ and loss is naturally present in resistive components.¹²⁷ 

• Precise control of component uncertainties and loss: that said, many circuit component manufacturers provide a wide variety of high-quality components with minimal deviations in component parameters and loss. In addition, for implementations that demand exceptionally low loss and deviation, it is feasible to preselect circuit elements due to their generally low cost, as a variety of low-loss components can be readily found to meet these stringent requirements.

• Direct observability and measurability: impedance measurements allow key band structure features, such as non-trivial topological zero modes, to be directly observed through straightforward profiling.⁴⁵ Such measurements can either measure the local transmissibility or the global, non-site-specific band structure properties.¹²⁸ 

• Ease of implementing synthetic dimensions: independence from physical geometric embedding allows the realization of lattice models in arbitrarily many synthetic dimensions, overcoming physical and spatial limitations.^{50,114,129,130}

• Broad frequency range compatibility: electrical circuits can operate across a wide frequency spectrum, from a few hertz to a few gigahertz.^131–133 Unlike photonic systems, which typically require microwave frequencies or higher,¹³⁴ electrical circuits can function efficiently at much lower AC frequencies (e.g., even a few hundred hertz), making them suitable for studying topological oscillations at timescales amenable to direct human observation. Similarly, while acoustic systems often face challenges in achieving realizations in the ultrasound range (above 20 kHz),⁶ integrated circuits can even extend their operation into the terahertz (THz) regime.¹³⁵ 

• Real-time reconfigurability: the modular nature of circuit components, such as varactors and digitally controlled amplifiers or analog multipliers, enables real-time localized adjustments.¹⁰² This capability is crucial for tasks requiring the dynamic modulation of specific couplings without needing to reconstruct the setup, particularly the simulation of Floquet media.^{101–103,132,136}

• Convenience in combining linear and non-linear components: electrical circuits provide a diverse range of linear components (such as resistors, capacitors, and inductors) and non-linear components (such as diodes, transistors, and non-linear amplifiers) that can be combined seamlessly. This versatility enables the study and realization of both linear and non-linear phenomena within the same platform.^82,137 In contrast, photonic and acoustical systems often require specialized materials to achieve non-linearity, making such juxtapositions more challenging.^138,139

Although some of these advantages may be even more prominent in certain platforms, i.e., strong intrinsic optical non-linearity in non-linear photonics,^140–144 electrical circuits remain a highly suitable medium for realizing a broad range of phenomena, as outlined above. In general, their versatility and practicality make them an excellent choice for many applications.

Many of the features listed above have already been effectively exploited in existing topolectrical circuit experiments.^145,146 Building upon these advantages, it can be anticipated that the field is moving toward real-world applications of the phenomena investigated. Early attempts, such as sensing device implementations based on topological non-Hermitian phenomena,¹⁴⁷ hint at the potential for further advancements.^97,148 In addition, the use of analog multipliers to realize complex couplings or temporal responses¹⁰¹ herald a new era when phenomena far beyond topology are realized with chip technologies that are integrated into TE circuits.^133,149,150

We present a concise review of the fundamentals of TE circuits, focusing on the Laplacian formalism and their impedance characteristics.

In electric circuit settings, both the Laplacian and Hamiltonian are frequently used. While both are used to describe aspects of the circuit dynamics, they are fundamentally different objects. The circuit Laplacian captures the graph connectivity structure of the circuit by representing the flow of the current in an electrical circuit through Kirchhoff’s law. By contrast, the Hamiltonian is formally defined as the generator of time evolution, as in Schrödinger’s equation. In the following, we examine how the Laplacian and Hamiltonian are related, particularly in the context of topolectrical circuit arrays,²⁵ where both acquire a momentum-space band description.

A significant obstacle in obtaining impedance results analytically lies in the implementation of open boundary conditions (OBCs), particularly for unit cells with multiple nodes. However, recent developments have demonstrated that the method of images can successfully yield analytical expressions for a wide range of OBC scenarios, for instance, 2D SSH circuits with four nodes per unit cell. The core idea involves tiling the space periodically with infinite copies of the actual lattice, such that the open boundary becomes an equipotential where no current flows through anyway. As illustrated in Fig. 1(c), symmetrically injecting current at the center nodes and extracting it from the intended boundaries converts the edges of the pink region into impenetrable equipotentials, such that the OBC problem can be treated as an infinite image current problem.

Importantly, Z(N) can be extraordinarily large for certain some values of certain circuit sizes N if there are circuit coupling elements that introduce opposite phases, i.e., capacitors and inductors [Fig. 1(e)]. This is because having at least two different w_i of opposite signs can allow the denominator to almost vanish at appropriate values of N. Such dramatic system size dependence in the impedance is not seen in homogeneous electrical media, where the dependence on N is smooth (logarithmic or power-law).^17,161–163

Behind the seemingly erratic occurrence of impedance peaks in Fig. 1(e) is a sophisticated fractal-like pattern in the distribution of resonance heights within the parameter space of system size N and ω—the dimensionless (normalized) frequency characterized by the LC components [Fig. 1(f)]. While the detailed pattern depends on the actual lattice model describing the circuit array, hierarchies of impedance peaks universally occur in a qualitatively similar manner. The intricacy of this pattern arises from the commensurability properties of the denominator in Eq. (10), in which some values of N can cause it to be especially close to vanishing [Fig. 1(g)]. Notably, there exists some branches that are particularly robust and hence experimentally detectable,¹⁵³ reminiscent of the more salient structures in the Hofstadter butterfly. In addition, resonances may also arise due to protected topological zero modes, as in a Chern lattice [bottom diagram in Fig. 1(f)].

The Laplacian eigenspectrum can be obtained either by reconstructing the Laplacian matrix and then diagonalizing it, or by directly measuring the impedance resonances.

In the former, one can perform N² impedance measurements by separately subjecting the circuit to N different input current configurations and measuring the electrical potentials at all the N nodes.^77,164–167 This yields N different current and voltage vectors, which can be written as N × N matrices I and V. This allows the Laplacian J to be recovered via J = IV⁻¹; for simplicity, one can just ground the circuit and choose to input the current at one node at a time, such that I is simply the identity matrix.

This is demonstrated in the work of Helbig et al.,⁴⁶ where the eigenvalue band structure of a 1D non-Hermitian SSH circuit under both open and periodic boundary conditions (OBCs and PBCs) was measured. By varying the injected signal frequency and measuring voltage responses, they extracted the system’s eigenvalue spectrum as a function of the driving frequency. For PBCs, translational symmetry reduces the independent degrees of freedom from N² to N, since it allows the spectrum to be computed in momentum space by exciting the nodes and Fourier transforming the measured node voltages.^115,168 However, OBC circuits require real-space voltage measurements across the entire array, yielding, for instance, additional spatially localized edge modes. Helbig et al.³⁹ first presented this framework, with a further implementation in a 2D topological Chern circuit by Hofmann et al.⁴⁰ 

Constructing the circuit Laplacian in full requires N² impedance measurements, in general, and that presents practical limitations in larger circuits. Recently, Franca et al.¹²⁸ developed a method to extract the Laplacian eigenenergies directly from impedance resonances using much fewer measurement configurations. As deduced from Eq. (9), any vanishing eigenvalue corresponds to a large impedance resonance. By tuning the system across a range of AC frequencies, various resonances can be observed, as demonstrated with high accuracy in their Fibonacci circuit implementation [Fig. 1(h)]. This approach bypasses the need for constructing the circuit Laplacian, reducing the measurement complexity from N² measurements to a single impedance measurement at various (easily tunable) frequencies.

To elaborate, this approach involves examining the second derivative of the impedance profile with respect to frequency, coupled with a fourth-order Butterworth filter to mitigate noise [Fig. 1(i)]. Advanced data processing techniques, such as the matrix pencil method and utilizing the symmetric energy profiles of chiral symmetric systems, further enhance the precision in determining admittance eigenvalues. In a similar spirit as other studies that utilize machine learning to infer the spectrum from limited measurements (to be discussed in subsequent sections, e.g., Ref. 169), this method represents a significant advance, which allows for the measurement of the Laplacian spectrum from a single measurement [Fig. 1(j)].

In the following, we highlight various studies, mostly experimental, on demonstrating non-Hermitian phenomena of contemporary interest. Emphasis is placed on recent developments in the past few years.

The eigenvalues of H_PT are given by $E=ω±κ2−γ2$⁠. When κ > γ, the eigenvalues are real, indicating the PT-symmetric phase. Conversely, when κ < γ, the eigenvalues become complex, signaling a transition to the broken PT-symmetric phase. At the critical point κ = γ, not only do the eigenvalues coincide, but the eigenvectors also coalesce (become parallel to each other). Such non-Hermitian critical points are known as exceptional points (EPs),^{174,176–182} which are special because the eigenspace is defective, i.e., not full rank. Furthermore, due to the square-root singularity of the eigenbands at the EP, the spectrum becomes highly sensitive to perturbations, with a divergent gradient as the EP is approached. This sensitivity increases for higher-order exceptional points, where n > 2 eigenvalues coincide and result in a E^1/n singularity. That the system’s response to perturbations becomes increasingly pronounced is particularly advantageous for sensing applications.^183,184 The sensitivity can be enhanced by incorporating multiple modes or alternative coupling mechanisms to enable higher-order EPs.^{182,185–190} Beyond isolated EPs, higher dimensional extensions such as exceptional lines¹⁹¹ and exceptional rings¹⁹² have also been observed, in analogy to nodal lines and rings.^193–198 

One of the earlier experimental realizations of a PT-symmetric circuit was performed by Schindler et al.,¹⁹⁹ where two LC oscillators—one with gain (achieved using an operational amplifier) and the other with loss (introduced by a resistor)—were coupled [Fig. 2(a1)]. These systems demonstrate the square-root branching behavior that leads to the bifurcation of both the real and imaginary parts of the eigenfrequency at the EPs [Fig. 2(a2)].

Integrated circuit implementations of PT-symmetric systems are particularly technologically promising, as they operate at high microwave frequencies and offer scalability. For instance, Cao et al.²⁰⁰ designed a fully integrated PT-symmetric two-component system [Fig. 2(b1)] where the oscillatory mode exhibits wideband microwave generation with significantly reduced noise aided by the PT-symmetric phase transition [Fig. 2(b2)]. The splitting of resonance peaks in different phases achieved a 0.35 GHz band isolation [Fig. 2(b3)], showcasing the exceptional sensitivity and practical utility of integrated circuit-based PT-symmetric systems.

Wireless electronic sensors²⁰⁵ primarily utilize the resonant behavior of LC circuits [Fig. 2(c1)], and enhancing their sensitivity can lead to promising applications such as wireless power transfer.²⁰⁶ In addition, implanted LC wireless circuits can monitor biological functions in living organisms [Fig. 2(c2)]. By leveraging on the highly sensitive readout of their microsensor implanted in vivo, Dong et al.²⁰¹ successfully measured physiological functions such as breathing rate, demonstrating superior sensitivity compared to standard LC sensory readers [Fig. 2(c3)].

Although an EP system becomes highly responsive at the EP, the accompanying defective eigenspace (where two or more eigenvectors coalesce) can also introduce undesirable susceptibility to noise,²⁰⁷ complicating the implementation of EP-based sensors.¹⁷⁷ However, appropriately designed non-linearity can mitigate that susceptibility while preserving the responsiveness, even leading to EP transitions.^208,209 Bai et al.²⁰² designed a two-component circuit that juxtaposes the non-linearity and non-Hermitian coupling between two oscillators [Fig. 2(d1)]. Introducing non-linear gain results in higher-order non-linear EPs (NEPs), such as third-order NEPs, as shown in Fig. 2(d2). They effectively suppress noise, even when the circuit is intentionally disturbed by a non-negligible external pulse.²¹⁰ With non-linearity, the Petermann factor can be prevented from diverging [Fig. 2(d3)]. In addition, the signal-to-noise ratio reaches its lowest value at the NEP and increases as the system moves away from the NEP [Fig. 2(d4)].

PT symmetry-based implementations can also enhance system stability. In semiconductor technologies, fast-switching devices are widely used, and achieving both high speed and stability is highly desirable.^211,212 Yang et al.²⁰³ demonstrated that transient PT symmetry, as implemented in their electric circuit [Fig. 2(e1)], can significantly reduce unwanted oscillations during fast switching. Noise-induced perturbations in the oscillations decay more rapidly at the EP compared to those in the broken or unbroken PT phases [Fig. 2(e2)]. For step-like dynamic switching, transient PT symmetry effectively suppresses unwanted oscillations [red in Fig. 2(e3)], as observed from the diminishing peak around 36 MHz in the transient PT phase [blue in Fig. 2(e4)] in the frequency domain.

Mathematically, EPs can also trace out interesting mathematical structures in either parameter space or real/momentum space. EPs underlie braiding in the complex eigenvalue plane^213,214 and define knots in the band topology.²¹⁵ By designing non-Hermitian circuits involving multiple bands, Cao et al.²⁰⁴ achieved higher-order EPs. The shape of knots or braids primarily depends on the winding paths around the EPs.²¹⁶ For example, a circuit with two nodes [Fig. 2(f1) and 2(f2)] or three nodes [Fig. 2(f3)] can exhibit two-component or three-component knot topology, with double or triple twisting in the complex eigenenergy plane, respectively. Momentum-space EP knots are also described in Refs. 217–219. In a larger parameter space, multi-band EPs can also describe mathematical singularities known as catastrophes, which have deep connections, i.e., McKay correspondence with diverse mathematical constructs such as the ADE classification of Lie algebras, finite subgroups of SU(2), platonic solids, and the monodromy group of simple singularities.^220,221 A recent circuit experiment has mapped out the swallowtail catastrophe in parameter space.²²² 

The defectiveness of EPs causes the occupied band projector to be singular, leading to substantial non-locality in its two-point correlation functions. A prominent consequence is the appearance of special isolated eigenstates [Fig. 3(a)] of the real-space truncated two-point correlator matrix known as exceptional bound (EB) states.^223–226 These EB states differ from topological modes, which are protected by non-trivial bulk topology, and non-Hermitian skin modes, which stem from non-trivial point gap band topology and are also boundary-localized in open-boundary systems.^227–234 While most eigenvalues $p̄$ of the truncated two-point correlator matrix remain clustered around 0 or 1, as in non-EB cases, a distinctive pair of EB states $p̄EB$ and $1−p̄EB$ additionally appears as long as the parent Hamiltonian is defective [Fig. 3(a)]. The value of $p̄EB$ scales strongly with the system size, as detailed in Refs. 223 and 224. Since the spectrum of the truncated two-point correlator matrix corresponds to fermionic occupation probabilities, EB states also lead to negative entanglement entropy in the context of free fermions,^223,224,235 with interesting ramifications in topological²²⁶ or competitive non-Hermitian skin effect settings.²³⁶ 

Experimentally, EB states can be simulated in classical platforms that mathematically describe the two-point correlator matrix. Electrical circuits are particularly suited for this purpose because any two nodes can be connected at will, free from the constraints of locality. As a pioneering demonstration, the circuit [Fig. 3(b)] designed by Zou et al.²²³ successfully demonstrated the robustness of EB states through measured voltage profiles [Fig. 3(c)]. Strong resonances at frequencies corresponding to the eigenvalues of the EB states were detected, demonstrating the direct measurability of these states. These EB states are shown to be much more robust than the non-EB resonances, which can be significantly disturbed by relatively minute perturbations of the larger hoppings.²²³ 

One of the most intensely studied non-Hermitian phenomenon is the non-Hermitian skin effect (NHSE), where states accumulate against spatial inhomogeneities, impurities, or boundaries^{165,227,229,230,237–269} challenging conventional condensed-matter insights into the correspondence between bulk and boundary properties.

Note that the GBZ condition, as well as the existence of a unique κ(k) that quantifies the skin accumulation, may no longer hold when more than one NHSE channel coexists in the system. A prominent example is the critical non-Hermitian skin effect (cNHSE),^{225,236,281–286} where weakly coupled chains with different NHSE directions produces feedback loops that amplify certain states and lead to qualitatively different band structure descriptions at different system sizes. Appropriately designed cNHSE systems may even undergo an exceptional phase transition at a particular system size, with special scaling-induced EP defectiveness and leading to unique entanglement dips.²²⁵ Fundamentally, the GBZ condition hinges on the assumption that the OBC eigenstate only comprise of the superposition of two bulk solutions; relaxing that generally leads to a “fragmented” GBZ in which more than one κ(k) participates in the physics substantially.^287,288

In a related vein, Shen et al.²⁹⁰ conducted a circuit experiment involving multiple asymmetric hoppings in coupled 1D SSH chains [Fig. 4(c1)]. They extended this setup by connecting multiple 1D SSH chains to form a spherical geometry, paving the framework for studying the interplay between band topology and real-space lattice topology.¹⁶⁶ While localization behavior in various lattice geometries, including kagome and honeycomb lattices,^295,296 has been widely studied, in the spherical model [Figs. 4(c2) and 4(c3)], localization occurs only when a defect is introduced [pointed by red arrow in Fig. 4(c4)]. In addition, Jiang et al.²⁹⁷ demonstrated how circuits with multiple non-reciprocal couplings can give rise to corner modes in 2D circuits.

Typically, the NHSE drags all the bulk states toward a boundary.²³² However, in two or more dimensions, Zou et al.¹⁶⁴ demonstrated that the NHSE can selectively pump topological modes while leaving the bulk unaffected, in what is known as the hybrid higher-order skin–topological (ST) effect.²⁵⁹ The idea is that, in a multi-dimensional lattice, it is possible for topological modes under mixed boundary conditions to possess a spectral point gap, but not the non-topological modes. Further opening up the remaining boundary(ies) will then result in the NHSE only for the topological modes. In Ref. 164, the 2D network consists of a unit cell with four nodes, with asymmetric intra-cell couplings implemented using INICs [Fig. 4(d1)]. With suitable parameter tuning, the 2D and 3D circuit implementations [Figs. 4(d2) and 4(d3)] exhibit skin–topological (ST) and skin–skin–skin (SSS) effects, respectively. These hybrid skin–topological modes arise from the interplay between topological and skin modes and cannot exist without either. Notably, these ST modes can exhibit localized skin behavior despite the absence of net non-reciprocity, since the reciprocity is spontaneously broken by the topological modes taking unequal amplitudes in the two sub-lattices.

The hybrid skin–topological effect²⁵⁹ has also inspired a scheme for switching on or off the NHSE by toggling the topological character of the perpendicular sub-chains, as was first proposed in the cold atom context.²⁹⁸ When the switch is activated by adjusting the loss parameter, non-reciprocal pumping prevails, driven by non-reciprocity introduced by the switch. Zhang et al.²⁹¹ experimentally demonstrated this topological switching using their 3D circuit [Fig. 4(e1)]. The voltage dynamics across four different unit cells revealed distinct eigenmode distributions, such as skin–topo–skin and skin–topo–topo modes [Fig. 4(e2) and 4(e3)]. The absence or presence of skin modes corresponds to trivial or non-trivial topological modes, which are tuned through relays that switch resistors on or off between specific nodes.

The interplay between topology and the NHSE was further explored by Tang et al.²⁹² through an experimental implementation of a rhombus honeycomb lattice electrical circuit [Fig. 4(f1)]. By tuning the non-reciprocity (controlled by C₂ in their design), the topologically localized corner modes at the acute angles of the rhombus lattice [Fig. 4(f2)] are significantly dragged through the bulk, allowing the NHSE to control the topological modes [Figs. 4(f3) and 4(f4)].

Despite being classical systems, electrical circuits can also simulate few-body interaction dynamics through mappings between many-body one-dimensional models and single-body multi-dimensional models.^48,299–301 In addition to their practical value, such mappings can also lend new physical insights,^302–304 such as the interpretation of two-body repulsion as 2-dimensional boundary effects, and 3-body non-Hermitian interactions as feedback mechanisms on a 2-dimensional non-orthogonal non-Hermitian lattice whose GBZ description is notoriously subtle.²⁴⁵ In particular, effective boundaries can be defined in the Hilbert space even when the lattice has PBCs along each direction. Reverse implementations of such mappings have also enabled small interacting quantum systems to simulate the physics in far larger multi-dimensional lattices.^305–307 

Zhang et al.²⁹³ designed a circuit based on mapping the eigenstates of their strongly correlated non-Hermitian few-body system [Fig. 4(g1)]. Through impedance measurements, they observed the simulated aggregation of bosonic clusters in their circuit. The impedance resonances in the spatial circuit [Fig. 4(g2)] correspond to eigenstate aggregation in the corresponding Hilbert space [Fig. 4(g3)]. This realization serves as a compelling example of the feasibility of topolectrical circuits to model many-body interactions in a very accessible manner.

Although paradigmatic examples of the NHSE—such as the Hatano-Nelson model³⁰⁸—are commonly realized using non-reciprocal (asymmetric) couplings, the NHSE can still manifest in reciprocal systems without requiring such asymmetry. In the so-called reciprocal non-Hermitian skin effect, a reciprocal system is designed such that it contains subsystems harboring equal and opposite NHSE, such that NHSE accumulation occurs in specific subsectors despite the system being reciprocal on the whole. This was first demonstrated by Hofmann et al.,¹²⁷ who measured eigenmode localization at the edges of their 2D circuit [Fig. 4(h1)]. A diagonal imaginary hopping within the square plaquette makes the Hamiltonian globally non-Hermitian but still reciprocal; with a basis transform, it can be shown that it contains “hidden” equal and opposite NHSE chains at different transverse momentum sectors or effective flux. This imaginary hopping—realized as a resistor in their model [Fig. 4(h2)]—results in different hopping probabilities in the presence of this effective flux, resulting in unequal amplitudes in opposite directions [Fig. 4(h3)].

In two dimensions, a reciprocal 2D lattice can also exhibit nontrivial system size-dependent NHSE in the presence of higher-rank chirality,³⁰⁹ characterized by a non-conserved charge current.³¹⁰ First demonstrated in Ref. 309, a higher-rank chiral mode (rank-2) was also demonstrated by Zhu et al.²⁹⁴ through gain/loss couplings that create an unbalanced chirality in the unidirectional chiral mode flow [Fig. 4(i1)]. This imbalance causes one specific mode to amplify and hence outlast others, leading to eventual mode localization. The long-lived rank-2 chiral mode dominates the system’s long-time dynamics and is observable in momentum-resolved measurements [Figs. 4(i2) and 4(i3)]. Remarkably, momentum-dependent input excitations lead to mode accumulation at specific driven locations in the lattice [Fig. 4(i4)]. This enables a versatile implementation of NHSE along a desired direction and position in a fully passive and reciprocal lattice, distinguishing it from the reciprocal skin effect presented in Ref. 127. (Even in the absence of gain/loss and non-reciprocity in the lattice bulk, non-Hermiticity can emerge through the interaction Hamiltonian in an electromagnetically engineered metamaterial that is inversely designed from a TE lattice;³¹¹ also see Ref. 312, where the NHSE is realized in reciprocal lattices via gauge fields without gain or loss.)

The extreme versatility of electrical circuit connections, which can directly couple any two desired nodes independent of their physical distance, makes them ideal for implementing intrinsically long-ranged models.³¹³ Since the NHSE is already highly non-local in that it leads to dramatic state accumulation in a spatially-distant edge, it is particularly susceptible to the effects of long-ranged hoppings that can change the overall real-space topology of the lattice. For instance, a single non-reciprocal impurity hopping in an otherwise homogeneous Hatano-Nelson chain is known to lead to enigmatic scale-free localization.^314–316 

Recently, Guo et al.¹⁶⁷ introduced the new notion of scale-tailored localization (STL), distinct from conventional NHSE or Anderson localization.³¹⁷ The STL is a spectacular demonstration of how singularities in the skin localization length subtly interplays with hopping non-locality, as was implemented via unidirectional couplings with INIC circuit elements, and a long-range coupling controlled by switches [Fig. 4(j1)]. For this circuit, some of the admittance eigenvalues encircle the unit circle, while others are isolated within it [Fig. 4(j2)]. The spatial distribution of the corresponding STL eigenstates exhibits varying localization lengths that depend on the long-range coupling distance [Fig. 4(j3)].

Since the NHSE is a dramatic phenomenon that arises from merely changing the boundary couplings, it is symptomatic of extreme sensitivity to spatial perturbations. In the following, we highlight some experiments that attempt to point toward potential sensing applications in electrical circuit settings.

The primary interest in the NHSE stems from its strong dependence on local perturbations, such as a coupling between the two ends of a chain, which can drastically alter the OBC vs PBC properties. Furthermore, a non-Hermitian topological lattice exhibits extreme sensitivity to its size due to the exponential growth of NHSE states, as formulated theoretically by Budich and Bergholtz³¹⁸ and popularized as non-Hermitian topological sensors.³¹⁹ This phenomenon was experimentally demonstrated by Yuan et al.⁹⁶ [Fig. 5(a1)]. The shift in resonance peak frequency occurs through two mechanisms: (i) control of the boundary coupling capacitance through the displacement and rotation of the copper electrodes [Fig. 5(a2)] and (ii) variations in the number of nodes in the circuit. The impedance resonances exhibit exponential behavior in response to circuit size N [Fig. 5(a3)].

Based on the exponential response to boundary perturbations in non-Hermitian circuits, Könye et al.³²⁰ designed a non-Hermitian topological ohmmeter with reciprocal coupling between the first and last terminals (reminiscent of Ref. 96), utilizing it as a sensor to precisely measure large resistances (Γ in Fig. 5(b1)), typically in the mega-ohm range. The measurement of large resistances often suffers from reduced precision due to significant losses. However, their multi-terminal non-Hermitian ohmmeter circuit demonstrated exponentially increasing sensitivity [Fig. 5(b2)] with the number of terminals, comparing favorably with standard measurement techniques [Fig. 5(b3)].

Enhanced response can also be achieved through various mechanisms, such as multiple non-local long-range asymmetric couplings, which lead to an Nth-order sensitive response, where N represents the number of nodes, as introduced in Refs. 95 and 97. Reference 148 also presents an integrated circuit sensor demonstrating an exponential response with respect to the device size, based on higher-order non-Hermitian topology.

In the previous non-Hermitian sensing experiments, the underlying mechanism is directional amplification, which results in a strongly non-local response. Indeed, exponential sensitivity intuitively requires a non-reciprocal non-Hermitian bulk, achieved through asymmetric couplings,³²¹ which necessitate active elements such as op-amps in INICs.^322,323 However, Zhang et al.¹⁰⁰ presented a fully passive circuit demonstrating a highly distinct impedance response dependent on the presence of a boundary coupling, even though the circuit uses only resistors as the non-Hermitian components.

Experimentally, the circuit’s response to non-local perturbations is characterized by the ratio of two-point impedance values under PBCs vs OBCs: Z_PBC/Z_OBC. While this impedance ratio across all two-point configurations within individual chains remains approximately equal to 1 [the two light blue plots on the left in Fig. 5(c3)], the impedance across inter-ladder pairs deviates significantly, with the ratio hovering around 6 [red in the right plot in Fig. 5(c3)]. This indicates substantial non-local impedance response to the boundary coupling, which is interestingly obtained in a purely resistive circuit, without using any amplificative components such as op-amps.

While this setup may seem to be just another demonstration of the reciprocal NHSE¹²⁷ [see Fig. 4(h) in Sec. III B 4], the effective inter-chain −1/r coupling is set to be relatively weak compared to the intra-chain terms, such that the NHSE from both chains does not simply cancel in the intended experimental setup. However, due to the exponential sensitivity of the NHSE, they still do cancel for sufficiently long circuits, as a variation of the critical NHSE^{236,281–283} (refer to the discussion related to Fig. 4(b) in Sec. III B 1). In this model, the circuit exhibits curious size-dependent isolated modes only in the presence of parasitic resistances [Fig. 5(c4)], when the OBC circuit is sufficiently long. Their spatial profiles are boundary-localized in a unique manner, as shown in the two rightmost plots in Fig. 5(c4). Despite being robust against perturbations, such isolated modes do not appear to be protected by any conventional topological invariant, appearing at certain ranges of system sizes.

In the following, we highlight other studies on topolectrical circuits, mostly experimental, which are centered around more traditional (mostly Hermitian) condensed matter phenomena. In this review, heavier emphasis is placed on the simulation of phenomena that are not easily achieved in conventional quantum matter.

Introducing non-linear dynamics into a system reveals intriguing phenomena that are unattainable in linear systems, such as higher harmonic generation,^140,324,325 soliton wave propagation,^326–329 non-linear bands,³³⁰ non-linear interface modes,³³¹ and various non-Hermitian interplays.^{208,332–337} In particular, in the presence of non-linear interactions, coherent collective dynamics can emerge among the autonomous units hosting non-linear oscillators. Of particular interest is the incorporation of a topological mechanism, which facilitates self-organized, protected boundary oscillations.^331,332,338 A common approach to realizing non-linear topological phenomena involves arranging non-linear oscillatory elements in a topological configuration, such as by dimerizing an array of oscillators in deference to the well-known SSH configuration.

A key aspect of such non-linear topolectrical circuit realizations is the ability to achieve a topological phase transition through the excitation intensity. This dependence eliminates the need for having separate lattice configurations for trivial or non-trivial phases and allows access to robust topological modes through external pumping [Fig. 6(c1)]. Zangeneh-Nejad and Fleury⁸³ subsequently realized a second-order non-linear TE [Fig. 6(c2)] that emulates second-order topological corner modes when the input intensity (P_in) exceeds a certain threshold. This implementation employs non-linear varactor diodes, similar to Ref. 137. For instance, when P_in is 25dBm, an admittance resonance occurs, corresponding to the second-order topological 0-dimensional corner mode [Fig. 6(c3)].

Moving away from using varactors as non-linear coupling components, Hohmann et al.³²⁷ employed varactors to create non-linear on-site potentials [Fig. 6(d1)]. This approach led to qualitatively different outcomes: injected sinusoidal waves were distorted into localized cnoidal waves—periodic, soliton-like waves [Fig. 6(d2)]. The localization behavior is attributed to topological exponential boundary localization, evident in the root-mean-square amplitudes decaying toward the bulk, demonstrative of the interplay between topological and solitonic localization. In addition, the non-linearity generates higher harmonics relative to the circuit’s driving frequency, although the amplitudes of these harmonics decrease with increasing harmonic order [Fig. 6(d3)].

Using back-to-back varactors also, Wang et al.⁸² introduced a non-linear transmission line circuit in which varactors were employed as intra-cell non-linear coupling components [Fig. 6(e1)]. This circuit exhibits third-harmonic signals [Fig. 6(e2)] that are much larger than the first harmonic. Interestingly, the higher-harmonic signal [blue in Fig. 6(e3)] propagates through the bulk without diminishing in amplitude. Instead, its amplitude is significantly enhanced, surpassing the edge-localized first-harmonic signal [red in Fig. 6(e3)]. Unlike in the setup of Ref. 137, this enhanced higher-harmonic behavior is attributed to the traveling-wave nature of higher-harmonic modes, which can excite the entire circuit. In contrast, the first harmonic remains localized at the edge in the deep non-linear topologically non-trivial regime [Fig. 6(e4)].

Hu et al.³³⁹ demonstrated greatly enhanced third-harmonic resonances in a mirror-stacked design consisting of two non-linear chains, each unit cell coupled by a linear capacitor [Fig. 6(f1)]. This mirror-stacked design enables the realization of both fundamental harmonics and third harmonics due to the presence of multiple edge nodes with two distinct frequency localized states. Under single-source excitation, the circuit exhibits prominent third harmonic resonances [Fig. 6(f2)], with identical voltage profiles in the first and third harmonics [Fig. 6(f3)].

While most non-linearity-related phenomena arise from non-linear elements such as varactors, Yang et al.³⁴⁰ observed a frequency comb generated by the non-linear behavior of op-amps under strong pumping. Frequency combs represent the proliferation of strong higher harmonics.³⁴⁴ The circuit by Yang et al. was designed to simulate skyrmion [Fig. 6(g1)] dynamics in the circuit unit within each node³⁴⁵ [Fig. 6(g2)]. Remarkably, distorted voltage oscillations resulting from op-amp saturation under a strong input signal gave rise to discrete, evenly spaced frequency combs [Fig. 6(g3)]. This observation highlights how non-linear behavior can emerge from devices inherently designed for linear operation, offering a controlled method for introducing non-linearity in circuit designs.

Lo et al.³⁴¹ recently presented a non-linear circuit experiment demonstrating NHSE that emerges depending on the intensity of the driving amplitude. Interestingly, this behavior arises not from asymmetric couplings but from Kerr-type non-linear dynamics.^346–348 The circuit comprises back-to-back onsite non-linear varactors [Fig. 6(h1)] assembled into a circuit array inspired by the physics of Bogoliubov modes³⁴⁹ [Fig. 6(h2)]. The non-linearity breaks the pseudo-Hermiticity, leading to a spectral point gap that, in turn, gives rise to NHSE localization. The sideband modes [Fig. 6(h3)], distinct from higher-order harmonics, exhibit localized profiles that depend on the auxiliary drive frequency. Due to the Kerr non-linearity, the NHSE appear only when the input signal amplitude exceeds a certain threshold [Fig. 6(h4)]. This work highlights how non-linearity can induce the NHSE in classical electrical circuits.

The complex emergent dynamical behavior in non-linear systems can sometimes pose significant challenges for achieving global synchronization.³⁵⁰ However, Zhang et al.³⁴² recently demonstrated a Hatano-Nelson circuit hosting on-site non-linear Stuart–Landau oscillators [Fig. 6(i1)]. Due to the asymmetric couplings in the Hatano-Nelson circuit, Hermiticity is broken, resulting in non-orthogonal eigenmodes. This facilitates more effective communication between modes, ultimately leading to global synchronization. This behavior manifests as collective voltage oscillations observed in the time domain [Fig. 6(i2)] and in the steady-state voltage profile [Fig. 6(i3)]. This study highlights how the NHSE can enable global synchronization in non-linear experimental circuits (also refer to Ref. 351).

A cornerstone of many non-linear systems is their ability to support chaotic dynamics.^{343,350,352,353} A paradigmatic chaotic system is the Chua oscillator which, unlike Stuart–Landau oscillators, exhibits irregular and chaotic orbits in some parameter regimes.³⁵⁴ Mathematically described by the Lorenz system of differential equations, the Chua oscillator possesses a well-known circuit realization known as the Chua circuit.^355–357 It can be constructed with four passive components and one active component known as the Chua diode³⁵⁸ [Fig. 6(j1)], which provides the requisite non-linear negative resistance. This non-linearity is evident in its non-linear current–voltage (IV) characteristic [Fig. 6(j2)], and typical chaotic behavior includes the famed double-scroll chaotic attractor [Fig. 6(j3)].

Of particular interest is how such chaotic behavior interplays with lattice band structure dynamics, such as when multiple non-linear chaotic oscillators are coupled to form an array. In Ref. 343, Sahin et al. investigated the consequences of coupling Chua oscillators in a topological SSH-like lattice structure [Fig. 6(j4)]. Since band topology is only rigorously defined in linear systems, one might intuitively expect the non-linearity, and especially its chaotic dynamics, to erode the topological localization. However, it turns out that topological edge localization persists robustly well into the non-linear regime, with the amplitudes of the chaotic scroll being very different at the bulk and edge oscillators [Fig. 6(j5)] only in the black parameter space region [Fig. 6(j6)]—the region extrapolated from the topological non-trivial parameter region defined in the linear limit. This extrapolation can be performed by tuning the I–V characteristic parameter b of the Chua diode [Fig. 6(j7)] and explicitly showcases how the topological edge oscillation suppression robustly carries over from the linear into the non-linear regime.

By externally perturbing the voltage oscillations by injecting current at the left edge node, one further observes that the chaotic oscillations, which are typically highly sensitive to external perturbations, persist robustly only when in the topologically non-trivial regime. In the regime connected to the trivial phase, the injected current reduces the chaotic double-scroll portraits to non-chaotic limit-cycle oscillations [Fig. 6(j8)]. However, in the non-trivial phase, the same current injection only disturbs the edge node oscillations, leaving the bulk oscillations mostly intact [Fig. 6(j9)]. This robustness is captured as a phenomenon termed “topologically protected chaos,” where chaotic dynamics counterintuitively become more robust by the remnants of topological localization deep inside the non-linear regime.³⁴³ 

Typical realizations of tight-binding condensed matter phenomena map the lattice model onto a real space physical array and are often met with challenges in implementing or maintaining a large number of lattice sites to within acceptance levels of error tolerance. One promising alternative approach is to instead realize the model in the frequency lattice, through a time-periodic driving approach known as Floquet engineering.^359–376 In analogy to the Fourier correspondence between a periodic Brillouin zone and its real-space lattice, a system that is periodically driven in time should also correspond to a lattice in frequency space. This powerful analogy enables the implementation of complicated hoppings and unit cells in terms of appropriately programmed temporal driving frequency components, all within a single (time-periodic) physical unit cell.

A key difference between Floquet-driven Hamiltonian systems and Floquet-driven electrical circuits is that in the latter (electrical circuits), the main physics is contained in the steady-state relationship between the current and the voltage, i.e., the circuit Laplacian J that encapsulates Kirchhoff’s law I = JV (refer to Sec. II A). As such, impedance measurements probe the properties of the Laplacian J, rather than the time-evolution generator H. As explained in Ref. 101, this fundamental difference is reflected in the frequency dependence of the Floquet operator: while ordinary Floquet Hamiltonians can only contain a linear “background” potential in the frequency lattice,^377,378 a Floquet Laplacian can depend on the frequency in arbitrarily complicated manner, depending on the frequency dependencies of its constituent components, i.e., iωC or 1/iωL for capacitive and inductive components. As such, Floquet circuit Laplacians can be engineered to possess a variety of background “potentials” in the frequency lattice, such that one can also probe impurity or edge phenomena, i.e., topological edge states in appropriately designed temporally driven electrical circuits.

Stegmaier et al.¹⁰¹ designed a circuit with two logical nodes that assumes the topological SSH form in the frequency domain [Fig. 7(a1)]. By employing a combination of AD633 analog multipliers and capacitors [Fig. 7(a2)], effectively time-varying capacitors were achieved, modulated at far higher frequencies (order of MHz) than what is possible with mechanical means. To recreate an SSH model in frequency space, the inter- and intra-node capacitances are time-modulated such that their temporal functional dependence resembles that of the momentum dependence of the conventional SSH model. Notably, the Floquet driving protocol is such that the effective background potential diverges at the zeroth “site” in the frequency lattice, such that it acts as a strong “barrier.” As conspicuously measured, topology boundary states exist at frequencies very near these barriers when the effective SSH parameters are tuned to topologically nontrivial values [Fig. 7(a3)]. These barrier states exhibit localized profiles in frequency space, as can be obtained by Fourier transforming the measured temporal voltage oscillations [Fig. 7(a4)]. Other than showcasing the utility of analog multipliers in effectively time-modulating circuit components at very high frequencies, this work (Ref. 101) also demonstrated that Floquet circuit Laplacians can exhibit highly customizable inhomogeneities in the frequency lattice, an important realization that has far-reaching implications in topological signal processing.

Although time-varying components enable the construction of a frequency-domain lattice through their temporal oscillations, directly observing Floquet modes can be challenging due to the substantial heating in rapidly time-modulated elements. For the goal of experimentally realizing (mathematical) Floquet behavior, one approach to circumvent this restriction is to directly simulate the frequency lattice in real space. An attempt along these lines is the experiment by Zhang et al.,¹³⁶ which observed Floquet modes along temporal dislocations, in analogy to the usual spatial dislocations that host topological modes. As shown in Fig. 7(b1), a (logically) time-dependent 2D higher-order topological lattice is realized as a 3D stack of 2D layers, where the third dimension represents frequency. The time-dislocation is spatially implemented as a sudden change in the inter-layer connections between the left and right halves of the setup. These inter-layer couplings represent the (logical) driving protocol. As a result, Floquet 0 and π topological modes exist in the bandgaps [Fig. 7(b2)]. Notably, these modes exist at the dislocation between the left and right halves of the 3D circuit setup [Fig. 7(b3)], even though the spatial connectivity within each 2D layer remains completely homogeneous across the dislocation. These Floquet 0 and π modes were directly measured through impedance measurements [Fig. 7(b4)].

Time-modulated electrical circuits can also be used to demonstrate dynamical phenomena such as Thouless pumping.³⁸⁰ Traditionally, the Thouless pump describes the adiabatic, quantized transport of charge across a bulk gap,³⁸¹ as can be deduced from spectral flow arguments. In the regime of classical electrical circuits, charge quantization is not accessible, but spectral flow can still be measured since the Laplacian spectrum corresponds to impedance resonances. Stegmaier et al.¹⁰² demonstrated such spectral pumping in an electrical circuit with inductors that are effectively time-modulated with analog multipliers [Fig. 7(c1)]. As in Ref. 101, the time modulation is controlled by an external voltage signal, which can be customized at will—here, the modulation is chosen to yield the Aubry–André–Harper (AAH) model. By measuring the impedances within the circuit, eigenfrequency bands were reconstructed as a function of the periodically pumped phase. Indeed, the expected AAH spectral flow was observed under PBCs [Fig. 7(c2)], and topological in-gap states can also be seen under OBCs [Fig. 7(c3)]. It is important to note that, unlike the quantized charge transport predicted by the Kubo formula in quantum settings, there is no occupied ground state here, and the observed state pumping is purely that of the spectral band resonances.

Even in a static electrical circuit, techniques used to resolve Floquet dynamics can also be used to measure the time-dependent behavior. In the non-Hermitian corner localization experiment by Wu et al.,³⁷⁹ the 2D circuit lattice, hosting non-Hermitian second-order topological phases, incorporates INICs to implement asymmetric intra-cell couplings [Fig. 7(d1)]. The modular design of the circuit, featuring detachable INIC modules, allows switching between Hermitian and non-Hermitian configurations [Fig. 7(d2)]. In the Hermitian configuration, a Gaussian pulse injected into the center of the circuit spreads across the entire lattice, as shown by the time-resolved voltage profile [Fig. 7(d3)]. In contrast, in the non-Hermitian setting, the pulse localizes at a corner of the circuit [Fig. 7(d4)]. These time-resolved voltage responses demonstrate how electrical circuits can serve as practical platforms for observing dynamical physical phenomena in real time (also see Ref. 382).

Another noteworthy result in Ref. 379 is the demonstration of the generalized Brillouin zone using the Laplace transform rather than the Fourier transform applied to the measured voltage distribution. The non-Hermiticity renders the wavevector complex, making it infeasible to obtain the Brillouin zone directly from the Fourier transform. However, Wu et al. utilized the Laplace transform method to effectively recover the generalized Brillouin zone, where different onsite admittances correspond to different contours of the band structure. Such site-resolved field distributions are highly accessible in topolectrical circuits, which is crucial for studying higher-dimensional lattices. For example, in another work,³⁸³ Wu et al. observed degeneracy conversions, including Weyl point-to-nodal line conversion and nodal line-to-entangled nodal line conversion.

Lately, there has been much interest in the band structure and topological properties of hyperbolic lattices, which can be regarded as a vast generalization of regular lattices in negatively curved space.^384–395 In 2D Euclidean (flat) space, regular p-gons can be tessellated such that every vertex contains q branches, if (p − 2)(q − 2) = 4. This leaves only three possibilities {p, q} = {3, 6}, {4, 4}, {6, 3} corresponding to the triangular, square, and hexagonal lattices, respectively. Relaxing this constraint to (p − 2)(q − 2) > 4 results in an infinite variety of so-called hyperbolic lattices, which, however, need to be embedded in negatively curved space.³⁹⁶ While this requirement of negatively curved space embedding makes them challenging to realize in most metamaterial platforms, electrical circuit connections can describe any desired graph structure and thus emerge as promising physical platforms for hyperbolic lattices. This negative curvature fundamentally alters the propagation of modes, enriching the behavior of fundamental phenomena such as topological propagation.³⁹⁷ Due to the relative proliferation of boundary terminations, a hyperbolic lattice structure also alters the nature of the non-Hermitian bulk-boundary correspondences in the presence of asymmetric lattice hoppings.³⁹⁸ The ability of electrical circuits to simulate lattices with high coordination numbers is also helpful in the realization of higher dimensional lattices.^{50,105,114,129,130,399–401}

Lenggenhager et al.⁴⁰² achieved one of the first experimental demonstrations of hyperbolic lattices on a circuit board [Fig. 8(a)]. By mapping the hyperbolic plane onto the Poincaré disk, the hyperbolic lattice takes the form of a tree with increasingly fine branches toward the boundary of the unit circle. Impedance resonances measurements reveal the eigenmodes of the hyperbolic lattice, which are distorted due to the negative curvature. By resolving the phase information at each node relative to a reference voltage, the structure of these “hyperbolic drum” eigenmodes can be mapped out.

A key feature of finite hyperbolic lattices is that the number boundary sites scales exponentially with the radius (i.e., number of lattice generations), reminiscent of fractal lattices.⁴⁰⁴ As such, unlike in Euclidean lattices, boundary sites can dominate the Hilbert space and greatly alter bulk-boundary correspondences, even in Hermitian setups. Zhang et al.³⁸⁴ demonstrated the exponential proliferation of zero modes in a hyperbolic circuit [Fig. 8(c)]. These topological zero modes give rise to a large number of impedance resonances at the edge nodes, while no resonances occur in the bulk nodes.

This boundary sensitivity is reflective of the small-world connectivity of a hyperbolic lattice, a property that has inspired their application in holographic duality.^405–411 The high connectivity also gives rise to heightened susceptibility to lattice defects or perturbations. Pei et al.⁴⁰³ experimentally constructed a hyperbolic Haldane circuit and showed that a single defect in the bulk can induce boundary-dominated one-way propagation. By introducing polygonal bulk defects into the Haldane hyperbolic lattice, robust boundary propagation was observed, even when induced by a single point defect in the bulk [Fig. 8(d)]. While the presence of zero modes was verified through impedance responses, the one-way propagation of topological edge states was demonstrated using dynamical voltage analysis.

Electrical circuits are also well-suited for simulating quantum media, both of conventional topological materials and more unconventional setups, such as Refs. 383 and ^412–414. In the following, some experimental demonstrations are described, particularly those of models that are hard to simulate on other platforms.

One class of models that are particularly amenable to electrical circuit realizations are the square-root models. Given a parent Hamiltonian, the square-root of it is a lattice model in which two successive hoppings yield the same connectivity structure as the parent model (up to an unimportant constant shift).^415–428 Mathematically, they may look contrived and complicated, but they inherit various properties from their parent Hamiltonians, such as midgap topological modes.⁴²⁹ The number of modes, including topological modes, is doubled in these lattices.⁴³⁰ Electrical circuits are suitable physical platforms due to their versatile connectivity, and Song et al.⁴³¹ constructed a 3D square-root higher-order Weyl semi-metal by stacking 2D square-root lattices along the third dimension [Fig. 9(a)]. From the admittance spectrum, non-zero midgap topological hinge states were observed.

Electrical circuits are also useful in demonstrating unconventional band structures, particularly in higher dimensions where fabrication may be challenging for some other metamaterials. For instance, Weyl points are crossings in 3D band structures and can tilt over to form type-II Weyl points,^{51,69–71,114,412,443–447} exhibiting great sensitivity during the so-called Lifshitz transition. Li et al.¹⁶⁸ experimentally demonstrated type-II Weyl points in their 3D circuit [Fig. 9(b)]. By exciting bulk states and measuring node voltages, topological surface states were measured, exhibiting tilted bands with two group velocities in the same direction—key properties of type-II Weyl systems.⁴⁴⁸ 

Beyond band structures, 2D or 3D electrical circuit arrays can also simulate few-body physics and other phenomena. By mapping the 2-photon degrees of freedom onto a 2D lattice,^48,299,449 Zhou et al.⁴³² simulated interaction-induced flat-band localizations and topological edge states, tuning the effective interaction strength through circuit grounding. By measuring impedance responses and voltage dynamics in the time domain, they were able to resolve two-boson flatbands and topological edge states [Fig. 9(c)]. On a different note, Yang et al.⁴³³ simulated Wilson fermions—a lattice discretization conventionally used in the context of lattice QCD to avoid fermion doubling—in a circuit lattice, tuning circuit parameters to achieve different Wilson fermion states. Through impedance measurements, they reconstructed the chiral edge domain wall separating two circuits with contrasting fractional Chern numbers and observed localized corner states arising from two Chern lattices with opposite chiralities [Fig. 9(d)]. Wang et al.⁴³⁴ realized the elusive Hopf insulator in a 3D electrical circuit systematically implemented with so-called pseudospin modules. Although this lattice required long-range spin–orbit couplings, the pseudospin modules aided the realization of Hopf bands, including both surface and bulk bands [Fig. 9(e)].

Electrical circuits have also been used to simulate non-Abelian systems, in which physical outcomes depend on the sequence of operations, i.e., twists.^450–455 Various recent experiments have attempted to classically simulate operations that are mathematically equivalent to that experienced by anyonic quasiparticles.^456–461 Wang et al.⁴³⁵ demonstrated non-Abelian Anderson localization using a 2D electrical circuit that represents the quasiperiodic AAH model with non-Abelian gauge fields. Both localization and delocalization behaviors were observed through site-resolved impedance spectra and voltage dynamics [Fig. 9(f)] (also see the experimental realization of non-Abelian inverse Anderson transition in Ref. 462).

Circuits are also particularly suitable for demonstrating the effects of lattice defects, since nodes are connected at will. Xie et al.⁴³⁶ experimentally demonstrated disclination states and their correspondence to real-space topology. Localized topological states at the disclinations—defects in rotational symmetry⁴⁶³—were observed in impedance maps and voltage distributions at the lattice nodes [Fig. 9(g)] (also see Refs. ^464–466).

The notion of bound states in the continuum (BICs),⁴⁶⁷ which are localized and isolated from the environment, is rapidly extending beyond photonics,^468–470 ring resonators,⁴⁷¹ and into electrical circuits.^428,458,472 Sun et al.¹⁰⁴ simulated boundary-localized many-body BICs by mapping the Fock states of three bosons into non-linear circuit networks. The three circuit nodes corresponding to the three bosonic states are connected by capacitors, and each node consists of a self-feedback module controlling the effective interaction strength. These modules, which involve op-amps and analog multipliers, allow for the simulation of few-boson BICs with different effective interaction strengths [Fig. 9(h)].

Blessed with the freedom in implementing any desired boundary condition, electrical circuit arrays can assume unconventional boundary conditions not possible in materials and most metamaterials, such as higher-dimensional real projective planes or even non-orientable manifolds, with the exception of some photonic systems.⁴⁷³ Combined with nontrivial momentum-space higher-order topology arising from the octupole moment of the bulk, Qiu et al.⁴³⁷ experimentally demonstrated a 3D lattice [Fig. 9(i)], which exhibits defect-like higher-order corner states through impedance analysis. The versatility of electrical circuit has enabled the realization of such unconventional states resulting from the interplay of non-trivial real-space and momentum-space topology.^47,474

Since electrical circuits commonly involve resistive and active elements, they can readily demonstrate how non-Hermiticity interplays with topology without much additional complication. Xie et al.⁴³⁸ demonstrated the non-Hermitian version of the graphene lattice, which exhibits two non-Hermitian Dirac cones, one experiencing amplification and the other experiencing decay. Site-resolved band measurements reveal unidirectional kink states^475,476 with dissimilar profiles at the two valleys [Fig. 9(j)]. Involving asymmetrical real next-nearest-neighbor couplings effectively results in a non-Hermitian single Dirac cone, leading to valley polarization^477–480 with a large bandwidth (see also Ref. 481).

The introduction of non-Hermiticity can also greatly alter the locality of topological modes. While topological modes are typically strongly confined to the boundaries,³⁰ non-Hermitian skin pumping in the opposite direction can delocalize them while preserving their eigenspectrum, in this case midgap zero-eigenvalues.⁴⁸² These delocalized topological states, still protected by lattice symmetries, can extend through the bulk, enabling their robustness to be harnessed not only at the boundaries but also within the bulk.^483,484 Lin et al.⁴³⁹ demonstrated these extended states in 1D, 2D, and 3D circuits by tuning the non-reciprocal coupling strength [Fig. 9(k)]. At a critical strength determined by both the topological and skin localization length, the topological modes become extended, a phenomenon that was also studied in Refs. 38 and ^485–488.

Beyond hyperbolic lattices (refer to Sec. IV C), electrical circuits can also simulate more esoteric network structures, such as fractals. On such networks, boundary terminations occur at various length scales, and topological modes localized unconventionally in the form of extended bulk states and inner states.^{404,489–492} He et al.⁴⁴⁰ demonstrated fractal topological circuits, which exhibit a rich spectrum of topological edge and corner modes. Reminiscent of hyperbolic lattices, the numbers of such boundary modes increase rapidly with system size, as directly observed through impedance analysis [Fig. 9(l)].

Topolectrical circuits can simulate not only classical systems but also quantum setups.^493–495 By mapping the Hilbert space of few-qubit systems to the nodes of suitably designed circuits, Zou et al.⁴⁴¹ demonstrated [Fig. 9(m)] the mathematical equivalences of Majorana-like edge states, T-junctions, two-qubit unitary operations, and Grover’s search algorithm. Through their resistor-capacitor-based circuit with segmented fixed resistances, braiding processes can be simulated without significant loss. Zhang et al.⁴⁴² demonstrated another example of quantum algorithm simulation using classical circuits. They simulated a two-player quantum game based on AND-OR tree structures [Fig. 9(n)]. Relays were employed to disconnect each node simultaneously after applying the initial voltage, allowing the system to evolve from the initial state.

Although most topolectrical circuit simulations of condensed matter have been built on printed circuit boards, a few experiments have demonstrated their implementation in integrated circuits (ICs). For instance, Iizuka et al.¹³¹ experimentally demonstrated topological Kitaev interface states⁴⁹⁶ in high-frequency ICs [Fig. 9(o)]. Through two-point impedance measurements, this IC realization successfully captured both topological and trivial phases at frequencies around 13 GHz. These IC realizations, with their much higher degree of miniaturization, demonstrate the great potential for topolectrical circuits in scalable high-frequency applications.

While various experiments have simulated knotted band touchings in 3D parameter space, in both Hermitian and non-Hermitian (exceptional knots) contexts,^105,204,497 realizing these knots in momentum space has the benefit of accessing their topological surface states at real-space terminations. Electrical circuits prove particularly suitable due to the ease in implementing long-range couplings, which are essential for the construction of nodal knots in momentum space.^{60,178,193,196,215,217,498–501} In general, the topological surface states of nodal knots lie within the interior of the 2-dimensional surface Brillouin zone projections of these knots, and are thus known as “drumhead” surface states.⁶⁰ 

As simulated and experimentally demonstrated in Ref. 502, nodal structures can be resolved from the admittance eigenvalues [Fig. 10(a)]. These nodal knots form intricate patterns in the 3D momentum space, while the corresponding drumhead surface states appear as projected flatbands in the surface admittance spectrum. While the edge-terminated direction was implemented in real space, the two other directions in the plane of the drumhead surface states were parametrically tuned by adjusting the inductance of the inductors to correspond to different momentum values [Fig. 10(b)]. The inductance was varied using ferrite rods or shorted wire loops to increase or decrease inductance, respectively [Fig. 10(c)], effectively reducing the undesired effects of parasitic and component uncertainties. By utilizing machine learning-powered optimization and microcontrollers to fine-tune the setup, the drumhead regions could be resolved with a minimal number of measurements [Fig. 10(d)].

Topolectrical circuit simulation of a condensed matter lattice usually entails the design, construction, and voltage measurement of a large number of nodes. To streamline the process, some studies have employed machine learning (ML) approaches, in particular deep learning, for optimizing their design and measurement process. In other contexts, ML approaches have already proven themselves as powerful tools for optimization and prediction based on incomplete information, for instance extrapolating the dynamics of complex systems and optimizing their design.^506–508 As such, they are particularly effective in addressing practical challenges such as managing a large number of circuit variables, designing printed circuit boards with minimal effective node-to-node connections, and streamlining circuit measurements.

To illustrate, fully acquiring the Laplacian spectrum of a circuit requires N × N impedance measurements in principle, where N is the total number of nodes. For large N, this process becomes increasingly tedious, especially in higher-dimensional circuits. To address this, Shang et al.¹⁶⁹ introduced the physics-graph-informed machine learning approach for reconstructing the circuit spectrum with incomplete data [Fig. 11(a1)], toward a similar objective as Ref. 128. Using physics-graph-informed ML, they conclusively demonstrated second-order NHSE spectra through impedance measurements only between selected ports [Figs. 11(a2) and 11(a3)]. Aided by the K-means clusters algorithm, only N²/K measurements were required, highlighting how embedding physics-informed priors into ML frameworks can significantly enhance data interpretation and analysis in topolectrical circuit experiments.

In addition to data analysis, deep learning can aid the design of the mathematical model itself. Chen et al.⁵⁰⁹ utilized multilayer perceptrons and convolutional neural networks to predict topological invariants and the circuit response. They simulated the topolectrical Chern circuit⁴⁰ [Fig. 11(b1)] with 1000 different sets of circuit component parameters to train their model. The topolectrical Chern circuit features chiral voltage edge propagation when a Gaussian pulse is injected at an edge node [Fig. 11(b2)]. Their trained model demonstrated the approximate phase diagram [Fig. 11(b3)] and captured aspects of the temporal evolution of chiral voltage propagation, even in the presence of defects [Fig. 11(b4)].

Chen et al.⁵¹⁰ proposed a framework that integrates text, image, and spectral data to assist in the design of two-dimensional SSH circuits. Their bidirectional collaborative design framework enables both forward prediction of measured quantities from the described circuit structure and inverse design of topolectrical circuits based on textual descriptions. For instance, when provided with an image of a fabricated 2D SSH circuit [Fig. 11(c1)], the trained model inferred circuit properties that aligned with experimental measurements [Fig. 11(c2)]. Conversely, given a target voltage profile {e.g., edge-localized voltage [Fig. 11(c3)]}, the model suggested a corresponding circuit design [Fig. 11(c4)]. The fabricated circuit based on this predicted structure exhibited a voltage profile similar to the intended design.

Electrical circuits, utilizing components such as conventional elements, varactors, op-amps, analog multipliers, and memristors,^511,512 enable the realization of features and phenomena that are hard to simulate in most other platforms, such as higher-order topological phases,^78,513 Floquet setups,^101,136 hyperbolic lattices,⁵¹⁴ and synthetic dimensions.⁵¹⁵ Recent advances have expanded the scope of electrical circuit experiments to include the simulation of flatbands,^516,517 interaction-induced relativistic effects,⁵¹⁸ quantum valley Hall effects,⁵¹⁹ exceptional Landau quantization,⁵²⁰ Hopf bundles,⁵²¹ inverse Anderson transitions,⁵²² and non-Abelian topological bound states,⁵²³ offering unprecedented control and tunability. These demonstrations highlight the adaptability of topolectrical circuits in bridging theoretical models with physical realizations. Their direct observability and scalability present unique opportunities for uncovering new physics. Furthermore, machine learning and deep learning have enhanced the potential of electrical circuits, enabling their efficient analysis, prediction, and design, even in systems with complicated topological features or incomplete data.¹⁶⁹ 

The integration of electrical circuits with advanced computational techniques and emerging technologies⁵²⁴ promises to deepen our understanding of condensed matter phenomena and push the boundaries of experimental physics.^145,146,525 Their adaptability^62,502 to simulate higher-dimensional systems,⁵⁰ such as those seen in magic-angle Moiré circuits,⁵²⁶ opens new avenues for investigating uncharted areas, including relatively esoteric condensed matter systems.^92,527–532

Potential future directions in topolectrical circuits include the integration of tunable active elements, such as digitally controlled inductors and analog multipliers, enabling real-time parameter adjustments for Floquet circuit implementations. Although not yet widely available commercially, memristors—often considered the fourth passive circuit element^512,533,534 alongside capacitors, inductors, and resistors—could facilitate memory-based physical phenomena such as axon-like transmissions⁵³⁵ or protected topological states even with a broken symmetry.⁵¹¹ The incorporation of other nonlinear elements, such as transistors and various types of diodes, remains an open avenue that could simplify circuit implementations and broaden potential applications. These advancements may transition topolectrical circuits from proof-of-concept research to real-world applications in signal processing, computation, and sensing. Further refinement and expansion of the scope of topolectrical circuits could lead to uncovering fundamental insights and advance the development of practical applications in condensed matter and beyond.

This work was supported by the Ministry of Education (MOE) under Tier-II Grant Nos. MOE-T2EP50121-0014 (NUS Grant No. A-8000086-01-00) and MOE-T2EP50222-0003 (NUS Grant No. A-8001542-00-00), as well as MOE Tier-I FRC Grants (NUS Grant Nos. A-8000195-01-00 and A-8002656-00-00).

The authors have no conflicts to disclose.

Haydar Sahin: Conceptualization (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Mansoor B. A. Jalil: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Writing – review & editing (equal). Ching Hua Lee: Conceptualization (lead); Formal analysis (lead); Supervision (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.