Topoelectric circuits: Theory and construction

We highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors. In particular, by supplementing each node with $n$ subnodes, where the phases of the current and voltage are the $n$ distinct roots of \emph{unity}, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the \emph{shift capacitor coupling} between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, \emph{topoelectric circuits}, where the robust zero-energy topological boundary modes manifest through a large boundary impedance, when the circuit is tuned to the resonance frequency. The simplicity of our circuit constructions is based on the fact that the existence of the boundary modes relies only on the Clifford algebra of the corresponding Hermitian matrices entering the Hamiltonian and not on their particular representation. This in turn enables to implement a wide class of topological models through rather simple topoelectric circuits with nodes consisting of only two subnodes. We anchor these outcomes from the numerical computation of the on-resonance impedance in circuit realizations of first-order ($m=1$), such as Chern and quantum spin Hall insulators, second- ($m=2$) and third- ($m=3$) order topological insulators in different dimensions, featuring sharp localization on boundaries of codimensionality $d_c=m$. Finally, we subscribe to the \emph{stacked topoelectric circuit} construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac and Weyl semimetals, respectively displaying surface and hinge localized impedance.

In this respect, topoelectric circuits, made of rather simple capacitance and inductance elements, yield a readily available route for the realization of a plethora of topological phases . The mapping between the response of a circuit to a locally applied voltage and a tight-binding Hamiltonian is facilitated by the frequencydependent admittance matrixĴ(ω). Its form depends on the connectivity of the circuit elements through the nodes, which is used to engineer the parameters of a hopping model. In this platform the impedance between * bitan.roy@lehigh.edu the two nodes, related to the admittance matrix, can be used to locally detect the boundary modes [33], and thus serves as an electric circuit analogue of a tunneling probe for topological crystals.

A. Summary of results
Motivated by these developments, in this paper we use a general framework for constructing arbitrary tightbinding models in topoelectric circuits to realize various gapped and gapless topological phases in one, two and three spatial dimensions. We first rederive in a rather transparent and independent way the known result [30,31,38] that a tight-binding model with arbitrary hopping amplitudes and phases can be constructed by extending a node in an LC circuit (see Fig. 1) to include n subnodes with the same amplitude of the input voltage but the phase factors representing n different nth roots of unity. This method relies on the fact that each of the n inequivalent connectivity configurations between the subnodes, realized with identical capacitors, directly maps into the phase factor equal to one of the nth roots of unity, see also Figs. 2 and 3 where, respectively, the cases n = 2 and n = 4 are displayed.
We show that a wide range of topological models can be realized in rather simple two-subnode topoelectric circuits. To this end, we use the fact that the existence of the topological modes relies only on the Clifford algebra of the Hermitian matrices entering the Hamiltonian, but not on their representation. This enables to implement arXiv:2008.11202v2 [cond-mat.mes-hall] 15 Sep 2020 various topological models so that the hopping elements are purely real. In turn, the corresponding topoelectric circuit can be constructed by supplementing each node (representing a lattice site) by only two subnodes, between which the phases of current and voltage differ by a factor of exp(iπ) = −1.
This method is exemplified by constructing circuit realizations of several toy models for topological phases. Some of them have already been discussed in the literature, as detailed below, and even though our explicit circuit constructions are often different, we arrive at qualitatively similar results. Furthermore, our framework allows for the topoelectric implementation of the models whose circuit realizations have not been achieved so far.
We start the discussion with circuit implementation of the paradigmatic one-dimensional SSH model (Fig. 4), experimentally realized in this platform in Ref. [33], and two-dimensional BHZ model (Fig. 6), featuring, respectively, the localized end-point and edge topological modes. We then compute the site resolved on-resonance impedance to show the hallmark boundary modes, as displayed in Figs. 5 and 7, and also to infer the global phase diagram of the models, shown in Fig. 8 for the square lattice topoelectric Chern insulator, previously studied on brickwall or honeycomb circuits [39]. The realization of the quantum spin Hall insulator (QSHI) is displayed in Fig. 9. The outlined general method can be also used to construct the phases beyond the tenfold way [70], such as second-order topological insulator in two dimensions (Fig. 10) [32] and third-order topological insulator in three dimensions (Fig. 11) [47], both featuring sharp corner localized on-resonance impedance. Besides demonstrating sharp corner localization of onresonance impedance, we also numerically demonstrate its sublatice polarization stemming from the representation of the generator of the particle-hole symmetry, which still remains to be observed in experiments.
We also show how the hierarchy of the topological insulators can be realized in the topoelectric platform in the case of two-dimensional second-order topological insulator by breaking the four-fold (C 4 ) rotational symmetry of the parent first-order phase (Fig. 12) and explore a related antiunitary symmetry protected generalized second-order topological insulator on electrical circuit (Fig. 13). Finally, various two-and threedimensional topological nodal semimetals can be implemented through stacking of the SSH circuits, as shown in Fig. 14(a) for two-dimensional Dirac semimetal (see also Ref. [33]), with its hallmark boundary impedance displayed in Fig. 14(b). The surface impedance corresponding to the drumhead states for the unknotted nodalline semimetal is shown in Fig. 14(c), while the Fermi arc surface states for the three-dimensional Weyl and Dirac semimetals, respectively obtained by stacking twodimensional Chern and quantum spin Hall isulators, are displayed in Fig. 15 and 16. Finally, the one-dimensional hinge mode impedance for the quadrupolar (or secondorder) Dirac and Weyl semimetals are shown in Figs. 17 Here C ab and L ab are the capacitance and inductance between these two nodes, respectively. The capacitance and inductance between the node a and the ground (G) are Ca and La, respectively. and 18, respectively.

B. Organization
The rest of the paper is organized as follows. In Sec. II we present the general framework for constructing tightbinding models with arbitrary parameters. In Sec. III we apply this method to the realization of the first-order topological phases in SSH and BHZ models, as well as to the topological Chern insulator. Sec. IV is devoted to higher order topological insulators in two and three spatial dimensions. In Sec. V, we present the construction of the C 4 -symmetry breaking second-order topological insulator from the quantum spin Hall insulator and a related antiunitary symmetry protected generalized higher-order topological insulator. In Sec. VI, we implement various topological nodal semimetals in the topoelectric platform, such as two-and three-dimensional Dirac semimetals, and three-dimensional Weyl, nodalline and quadrupolar Dirac semimetals. In Sec. VII, we summarize and discuss our results.

II. GENERAL SETUP
In this section, we highlight a general procedure for implementing an arbitrary tight binding model in LC electrical circuits. For completeness, we first discuss the relation between the admittance and the experimentally measurable impedance [33]. In particular, we lay out the connection between the divergence of this observable and the existence of the admittance zero modes, which in turn captures the hallmark of topological phases in the circuit setup. Second, we rederive a general rule for devising a tight-bonding model with an arbitrary phase of the hopping in an LC circuit network [30,31,38] in an independent and transparent approach, which will be subsequently used to demonstrate the realization of different topological phases in topoelectric circuits.

A. Admittance and impedance matrices
To make the connection between the electric circuit and a tight-binding lattice model, we start by considering a representative LC circuit, shown in Fig. 1. The electric current that flows into the ground and the voltage at a particular node a of an LC circuit are related by the Kirchhoff's law according tȯ Here,Ẋ ≡ dX/dt, the roman letters a, b, . . . label the nodes in the circuit, while I a and V a denote the current and voltage at the node a, respectively. The capacitance and inductance between nodes a and b are respectively C ab and L ab , while C a and L a represent these parameters between the node a and the ground.
To characterize the response of an LC circuit to an applied voltage, we Fourier transform Eq. (1) to obtain a nonlocal relation between the voltage and the current at frequency ω The admittance matrix J ab (ω) reads as where and N ab and W a depend on the network structure of the circuit and on the grounding, respectively. The experimentally measurable quantity in this context is the impedance, defined through the response of the circuit to an applied current. In particular, we are interested in the voltage response when a current I a (ω) ≡ I(ω) is injected into a node a, yielding an outgoing current from node b, given by I b (ω) = −I a (ω) (following the Kirchhoff's sign convention), and with no current inflow or outflow at any other nodes. The voltage difference between nodes a and b then defines the two-point impedance between these two nodes Taking the spectral form of the admittance matrix and introducing its regularized inverse matrix defined to exclude the zero eigenmodes of J ab , the impedance Z ab reads as [71][72][73] [33] Therefore when at least single eigenvalue j n of the admittance matrix is very small, we expect a large response in the impedance Z ab . Particularly, in the circuit realizations of topologically nontrivial phases, such a large signal corresponds to the gapless modes at the boundary of the system. Furthermore, in spatially extended circuits with translational symmetry in the bulk, the zero modes of the admittance are localized at the boundary, as we will show later in concrete examples, and directly identify the topological nature of the system.

B. Arbitrary hopping phases
To establish the correspondence with tight-binding models, we first notice that the admittance matrix in Eq. (3) up to the factor of iω is Hermitian, namely J ab (ω) = iωH ab (ω), with H ab (ω) = H * ba (ω). Furthermore, if the nodes are thought of as the lattice sites, the matrixĤ(ω) can be associated with a Hamiltonian of a tight-binding model on this lattice, with the off-diagonal elements that correspond to hoppings, while the diagonal ones representing the on-site chemical potential. Hereafter, the matrixĤ(ω) is referred to as the Hamiltonian. As can be seen from the form of the admittance matrix in Eq. (3), the corresponding Hamiltonian is completely real. Nevertheless, a hopping with an arbitrary complex phase can be implemented in this setup, as shown in (d) Schematic representation of the one-shift capacitor connection from (c). Two-shift, three-shift, and no-shift or direct capacitor connections between nodes a and b are schematically shown in (e), (f), and (g), respectively. The effective hopping elements between these two nodes or sites are then Refs. [30,31,38], which we rederive in a rather straightforward and independent manner below.
To this end, we first notice that the voltage and current at a node are defined up to a phase factor. This observation allows one to enrich the node structure to include more subnodes within the same node with the same magnitude of the voltage and current, but the phase factors shifted with respect to each other, see Figs. 2 and 3.

Two subnodes
To illustrate this protocol, let us start with the simplest example of a node featuring the two subnodes (±) at which the voltages are shifted by a relative phase factor exp[iπ] = −1, see Fig. 2. Two possible configurations of connection between the neighboring nodes, shown in Figs. 2(a) and 2(b), correspond to the admittance matrices with the following effective Hamiltonian HereĈ α = Cτ α , α = 0 (α = 1) corresponds to the circuit configuration in Fig. 2(a) [ Fig. 2(b)], and ⊗ represents direct or tensor product. The Pauli matrices {σ ν } and {τ ν } act on the node and the subnode spaces, respectively, where ν = 0, · · · , 3. Here, the diagonal part in the node space iŝ where C g and L g are the capacitance and inductance of the grounding elements (displayed in Fig. 1, but not shown in Fig. 2), respectively, which can tune the circuit to the resonance. Since we are interested only in the configuration with the voltages tuned out of phase on the two subnodes within the same node, to obtain the hopping and the on site chemical potential the subnode matrix is projected onto the relevant subnode subspace spanned by the vector v 2 = 1/ √ 2[1, −1] . The projector onto this subspace P 2,ij = v 2,i v * 2,j explicitly readŝ Consequently, the hopping matrix element for the two configurations in Fig. 2 respectively read for α = 0 and 1, yielding after using Eq. (10). On the other hand, the on site chemical potential for these two configurations are As a consequence of Eq. (10), we find for both circuit network configurations µ 0 = µ 1 ≡ µ, where Obviously, by applying different grounding elements on the two sites, the difference of the onsite chemical potentials can be realized. Consequently, by periodically repeating two subnode circuit configurations an arbitrary tight-binding model with pure real hopping can be realized in topoelectric circuits. In fact, as we show, a wide class of topological models can be cast in such a form since the topological features do not depend on the representation of the Hermitian matrices entering the Hamiltonian, but only on their Clifford algebra. Finally, by generalizing the above construction to include more than two subnodes on a single node a hopping with an arbitrary phase factor can be obtained.

Four subnodes
Next we consider the four subnode configurations, as shown in Fig. 3(a), where the phase differences between the subnode a(+) and the rest of the subnodes [including the subnode a(+)] are ±1, ±i. Therefore the relevant subnode subspace is spanned by the vector v 4 = When |λ/t| < 1 the system is in the topological phase. (b) The hopping pattern in the limit t = t1, see Eq. (21). (c) A circuit realization of the SSH model in the limit t = t1, with two subnodes at each node or orbital. Two nodes residing within the dashed rectangle constitute the unit cell of the SSH model. The corresponding capacitances in the circuit are λ (red) and t (blue). The inductors connecting each subnode to the ground (not shown explicitly here, see Fig. 1) and two subnodes at each node are all with the same inductance equal to L. (d) The circuit realization of the SSH model with an effective negative nearest-neighbor hopping amplitude due to the crossed or one-shift connection (see Fig. 2), yielding access to the parameter regime λ/t < 0, which is not possible in a one subnode structure with only capacitor connections, see also Ref. [33].
(1/2)[1, −i, −1, i] , graphically represented in Fig. 3(b). The corresponding projector P 4,ij = v 4,i v * 4,j , with the explicit form given bŷ The hopping element between the nodes a and b is then the projection of the connectivity matrix between the subnodes onto the subspace spanned by the vector v 4 , namely t α = Tr(P 4ĈαP4 ), (18) withĈ α as the connectivity matrices for four possible configurations, schematically shown in Figs. 3(d)-(g). For instance, the circuit in Fig. 3(c) corresponds to the graph in Fig. 3(d), and the corresponding connectivity matrix isĈ where C is the capacitance between two neighboring nodes. Then Eq. shows that the corresponding on site potential when the grounding elements on all four subnodes are the same, is given by µ = C g − 1/(ω 2 L g ) − 1/(ω 2 L). Again, by choosing different grounding elements for different nodes, one can tune difference of the on site chemical potentials. Therefore, the four subnode LC circuit can realize an arbitrary tight-binding model with hopping phases being four fourth roots of unity, namely exp[2πim/4], with m = 1, 2, 3, 4, i.e. either purely real or purely imaginary.

n subnodes
This construction can now be further generalized to a hopping with phase factors Φ n,m = exp[2iπ(1 − m)/n], with m = 1, ..., n, which are the nth roots of unity. Now each node contains n subnodes. The hopping elements are obtained by projecting the connectivity matrices onto the subspace spanned by the n-component vector v n , with the elements v n,m = Φ n,m . The matrix corresponding to the "one-shift" in the connectivity 1 → n, 2 → 1, 3 → 2, 4 → 3, ..., n → n − 1 generates the hopping equal to t (1) = exp[2iπ/n]. See Appendix A for the proof. More generally, an s-shift results in the hopping phase equal to t (s) = exp[2iπs/n], as also shown in Appendix A.
In the following, we apply this general protocol to construct various insulating and gapless or nodal topological phases, both more conventional first-order ones and the higher-order ones, in one, two, and three spatial dimensions.
The hallmark of any topological phase of matter is the bulk-boundary correspondence, which assures the existence of topologically protected robust boundary modes [4][5][6][7][8][9][10][11][74][75][76][77][78][79]. Typically, a d-dimensional topological phase supports such modes that reside on (d − 1)dimensional boundaries, which are also characterized by the co-dimension d c = d − (d − 1) = 1, yielding firstorder topological phases. Some well known examples of topological modes include the endpoint modes in onedimensional SSH topological insulator, one-dimensional edge modes for two-dimensional Chern and quantum spin Hall insulators, and the surface states of threedimensional topological insulators. In this section we demonstrate realizations of some of these insulating phases in topoelectric circuits. In Secs. VI A, VI B and VI C we demonstrate circuit realizations of first-order nodal or gapless topological phases.

A. Su-Schrieffer-Heeger model
The Su-Schrieffer-Heeger (SSH) model is a paradigmatic example of a topological state in one dimension [1][2][3]. Here we show its circuit realization within the general construction principle, discussed in the previous section. The Hamiltonian of the SSH model in the real space takes the following form where the annihilation operators a i and b i act on the orbitals A and B localized at the site i, respectively. The hopping amplitudes λ andt 1,2 = (t±t 1 )/2 are purely real, see Fig. 4(a) and 4(b). The corresponding momentum space representation of the SSH model reads as where the Pauli matrices {τ µ } with µ = 0, · · · , 3 act on the orbital space. For the range of parameters −1 < λ/t < 1, the SSH model is in the topological regime and features a localized zero mode at each end of the system. Furthermore, the matrix τ 3 anticommutes with the Hamiltonian H mom SSH and ensures the spectral symmetry. Consequently, all zero-energy modes are eigenstates of the matrix τ 3 , and as such they are completely localized on either of the two orbitals. By contrast, the system becomes a trivial insulator when |λ/t| > 1. This model belongs to class BDI in the ten-fold classification [70].
The form of the Hamiltonian in Eq. (20) in terms of only real hopping parameters allows a realization of the SSH model through an electric circuit with two nodes corresponding to the two orbitals per lattice site. Furthermore, when each node contains two subnodes, it allows access to the parameter regimes λ/t > 0 as well as λ/t < 0, which is not possible in a one subnode structure with pure capacitor (or inductor) connections, see (a) On-resonance logarithm of the impedance ln(Z) in an SSH circuit of linear dimension = 10 with input point on the orbital A at site 1 and output point on the orbital B at various sites of the circuit for λ/t = 0.6 (top) and λ/t = −0.6 (bottom). (b) Same as (a), but with the output point on the orbital A at various sites of the circuit. Frequency dependence of the impedance in an SSH circuit of linear dimension = 10 for (c) λ/t = ±0.55 (topological phase) and (d) λ/t = ±2.00 (trivial phase). Here we compute the impedance with the orbital A at site 1 as the input point and the orbital B at site 10 as the output point. While the peak (red dots) of impedance in (c) at the resonance (ω = ωR = 1/ L(λ + t)) indicates the topological nature of the state with endpoint zero modes, the dip (blue dots) at the resonance in (d) confirms that the phase is a trivial insulator. Note that the SSH model has been studied in topoelectric circuit extensively in Ref. [33], as well as in other circuit setups [35,37,42,43,45]. Here we discuss this model for the sake of completeness and to establish generic features of a topoelectric circuit. Now consider a periodic SSH circuit [see Fig. 4(a)] with the Hamiltonian matrix,Ĥ SSH (ω) ≡Ĵ(ω)/iω, which according to Eq. (3), is given bŷ It can be seen that this Hamiltonian matrix represents an SSH Hamiltonian with an overall shift in chemical potential equal to λ + t − 1 ω 2 L . Therefore, the SSH circuit is at resonance when the a.c. frequency ω = ω R = 1/ L(λ + t). We notice that the two zero-energy modes localized on one particular orbital mix due to a finite size effect. Because of such an orbital mixing between the zero-energy modes, the measured impedance with one orbital (or sublattice) as the input point and the other one as the output point shows a large orbital selective on-resonance peak in the topological regime. This mechanism for the observed large orbital or sublattice selective on-resonance topological impedance is operative on all the topoelectric circuits we discuss in this paper. Furthermore, in a finite system zero energy states also split symmetrically about the zero admittance eigenvalue. Therefore the impedance between the same sublattice (or same orbital) at two sites is extremely small, as can be seen from Eq. (9), which approaches zero with the increasing system size. On the other hand, the impedance between two distinct sublattices or orbitals at the two ends of the circuit is large, since the zero admittance state is predominantly localized on only one of the two orbitals at a given end. These outcomes are demonstrated through a numerical simulation of the SSH circuit in the topological and trivial phases, shown in Fig. 5(a) and 5(b), respectively. One can also see that the impedance grows with the separation between the two sites, consistent with the topological mode being localized at the endpoint of the system.
To further corroborate the topological nature of the circuit, we measure impedance over a wide range of frequencies between orbital A from site 1 and orbital B of site 10, located at two opposite ends of the circuit, see Fig. 5(c). The peak of the impedance at the resonance frequency (ω = ω R ) signals that the system is topological when |λ/t| < 1. On the other hand, for |λ/t| > 1, the dip in the impedance at the resonance frequency is a signature of the topologically trivial nature of this phase, as shown in Fig. 5(d).

B. Chern insulator
We now consider the circuit realization of the twodimensional Chern insulator. The corresponding Hamiltonian in momentum space is given by Pauli matrices {τ µ } with µ = 0, · · · , 3 act on two orbitals in a unit cell. While t and t 0 are the inter-unit cell hopping parameters, m denotes the intra-unit cell hopping. This model breaks the time-reversal symmetry. The system is in the topological regime when −2 < m/t 0 < 2, describing Chern insulator with one-dimensional edge mode. By contrast, the system describes trivial insulators when |m/t 0 | > 2. The band gap closing between the Chern and trivial insulators occurs at m/t 0 = ±2. On the other hand, the transition between two distinct Chern insulators takes place at m/t 0 = 0, also through a band gap closing. We note that the system possesses an antiunitary particle-hole symmetry represented by the operator A = τ 3 K which anticommutes with the Hamiltonian of the system H CI , where K is the complex conjugation. Consequently, the zero-energy edge mode of the Chern insulator is an eigenstate of the antiunitary operator A.
Since the unitary part of A is τ 3 , the edge modes reside on either one of two orbitals. Previously a Chern insulator has been constructed in a brickwall or honeycomb topoelectric circuit [39]. Here we realize Chern insulator in a square topoelectric circuit [see Eq. (23) and Fig. 6] and besides probing its topological edge mode [see Fig. 7], we also construct its phase diagram [see Fig. 8].
The tight-binding Hamiltonian in Eq. (23), according to the previously discussed correspondence between the hopping amplitudes and the circuit elements, is realized on the circuit consisting of the components shown in Fig. 6. These elements are then connected so that the two-dimensional circuit network features a two-node (A and B) unit cell (the dashed box in Fig. 6). Since the lattice model H CI involves both real and imaginary hopping elements in the real space, each node possesses four subnodes. Then the subnodes from the nearest-neighbor nodes are connected by capacitors with appropriate capacitance following the general prescription discussed in Sec. II B 2, see also Fig. 3. All four subnodes of A and B nodes are grounded with inductor of inductance L, so that the resonance frequency of the Chern circuit is ω R = 1/ √ Lm. The numerical computations of the impedance are performed for t = t 0 = 1.
To capture the topological edge modes in a Chern topoelectric circuit, we choose either A or B node from one of the four corners of the circuit as the input point and numerically compute the impedance with B or A node as the output point. For a fixed input A or B node, we scan all the B or A nodes in the system to capture the spatial resolution of the on-resonance impedance. This measurement is then repeated for different corner choices of the A or B nodes. The results are summarized in Fig. 7. We find that only when m < 2 the impedance increases sharply at the edges of the circuit, signaling the appearance of the boundary modes, which is consistent with the system becoming a Chern insulator. Also when we scan the impedance over a wide range of frequency, it shows an on-resonance (ω = ω R ) peak only when the system is in the topological regime. On the other hand, in the trivial phases the on-resonance impedance shows a dip. These features are qualitatively similar to the ones we previously reported for the SSH chain, see Figs. 5(c) and 5(d). Hence, we do not display these results in the paper.
Finally, we construct the global phase diagram of the model H CI from the scaling of the on-resonance impedance (Z) as a function of m, see Fig. 8. While in the Chern insulator phases Z is finite, it (almost) vanishes in the trivial insulating phases. Finally, at the transition point between two Chern insulators (m = 0) the impedance Z (almost) vanishes. Therefore, measurement of the on-resonance impedance in topoelectric circuit can be instrumental not only to identify gapless topological modes but also to construct the global phase diagram of various topological models, and the topological phase transitions therein.

C. Quantum spin Hall insulator (QSHI)
Next we focus on the lattice model of QSHI and demonstrate its realization on topoelectric circuits. The corresponding tight-binding model takes the following form in the momentum space where Γ are mutually anticommuting four-component Hermitian matrices. When the Γ matrices belong to the representation one can identify two sets of Pauli matrices {σ µ } and {τ µ } operating on the spin (↑ and ↓) and sublattice or orbital (A and B) indices, respectively, with µ = 0, · · · , 3. The above model is in the topological regime for |m/t 0 | < 2, where it describes a QSHI supporting counter-propagating one-dimensional edge modes for opposite spin projections. However, topology of the above quadratic Hamiltonian is insensitive to the representation of the Γ matrices. We judiciously commit to the following representation such that in the real space all the hopping elements associated with H QSHI are completely real. This is so, because in this representation purely imaginary (real) Γ matrices multiply the sine (cosine and real constant) functions. In the following sections, we will subscribe to such representation whenever possible. In this representation, two sets of Pauli matrices {σ µ } and {τ µ } respectively operate on the orbitals or sublattices (C,D) and (A,B). Also note that H QSHI anticommutes with the unitary operator Γ 5 , which in turn generates the spectral or particle-hole symmetry of the system. Consequently, the one-dimensional edge modes at zero energy are eigenstates of Γ 5 , and reside either on (A,B) or (C,D) sublattices. By virtue of committing to such representation of the Γ matrices, we can engineer H QSHI in an electric circuit by supplementing each sublattice or node by two subnodes (consult Sec. II B 1 and see Fig. 2). A concrete circuit realization of H QSHI is shown in Fig. 9(a). By numerically computing the on-resonance (ω = ω R = 1/ √ Lm) impedance, we confirm that only in the topological regime impedance is sharply peaked around the edges of the circuit. Furthermore, by performing node or sublattice resolved computation of the on-resonance impedance, we find that the edge mode is either localized on the sublattices (C,D) [see Fig. 9(b)] or (A,B) [see Fig. 9(c)]. Later we will show (in Sec. V A) how one can break the discrete four-fold (C 4 ) rotational symmetry of this circuit to realize a higher-order topological (HOT) insulator, supporting corner impedance. Prior to that we first discuss some standard models for HOT phases in two and three dimensions, and their realizations in topoelectric circuits.

IV. HIGHER-ORDER TOPOELECTRIC CIRCUITS
Recently, notion of the bulk-boundary correspondence has been generalized to include low-dimensional topolog- modes. Namely, an nth order topological phase supports boundary modes of co-dimension d c = n. So far, we discussed first-order topological insulators and their realizations on topoelectric circuits. Next we present realizations of HOT insulators on topoelectric circuits. In Sec. VI D, we will discuss a circuit realization of a HOT semimetal.

A. Two-dimensional HOT insulator
We start by considering the example of twodimensional Benalcazar-Bernevig-Hughes (BBH) model [80], a second order topological insulator featuring four corner modes with d c = 2. The model is defined on a square lattice with four sublattices (A, B, C, and D) per unit cell. The lattice Hamiltonian reads as H 2D The mutually anticommuting Γ matrices ({Γ i , Γ j } = 2δ ij ) acting on the sublattice space can be chosen as Namely, the sets of Pauli matrices {τ µ } and {σ µ } respectively operate on the sublattice space spanned by (A,B) and (C,D), with µ = 0, · · · , 3. Notice that h x (h y ) assumes the form of the SSH model along the x (y) direction [see Sec. III A]. Therefore, h x and h y individually support a string of zero-energy end modes along the y and x directions, respectively, when |λ/t| < 1. However, {h x , h y } = 0, i.e., h x acts as a mass for the end modes of h y and vice-versa. Consequently, only the four corner modes survive in the spectra of H 2D BBH and we realize a two-dimensional second-order topological insulator.
This specific representation of the Γ matrices is convenient for our purposes as it features three real and two imaginary Γ matrices. It allows to construct the BBH model with all the hopping elements being real [see Fig. 10(a)] and therefore the nodes (representing sublattices) with only two subnodes are sufficient to be implemented in the circuit realization of the BBH model, as shown in Fig. 10(b). Each subnode is grounded by an inductor of inductance L. Consequently, the resonance frequency of the circuit is ω R = 1/ L(λ + t). Furthermore, note that the matrix Γ 5 = σ 3 ⊗τ 0 anticommutes with the Hamiltonian H 2D BBH . Hence, the spectrum is particle-hole symmetric and the zero-energy corner modes are eigenstates of Γ 5 . Therefore, the topological corner modes are localized either on the sublattices A and B or on the sublattices C and D, as we also explicitly demonstrate from the concrete circuit realization of the BBH model.
The spatial distributions of the impedance are shown in Figs. 10(c) and 10(d) on a circuit with = 15 sites in each direction. In the topological regime (|λ/t| < 1) of the BBH model, we first fix the input point at (1,1) and on node A. The spatial variation of the on-resonance impedance on four individual nodes over the entire system then shows sharp corner localization on the C and D nodes, see Fig. 10(c). We arrive at similar conclusions by choosing the B node at (1,1) as the input point (results are not shown here explicitly). On the other hand, when the input point is fixed on the C node located at (1,15), the corner localized peaks of impedance appear on nodes A and B, see Fig. 10(d). Similar conclusions are found with the D node at (1,15) as the input point, for which the results are not displayed here. These features of the node (or sublattice) and site resolved impedance confirm the realization of a 2D HOT insulator on topoelectric circuit and the sublattice or node selection of the corresponding corner impedance. We note that the 2D topoelectric BBH circuit has already been realized in Ref. [32]. Even though our explicit circuit construction is distinct from the one engineered in Ref. [32], here we arrive at qualitatively similar results. On the other hand, the observed sublattice polarization of the on-resonance corner impedance is yet to be demonstrated in experiments.

B. Three-dimensional HOT insulator
Similar to the two-dimensional HOTI, one can construct a BBH Hamiltonian for its three-dimensional coun-terpart [80], yielding a thrid-order topological insulator that supports eight corner modes with d c = 3. Each unit cell then contains eight sublattices (A, · · · , H), see Fig. 11, and in the Fourier space the corresponding Hamiltonian reads as where Here Γ are mutually anticommuting eight-dimensional Hermitian matrices that satisfy the Clifford algebra {Γ i , Γ j } = 2δ ij . Even though topology of H 3D BBH is insensitive to the representation of the Γ matrices, for the sake of convenience (about which more in a moment), we choose where {Σ µ },{σ µ } and {τ µ } are three independent sets of Pauli matrices. Since the maximal number of mutually anticommuting eight-dimensional Hermitian matrices is seven, we can always find an Hermitian matrix, namely Γ 7 , that satisfies {Γ 7 , Γ j } = 2δ 7j for j = 1, · · · , 7, and therefore anticommutes with H 3D BBH . The Γ 7 matrix generates a unitary particle-hole symmetry of H 3D BBH , and in the announced representation Γ 7 = Σ 3 ⊗ σ 0 ⊗ τ 0 . The above model is in the topological regime for |λ/t| < 1 for which it supports eight corner localized zero energy modes. This is so because all three components of H 3D BBH , namely h x , h y and h z assume the form of the one-dimensional SSH model. Consequently, h x supports a collection of end-point zero-energy modes, localized on the yz planes, and similarly h y on the zx planes and h z on the xy planes. However, these three components of H 3D BBH mutually anticommute with each other. Consequently, only the zero modes at eight corners survive, where xy, yz and zx planes meet, yielding the corner modes. Due to the particle-hole or spectral symmetry of H 3D BBH , the corner modes are eigenstates of Γ 7 . With the above specific choices of the Γ matrices, the zero-energy corner states are therefore localized on either the sublattices (A,B,C,D) or (E,F,G,H).
By virtue of the above representation of the Γ matrices, all the hopping elements associated with H 3D BBH are purely real, see Fig. 11(a). Hence, this model can be implemented on a topoelectric circuit by supplementing each node with only two subnodes, as shown in Fig. 11(b). Each subnode is grounded by an inductor of inductance L, and therefore the resonance frequency of the circuit is ω R = 1/ L(λ + t). The numerical computation of the on-resonance (ω = ω R ) impedance then shows that in the topological regime (realized on the circuit for t = 1 and λ = 0.4) it is highly localized at the corners of the cubic circuit, see Fig. 11(c). Moreover, when we fix the input point for the measurement of impedance on the A node at (1,1,1), the sharp corner localized impedance is realized on the E, G and H nodes, confirming its node (or sublattice) selection, stemming from the particle-hole symmetry of H 3D BBH . Experimental realization of threedimensional BBH model in the topoelectric platform has been recently reported in Ref. [47], where an enhanced impedance was detected at the corners, but no sublatticeselective measurement was carried out (despite the differences in the explicit circuit construction). Our discussion should therefore motivate future experiments to search for the predicted sublattice selection rule of corner impedance.

V. DISCRETE SYMMETRY BREAKING AND ANTIUNITARY SYMMETRY
So far we have shown realizations of various topological phases in one-, two-, and three-dimensional topoelectric circuits. In this section, by focusing on the specific example of two-dimensional QSHI, we first demonstrate how one can introduce a discrete four-fold (C 4 ) symmetry breaking Wilson-Dirac mass on a circuit to convert the QSHI into a 2D HOTI, supporting corner localized impedance. Subsequently, we also show how one can exploit an underlying antiunitary spectral symmetry to generalize the HOTI model [98,99] in a circuit.
A. Discrete symmetry breaking: QSHI to HOTI Recall that a QSHI supports two counter-propagating edge modes for opposite spin projections. If we now add a term, namely However, notice that H ∆ changes sign under the C 4 rotation, and as such breaks the four-fold rotational symmetry of H QSHI . Therefore, H ∆ acts as a mass for onedimensional edge modes of H QSHI , with the profile of a domain wall mass that changes sign across each corner of the system. Consequently, the edge modes are only partially gapped, and according to generalized Jackiw-Rebbi mechanism give rise to four sharply corner localized modes [89,100]. We then realize a 2D HOTI by lifting the C 4 symmetry of the system via the Wilson-Dirac mass H ∆ . We now show how to break such discrete rotational symmetry in topoelectric circuit and implement H ∆ to realize HOTI from QSHI. Notice that Γ 4 is a real Hermitian matrix [see Eq. (26)]. Therefore, hopping matrix elements associated with H ∆ in the real space are completely real. We can then introduce H ∆ in a circuit consisting of nodes that are accompanied by only two subnodes, as shown in Fig. 12(a). Note that the total Hamiltonian H QSHI + H ∆ anticommutes with Γ 5 = σ 3 ⊗ τ 0 . Consequently, the on-resonance (ω = ω R = 1/ √ Lm) impedance shows a node or sublattice selective sharp corner localization, displayed in Figs. 12(b) and 12(c), confirming the realization of C 4 symmetry breaking HOTI in a topoelectric circuit.
Note that here we present an alternative realization of HOTI in two dimensions, in comparison to the previous one from the BBH model in Sec. IV A. However, these two seemingly distinct realizations of 2D HOTI are equiv-alent [98]. The main purpose of the present discussion is to highlight a concrete path to explore the hierarchy of orders for topological states (such as first and second in this case) in topoelectric circuits. Next we discuss a further generalization of 2D HOTI by using its antiunitary spectral or particle-hole symmetry in a topoelectric circuit.

B. Antiunitary symmetry protected HOTI in 2D
So far we discussed a plethora of topological phases in different spatial dimensions, in which robust zero-energy boundary modes are protected by the spectral or particlehole symmetry, typically generated by unitary operators. A much less explored situation is when the zero-energy modes are protected by antiunitary operator. We have already encountered one such example, the edge mode of the Chern insulator in Sec. III B, that is protected by antiunitary spectral symmetry. The notion of the antiunitary spectral symmetry is also germane for 2D HOTI and its corner modes, which we discuss next.
Recall first that the model Hamiltonian for 2D HOTI, namely H QSHI = H QSHI + H ∆ [see Eqs. (24) and (32)] possesses unitary particle-hole symmetry generated by Γ 5 . This model also enjoys an antiunitary particle-hole symmetry generated by A = Γ 5 K, as {H QSHI +H ∆ , A} = 0. Here K is the complex conjugation, and by virtue of the Γ matrix representation in Eq. (26), the unitary component of A is identical to the unitary particle-hole symmetry generator (Γ 5 ).
The model Hamiltonian for 2D generalized higherorder topological insulator (GHOTI) reads [98]  where with ∆ 1 and ∆ 2 as real parameters. Notice that {H GHOTI , A} = 0. Hence, H GHOTI enjoys antiunitary particle-hole symmetry, but looses the particle-hole symmetry with respect to the unitary operator Γ 5 . The global phase diagram of this model has already been analyzed in Ref. [98], which we do not discuss here in details.
In brief, H GHOTI supports four corner localized zeroenergy modes for (1) small ∆ 1 or ∆ 2 , and (2) arbitrarily large ∆ 1 = ∆ 2 . Next we implement H GHOTI on topoelectric circuit and test the validity of these predictions from the numerical measurement of the on-resonance corner impedance. First note that we cannot find any representation of the Γ matrices in which all the onsite and hoppoing elements of H GHOTI are real. So we stick to the old representation of the Γ matrices [see Eq. (26)], and first supplement each node associated with the circuit realization of H QSHI + H ∆ by four subnodes. Such a doubling of HOTI circuit does not alter any outcome we discussed so far. Nevertheless, when each node (or sublattice) contains four subnodes one can implement H p in a topoelectric circuit following the design shown in Fig. 13(a).
As the corner modes of GHOTI are protected by A = Γ 5 K, with Γ 5 = σ 3 ⊗ τ 0 , their sublattice selection for finite ∆ 1 and/or ∆ 2 remains unchanged. From the measurement of the on-resonance impedance with the B node at (1,1) as the fixed input point, we find the following. Node (or sublattice) selective sharp corner localization of impedance does not change for (1) small ∆ 1 [see Fig. 13 Fig. 13(d)]. These findings are in agreement with Ref. [98], and we find a circuit realization of GHOTI for which the corner impedance is protected by an antiunitary operator. When ∆ 1 = ∆ 2 , the system always remains in the GHOTI phase, while for large ∆ 1 or ∆ 2 it undergoes a transition into a trivial phase, without any corner modes. In this regime the on-resonance impedance does not show any corner localization, which we do not display here.
We point out that the same generalization is also applicable for the 2D BBH model for HOTI, discussed in Sec. IV A. Furthermore the 3D BBH model from Sec. IV B also possesses an antiunitary spectral symmetry, generated by the eight-dimensional operator Γ 7 K, with Γ 7 = Σ 3 ⊗ σ 0 ⊗ τ 0 . In the future, it will be interesting to find a generalized model for 3D HOTI.

VI. TOPOELECTRIC NODAL SEMIMETALS
So far we discussed realizations of first-, second-and third-order topological insulators on topoelectric circuits and their identification from the boundary localized onresonance impedance. On the other hand, there exists a whole family of topological phases of matter, known as topological semimetals, where the bulk quasiparticle spectra are gapless, but the bulk-bounadary correspondence remains operative therein [8][9][10][11]. Typically, topological semimetals are constructed by stacking lowerdimensional topological insulators, while preserving the translational symmetry in the stacking direction, which is the approach we use to construct their topoelectric realizations. In this section, we show that some prominent nodal topological phases can be realized in topoelectric circuits and identify them from the boundary localized on-resonance impedance.
A. Stacked SSH chain: 2D Dirac and 3D nodal-line semimetals In contrast to Weyl and Dirac semimetals where the valence and conduction bands touch at isolated points, in nodal-line semimetals (NLSMs) the band touching takes place along a closed curve in momentum space [101][102][103][104][105][106][107][108]. In this section, we first construct a two-dimensional (2D) Dirac semimetal (DSM) by stacking a collection of onedimensional (1D) SSH chains in the y-direction in a translationally invariant fashion, such that the symmetry class of the system, namely BDI, remains unchanged. The corresponding Hamiltonian in the momentum space takes the form where t 2 denotes the strength of the inter-SSH chain hopping along the y direction. Depending on the relative strength of various hopping parameters (λ, t and t 2 ), the system supports a 2D DSM, topological and trivial insulators. Specifically for λ/t > 0, the Dirac points are located at k = (π, ±k y ), where As one can see, depending on the values of t, λ and t 2 , Eq. (36) has a real solution for k y , which then describes a 2D DSM. Otherwise, the system is an insulator. Furthermore, each of the 1D SSH topological insulators stacked between two Dirac nodes supports endpoint zero modes. The collection of these zero modes ultimately constitutes Fermi arc states connecting two Dirac points in the momentum space. On the other hand, in the real space the Fermi arc states are localized along only two edges in the y direction. This scenario is supported from the numerical computation of the on-resonance impedance in a 2D circuit constructed by stacking 1D SSH circuit in the y direction. We note that two-dimensional DSM has been realized in topoelectric circuits in Refs. [33,36,43], and our results are in qualitative agreement. The main purpose of this discussion is to develop a concrete path for stacked topoelectric circuits to realize various gapless topological phases, which we systematically explore next. As we show now, by extending this construction in the z direction, we can find three-dimensional unknotted nodalline semimetal. We point out that even though various topological knots in the momentum space have been engineered in topoelectric circuits [46], explicit demonstration of simple nodal-line semimetal has not been reported so far.
Since the hopping parameter in the y direction is completely real, one can construct the circuit corresponding to Eq. (36) by coupling 1D SSH circuits, shown in Fig. 4(c), by capacitors of capacitance t 2 in the y-direction between nodes A and B, as shown in Fig. 14(a). Next we numerically compute the onresonance impedance with either an A or B node of the circuit as an input point, and scanning all the sites as the output point. Results are shown in Fig. 14(b), displaying a sharp y-edge localization of the on-resonance impedance, which in turn captures the existence of Fermi arcs states and a 2D DSM in a topoelectric circuit.
Next we continue with the stacking protocol and extend it along the z direction. For the sake of simplicity we consider the inter-layer hopping amplitude along the z direction to be t 2 as well. Then the corresponding Hamiltonian in the momentum space reads as This model supports a NLSM, in which the valence a conduction band touch each other over a closed curve in the k x = π plane when t = 1, λ = 2 and t 2 = 1, for example. The implicit form of the closed curve in the k x = π plane is then given by cos(k y ) + cos(k z ) = 1. A NLSM is therefore constructed by stacking 1D SSH topological insulator in the y and z direction within the perimeter of the closed curved mentioned above. A collection of endpoint zero modes associated with the SSH insulator within the perimeter of such closed curve ultimately constitutes the drumhead surface states on the (k y , k z ) planes, which are the surface projections of the bulk nodal loop. On the other hand, in the real space the surface zero-energy modes are localized on the entire yz plane. A circuit realization of NLSM can be engineered by extending the construction from Fig. 14(a) in the z direction, along which the hopping elements are also real. Therefore, inter-layer hopping in the z direction is accomplished by capacitor connections with capacitance t 2 . The explicit circuit construction for NLSM is, however, not shown here. The numerical computation of the onresonance impedance clearly displays a strong surface localization on the top and bottom yz planes, capturing the imprint of the drumhead surface states associated with a NLSM in a three-dimensional topoelectric circuit, as shown in Fig. 14(c).

B. Stacked Chern insulator: 3D Weyl semimetal
Next we construct a 3D Weyl semimetal (WSM) by stacking 2D Chern insulators in the z-direction in a translationally invariant fashion. The resulting Hamiltonian in the momentum space then reads as where t z denotes the interlayer hopping amplitude in the z direction. The global phase diagram of this model has been reported in Refs. [109,110], which we do not discuss here in details. Conveniently, we set t = t 0 = 1, m = 2 and t z = 2.5 for the rest of the discussion. For this set of parameters the system supports a WSM, with two Weyl nodes located at k = (0, 0, ± cos −1 (4/5)). The two Weyl nodes represent the bandgap closing points for the collection of Chern insulators stacked in the direction connecting them, and the zero-energy states associated with each Chern insulator layer ultimately constitute the Fermi arc surface states. In the surface Brillouin zone on the (k j , k z ) plane the Fermi arc connects two Weyl points at k z = ± cos −1 (4/5) and placed along the k j = 0 line, where j = x or y [11,111,112]. In the real space, the Fermi arc occupies the xz and yz planes. Next we demonstrate these features in a circuit realization of the WSM. The existing literature reporting topoelectric circuit realizations of WSM concerns with systems, possessing an even number of pairs of Weyl nodes [33,54,61,63,65]. By contrast, here we demonstrate circuit realization of the minimal WSM, supporting only two Weyl points, by stacking layers of 2D Chern insulators.
The circuit construction of the WSM follows the spirit of stacked SSH chain, shown in Fig. 14(a). To engineer a WSM we couple layers of Chern circuits, schematically shown in Fig. 6, by capacitor connections with capacitance t z . The construction of the Weyl topoelectric circuit is not shown here explicitly. The numerical computation of the on-resonance impedance (Z) shows a strong localization on the xz and yz surfaces, see Fig. 15. Selective surface localization therefore bears the signature of the Fermi arc surface states, in turn confirming the realization of a 3D WSM in a topoelectric circuit.

C. Stacked QSHI: 3D Dirac semimetal
Following the spirit of constructing a 3D WSM from the last section, one can also realize a 3D DSM by stacking two-dimensional layers of quantum spin Hall insulators (QSHIs) in the z direction in a translationally invariant manner. The corresponding Hamiltonian in the momentum space then takes the form The Γ matrices follow the representation from Eq. (26), and t z denotes the hopping between the nearest-neighbor layers of 2D QSHI in the z direction. For t = t 0 = 1, m = 2 and t z = 2, the system supports a pair of Dirac points separated along the k z direction and located at k = (0, 0 ± π/2). The resulting Fermi arc surface states in a 3D DSM possess two-fold Kramers degeneracy and connect the Dirac points in the momentum space, similar to the Fermi arc state in a 3D WSM. As {H 3D DSM , Γ 5 } = 0 and Γ 5 = σ 3 ⊗ τ 0 , the Fermi arc states are sublattice polarized. Specifically, they are localized either on the A and B sublattices or on the C and and D sublattices, similar to the edge modes of the underlying 2D QSHI layers, shown in Fig. 9.
A circuit realization of 3D DSM is similar to the other cases we discussed so far. Specifically, in a Dirac topoelectric circuit the layers of 2D QSHI are connected by the capacitors with capacitance t z . Numerical computation of the on-resonance impedance then reveals sublattice polarization and strong surface localization on the xz and yz planes, see Fig. 16. These observations confirm the existence of the Fermi arc surface states in a 3D Dirac topoelectric circuit.

D. Stacked 2D HOTI: 3D quadrupolar Dirac semimetal
As a penultimate topic, we discuss the topoelectric circuit realization of a 3D second-order or quadrupolar DSM [87,89]. In an electronic system, a 3D quadrupolar DSM is obtained by stacking 2D HOTIs with corner modes in the z direction, for example, while preserving the translational symmetry. The corresponding Hamiltonian in the momentum space assumes the following form where H 3D DSM is defined in Eq. (39) and Γ 4 = σ 1 ⊗ τ 3 [see Eq. (26)]. Note that the second term in the above equation, proportional to ∆, causes dimensional reduction of the edge modes associated with each layer of QSHI, yielding four corner modes (see Sec. V A). The underlying insulating layers then correspond to 2D HOTI. Stacking of such layers of 2D HOTIs produces a pair of higher-order Dirac nodes located at k = (0, 0, ±π/2) for t = t 0 = 1, m = t z = 2 and arbitrary value of ∆. The corner states connecting these two Dirac nodes produce one-dimensional hinge modes.
A circuit realization of this model follows the general stacking protocol we discussed so far. Specifically, we Numerical computation of the on-resonance impedance (Z) in a 3D second-order Dirac topoelectric circuit with the A node at (1,1,5) as the fixed input point for t = t0 = 1, m = tz = 2 and ∆ = 0.8, see Eqs. (40) and (39). The output point is then chosen on four individual nodes or sublattices separately (mentioned in each subfigure), which we scan over the whole system. The on-resonance impedance shows sublattice selective sharp hinge localization, which in turn confirms realization of a 3D higher-order (namely, second-order) DSM in a topoelectric circuit.
couple 2D layers of HOTI circuits, obtained by combing the components shown in Figs. 9(a) and 12(a), by capacitor connections of capacitance t z . The numerical measurement of the on-resonance impedance then shows sublattice selective sharp hinge localization for the above mentioned parameter values, which we display in Fig. 17 for a specific choice of ∆ = 0.8. These results demonstrate the existence of 1D hinge modes in a second-order Dirac topoelectric circuit.

E. Stacked 2D GHOTI: 3D HOT Weyl semimetal
Finally, we demonstrate the topoelectric circuit realization of 3D HOT Weyl semimetal. To this end we consider layers of 2D GHOTI [see Sec. V B] coupled in the z direction along which the translational symmetry is preserved. The resulting Hamiltonian in the momentum space takes the form where the Γ matrices are defined in Eq. (26) and H 3D DSM is shown in Eq. (39). Notice that additional terms proportional to ∆ 1 and ∆ 2 lift the Kramers degeneracy from the Numerical computation of the on-resonance impedance (Z) in a 3D second-order Weyl topoelectric circuit with the A node at (1,1,5) as the fixed input point for t = t0 = 1, m = tz = 2, ∆ = 1.0 and ∆1 = ∆2 = 2.0, see Eq. (41). The output point is then chosen on four individual nodes or sublattices separately (mentioned in each subfigure), which we scan over the whole system. The onresonance impedance shows sublattice selective sharp hinge localization, which in turn confirms the realization of a 3D second-order WSM in a topoelectric circuit.
valence and conduction bands [98], yielding 2D GHOTI in the absence of the interlayer coupling. When such layers of GHOTI are coupled by translationally invariant interlayer hopping (t z ), specifically for t = t 0 = ∆ = 1.0 and m = t z = ∆ 1 = ∆ 2 = 2.0 Kramers nondegenerate band touch each other at k = (0, 0, π/2), yielding a pair of higher-order Weyl nodes. Then each insulating layer of GHOTI between these Weyl nodes hosts four corner modes. By virtue of the translational symmetry, the collection of such corner modes between these two points then constitutes the one-dimensional hinge modes, yielding a second-order WSM.
In an electric circuit this model is realized by combining the circuit components, shown in Figs. 9(a), 12(a) and 13(a), where each node is supplemented by four subnodes, and subsequently coupling each layer of 2D circuit by capacitor connections of capacitance t z . When the capacitor values are chosen to be the ones mentioned above for the lattice model, we realize a higher-order Weyl topoelectric circuit. The numerical measurements of on-resonance impedance, shown in Fig. 18, display sublattice selective sharp hinge localization, which in turn confirms the circuit realization of a second-order WSM.

VII. SUMMARY AND DISCUSSION
To summarize, here we present an alternative derivation of the construction of an arbitrary hopping element, stemming from an underlying lattice tight-binding model, in LC electric circuits, where the electrical nodes play the role of lattice sites. Subsequently, we apply this general protocol to engineer a plethora of topological lattice models in topoelectric circuits. In particular, we identity first-order SSH model (Sec. III A), Chern (Sec. III B) and quantum spin Hall (Sec. III C) insulators from the highly endpoint and edge localized on-resonance impedance (Z) respectively in d = 1 and d = 2. We also propose simple circuit realizations of twoand three-dimensional higher-order topological insulators (Secs. IV A and IV B), supporting corner localized and sublattice polarized on-resonance impedance. In addition, we also demonstrate a concrete route to break discrete rotational symmetry and implement Wilson-Dirac mass in topoelectric circuits (Sec. V A). Such a construction allows us to convert a two-dimensional first-order quantum spin Hall insulator (with edge modes) into a higher-order topological insulator (with corner modes). Finally, we construct a generalized second-order topological insulator for which the corner impedance is protected by an antiunitary operator (Sec. V B).
The simplicity of our circuit constructions is based on the representation theory of the Clifford algebra. In particular, throughout we exploit the fact that the Clifford algebra of 2 N -dimensional Hermitian matrices is closed by (2N + 1) mutually anticommuting Hermitian matrices, among which N + 1 (N ) are purely real (imaginary) [113,114]. As the existence of topological boundary modes relies on the anticommuting nature of the involved matrices (not on their explicit representations), we choose (whenever possible) matrices multiplying the sine (cosine and constant) functions to be purely imaginary (real), such that the hopping elements in the real space are completely real. Exceptions from this scenario are rather sparse, see Secs. III B and V B, for example. One can then implement a lattice topological model on an electric circuit by supplementing each node (representing a lattice site) with only two subnodes, between which the phases of current and voltage differ by a factor of exp(iπ) = −1, see Sec. II B 1. We also highlight a generalization of this construction involving four (Sec. II B 2) as well as n (Sec. II B 3) subnodes.
Subsequently, we present electric circuit realizations of various gapless topological phases, such as two-and three-dimensional Dirac semimetals, in Secs. VI A and VI C, by respectively stacking 1D SSH and 2D QSHI circuits, while preserving the translational symmetry in the stacking direction. In addition, we also show a concrete realization of Weyl topoelectric circuits by stacking 2D Chern insulators (Sec. VI B). These topoelectric semimetals are then identified from the on-resonance impedance that mimics the Fermi arc states in the real space. On the other hand, a nodal-line topoelectric circuit is identified from on-resonance impedance, localized on the top and bottom surfaces, bearing the signature of drumhead surface states in the real space (Sec. VI A). Finally, we also show realization of higherorder Dirac (Weyl) topoelectric circuit, featuring hinge localized impedance, in Sec. VI D (Sec. VI E).
By focusing on the specific example of the Chern circuit (Sec. III B), we show that the measurement of onresonance impedance can be instrumental in mapping the global phase diagram of topological lattice models in topelectric circuits. To this end, we use that the onresonance impedance is finite only inside the topological phases, while it vanishes in the trivial phase as well as at the topological quantum critical point between two topologically distinct phases (Fig. 8). Therefore, our findings can be experimentally consequential for understanding various paradigmatic toy models of topological phases.
Finally, we comment on a subtle issue regarding the time-reversal symmetry in topoelectric circuit. Note that we propose circuit realizations of various topological models that in electronic systems are protected by the time-reversal symmetry (T ), satisfying T 2 = −1 (such as the quantum spin Hall insulator). However, topoelectric circuits are constituted by capacitors and inductors, and all these elements are real. Therefore, in topoelectric circuits T = +1. Nevertheless, due to high precision tunability of the circuit elements we believe that these models can still be engineered in topoelectric circuits and their topological modes can be observed through on-resonance boundary impedance. This should be so at least when sufficient care is taken to minimize circuit disorder in the setup, given that all the topological phases we discuss here are robust against sufficiently weak randomness.
In the future, our setup should be instrumental to systematically investigate the role of disorder in topoelectric circuits. Disorder can be implemented in this setup by randomly and independently varying the grounding elements (the capacitor (C a ) or inductor (L a ) in Fig. 1, for example) at each node of the circuit, such that the resonance frequency (ω R ) displays a random spatial variation δω R (x), with the spatial average δω R (x) = 0. Such disordered topoelectric circuits can mimic a variety of fundamentally important phenomena in dirty topological systems, among which possibly the most interesting are topological Anderson insulator in electrical circuits [64], gradual melting of the Fermi arc [112] and hinge [115] impedance, respectively, in 3D Weyl and higher-order Dirac topoelectric circuits. Furthermore, the jurisdiction of topoelectric circuits can be further extended by engineering lattice defects to probe topological phases within this setup. Even though lattice defects have been realized in other topological metamaterials, such as photonic [116,117] and phononic [118][119][120] crystals, their realizations in topoloelectric crystals remain to be studied. ACKNOWLEDGMENTS V.J. acknowledges the support of the Swedish Research Council (VR 2019-04735). B.R. was partially supported by the Startup grant from Lehigh University. We are thankful to Ronny Thomale for useful correspondence.