Realization of Weyl semimetal phases in topoelectrical circuits

In this work, we demonstrate a simple and effective method to design and realize various Weyl semimetal (WSM) states in a three-dimensional periodic circuit lattice composed of passive electric circuit elements such as inductors and capacitors (LC). The experimental accessibility of such LC circuits offers a ready platform for the realization of not only various WSM phases but also for exploring transport properties in topological systems. The characteristics of such LC circuits are described by the circuit admittance matrices, which are mathematically related to the Hamiltonian of the quantum tight-binding model. The system can be switched between the Type-I and Type-II WSM phases simply by an appropriate choice of inductive or capacitive coupling between certain nodes. A peculiar phase with a flat admittance band emerges at the transition between the Type-I and Type-II Weyl phases. Impedance resonances occur in the LC circuits at certain frequencies associated with vanishing eigenvalues of the admittance matrix. The impedance readout can be used to classify the Type-I and Type-II WSM states. A Type-I WSM shows impedance peaks only at the Weyl points (WPs) whereas a Type-II WSM exhibits multiple secondary peaks near the WPs. This impedance behaviour reflects the vanishing and non-vanishing density of states at the Weyl nodes in the Type-I and Type-II WSM phases, respectively.


Introduction
Exotic topological phases of matter have emerged as one of the most exciting branches of condensed matter physics in the past decades due to their exceptional electronic properties [1][2][3]. Although gapped topological materials such as topological insulators [4][5][6], integer quantum Hall insulators [7,8] and topological superconductors [9] have attracted a lot of attention, the physics of topological gapless materials [10] has recently gained more prominence due to their novel properties. A three-dimensional topological gapless system can be characterized by the nature of its band degeneracy points where two bands touch each other in momentum space. These band degeneracy points are classified as either Dirac [11] or Weyl points (WP) [12,13] depending on their symmetries. Dirac points appear only when both time-reversal [14] and inversion symmetry [15] are conserved in a system. On the contrary, WPs emerge if either or both symmetries are broken. Although both types of band touching points appear and annihilate pair-wise, WPs are more robust against perturbations than Dirac points. One important class of topological system that hosts WPs are called Weyl semimetals (WSMs) [16][17][18][19]. WSMs disperse linearly in all three spatial directions in the vicinity of the WPs. Since the linear dispersion around the WPs can be described by the Weyl Hamiltonian involving all three Pauli matrices, small perturbations do not lift the energy degeneracy but only displace the WPs in momentum space. Besides fundamental properties such as the massless and chiral nature of the bulk carriers and large carrier mobility [20,21], WSM states also exhibit unusual transport phenomena like the quantum anomalous Hall effect [22], large positive magnetoresistance [23], Klein tunnelling [24,25], chiral anomaly [26,27] and novel quantum oscillations [28,29]. These exotic features and inherent robustness against disorder make WSMs promising candidate for future generation nanoelectronics, spintronics [30] and valleytronics [31] devices. WSMs can be further classified as Type-I and Type-II based on the tilt of the Weyl cones around the WPs [32,33]. Type-I WSMs are marked by the existence of point-like iso-energy surfaces at the WPs, and the simultaneous presence of carriers with both signs of group velocities near the WPs [34,35]. In contrast, in the Type-II WSM phase, potential energy terms dominate over the kinetic energy terms in the energy relation. This modifies the dispersion relation in such a way that only one sign of group velocity exists along certain directions near the WPs. Moreover, there is a finite density of states (DOS) at the WPs. When the kinetic and potential energy contribute equally, the transition between the Type-I and Type-II phases arises where one of the bands become completely flat with vanishing group velocity. This is the so-called Type-III [36] or Critical-type WSM phase [37]. However, realizing WSM states in condensed matter systems and tailoring their properties are experimentally challenging. For instance, carrier doping in WSMs may compromise the stability of the Weyl phases due to broken translational symmetry [38]. Additionally, it is usually difficult to achieve transitions between different topological WSM phases (Types-I, II, III) in a given material, and thus to form heterojunctions of WSMs of different phases [23,39,40]. There is hence a need for alternative platforms to realize gapless topological states. Researchers have studied various artificial systems such as photonic crystals [41], metamaterials [42,43] and quantum resonators [44]. However, these methods all come with their own experimental complexities and limitations. Recently, topological states were realized in periodic electrical circuits consisting of inductors and capacitors (LC networks) known as topoelectrical (TE) circuits [36,45]. Under the TE framework, many striking phenomena have been demonstrated such as the quantum spin Hall insulator state [46], magnet-less Floquet topological insulator state [47], topological photonic state [48], and edge modes in the SSH model [49]. A key advantage of electrical networks is the flexibility of experimental realization, as the circuits can be implemented even on simple printed circuit boards. Additionally, circuit parameters in electrical networks can be more readily adjusted and tuned, compared to lattice model properties of real materials.
In this paper, we demonstrate the realization of Type-I and Type-II WSM phases in a three-dimensional TE circuit network. We relate the admittance matrix of a TE circuit to the tight-binding Hamiltonian matrix of a condensed matter system. The resultant admittance band structure in a LC model resembles the energy dispersion [50]. The transition between Type-I and Type-II WSM phases can be readily effected by tuning the coupling between certain nodes. Moreover, we realize a distinct topological phase at the transition between the Type-I and Type-II phases, by isolating these nodes. This so-called Critical-type phase is characterized by the emergence of a flat admittance spectrum for one of the bands. We additionally derive the impedance spectrum and relate it to the circuit Green's functions [51]. The impedance spectra of the TE circuits corresponding to different WSM phases show significant differences. The Type-I WSM impedance spectrum is marked solely by distinctive peaks at the WPs. On the contrary, the Type-II WSM impedance dispersion exhibits not only impedance peaks at the WPs but also multiple secondary peaks along the tilt direction in wave-vector space. Finally, the intermediate Critical-type phase is characterized by a high impedance region between two WPs which signifies the flat dispersion of the electron or hole bands there. In short, TE circuits provide an accessible and tunable platform to design and model distinct topological phases and transitions in WSMs, while their impedance spectra provide a signature of the different phases.

Topoelectrical circuit model
Here, we consider a periodic (along all directions) LC circuit, shown in figure 1, which comprises of two sublattices A and B. In an AC circuit, the admittance Y of any two-terminal device component (resistor, capacitor, inductor etc) can be expressed as = a a -Y I V 1 where I α and V α are the current and the potential difference across the circuit component, respectively. In general, I α and V α are complex numbers [52]. By applying Kirchhoff's current law (KCL) at nodes A and B in the unit cell at = r x y z , , ( )  of the circuit model shown in figure 1, we have: where ω is the frequency of the alternating current, C 1 and C 2 are the intra-and inter-cell coupling capacitances along the x direction and C y is the coupling along the y direction between lattice points on the same A/B sublattice. The lattice circuit model in the x-y plane is connected symmetrically with capacitors C Az and C Bz , respectively, between A-A and B-B sites of adjacent layers along the z axis. C n denotes the next nearest lattice interaction between A-A or B-B sub-lattice nodes along both the x and y directions. The A and B lattice points are grounded by capacitors 2C Bz and 2C Az respectively, and an additional inductance L that serves to adjust the offset 'potential'. Note that the capacitors C Bz can be replaced by inductors ( ) ) to switch the WSM phase of the circuit (see later). The schematic diagram of the various capacitive and inductive couplings in the LC circuit is depicted in figure 1. Using the Fourier transformation , equations (1) and (2) can be expressed in momentum k space, in analogy to the tight binding (TB) approach in a crystal lattice model [53]. The relation between voltage and current distribution in k space can then be expressed in terms of the Laplacian matrix equation [36,49]    are the vectors representing the potential and current distribution at the A and B sublattice nodes in k space, respectively. k L Weyl ( )  denotes the admittance matrix (Laplacian matrix) which is analogous to the Hamiltonian in condensed matter physics [36], and can be expressed as w w s s s s where s s s , , x y z ( )are the Pauli matrices corresponding to the A/B sublattice pseudospin degree of freedom and σ 0 is the 2×2 identity matrix. The admittance spectrum, which corresponds to the energy dispersion in the quantum TB model [36], is obtained from the eigenvalues of in equation (4): Here, the 'energy' E refers to the admittance (divided by w i ) of the circuit. The energy spectrum is gapless at the charge neutrality point or WP in which the two eigenvalues of equation (5) are equal. This equality is satisfied by the following conditions Without loss of generality, we consider all capacitances and inductances as having non-negative values, . The solution set of equation (6) is then However, for our parameter range, no real solution of k y exists in the first Brillouin zone for η=1. For η=−1, we obtain two pairs of WPs in k space which are located at , no band touching points exist. The type of WPs hosted by the Laplacian in equation (4), either Type-I or Type-II, depends The circuit representation in the x-y plane, where each A-type (red dot) node is connected to a B-type node (green dot) by C 1 , and the repeating units are connected by C 2 in the x direction. All alternate lattice sites along the y direction are connected by C y . Nodes of the same type are coupled to each other by C n in the x-y plane. (b) The circuit representation in the x-z plane. All A-A couplings are capacitive in nature with magnitude C Az , while the B-B coupling can be tuned to be in-phase (out-of-phase) with respect to the A-A coupling by selecting capacitive (inductive) hopping strengths , respectively, along the z direction. (c) Additionally, all A(B) nodes are connected to ground by common inductor L in parallel with capacitor 2C Bz (2C Az ) respectively. The common inductor L determines not only the resonant frequency of the circuit but also the effective offset 'potential' of the TE circuit.
on the values of C n , C Az and C Bz , which determine the d 0 term in equation (5). These parameters do not shift the WPs in the k-space but only modify the tilt of the Weyl dispersion cones. Figure 2 shows the evolution of the resonant admittance band structures and iso-admittance contour plots for different WSM phases. The resonant frequency is given by where ω r corresponds to the frequency at which the σ-independent terms in the Hamiltonian vanish. As shown in figure 2(a), when the hopping parameters between the A and B nodes along the z direction have opposite signs, a Type-I WSM dispersion is obtained. In contrast, when the coupling between two similar sites along the z direction has the same signs, the WSM phase changes to Type-II (figure 2(g)). The Critical-type phase between two WSM phases is depicted in figure 2(d), corresponding to zero coupling between two adjacent B-B nodes along the z axis. As shown in figure 2(b), for the Type-I WSM phase, the constant admittance (iso-admittance) contour is a closed loop in the k y -k z plane that encompasses the WPs. In contrast, the iso-admittance contours for Type-II WSM are hyperbolic and connect the WPs with different chiralities, as shown in figure 2(h) for both energy bands. The contours for the transitional Critical-type phase between Type-I and Type-II WSMs are distinct from the other two types, as depicted in figure 2(e). For this phase, the WPs are connected by the zeroadmittance contour lines for the electron (upper) band while for the hole (lower) admittance band, vanishing Admittance spectrum corresponding to Type-II WSM phase with capacitive coupling between B-B and A-A also sites along the k z direction, C Az =0.5 mF, C Bz =0.2 mF and k x =π. Both upper and lower bands show group velocity of the same sign around the WPs in contrast to the Type-I dispersion of (a), opposite velocities. (h) Iso-admittance contour plot of the admittance bands of a Type-II WSM in (g), projected on the k y -k z plane. Due to the hyperbolic dispersion of Type-II WSM, the iso-admittance contour lines do not enclose the WPs. However, the number and position of WPs have not changed with respect to Type-I and Critical-type WSM. (c), (f), (i) Constant admittance cross section at E=0, for admittance spectra in (a), (d) and (g) for the Type-I, Critical-type and Type-II WSM, respectively. Type-I WSM hosts no states at the nodal energy. Critical-type WSM phase exhibits straight line bulk-like states between WPs. In Type-II WSM, electron and hole pockets emerge at Fermi energy which touches the WPs. admittance is obtained for the rest of Brillouin zone along the k z direction for a specific value of k y that satisfies equation (7).

Low admittance general Laplacian and classification
(( ) ) and l = 1 for the k and ¢ k valleys respectively. v 0 is a constant 'potential' offset which can be tuned to zero or any other arbitrary value by varying the inductance L. The tilt of the Weyl dispersion cones along the k i direction increases with t v i i | |. The Weyl phases can be classified based on this ratio: Type-I with figure 3 (b)). In the vicinity of the WP, the slope of the dispersion cone, which represents the group velocity, has opposite (same) signs for opposite signs of q z in a Type-I (Type-II) WSM, while at the transition value = t v i i | | | |, one of the electron or hole bands has zero group velocity and hence a flat dispersion relation, which corresponds to the Critical-type phase. From equation (10), the linearized admittance spectra of the two bands are given by where χ=±1 for electron and hole bands respectively and = + + A v q v q v q x x y y z z , . In the vicinity of As can be seen in equation (12), electrons and holes can propagate along the opposite or the same directions depending on the respective circuit parameters. In the x-direction, the velocity is purely determined by the kinetic term (coefficient of σ i ), and the particle and hole bands show opposite gradient with respect to q x . However, for the y and z directions there is a combination of kinetic and potential terms (coefficient of σ 0 associated with the dispersion tilt) in the group velocity expressions. If the potential term dominates over the kinetic term along any direction, the upper and lower bands will have the same sign of group velocity resulting in  | | | |, the amplitudes of the potential and kinetic hopping terms along the y axis are equal. This corresponds to the Critical-type phase, with a flat hole band close to the WPs (with vanishing velocity -V g y ), while the electron band retains a finite group velocity + V g y . Interestingly, the second-nearest hopping along the x axis via the coupling capacitance C n does not contribute to the tilt in that direction (n.b. t x =0 in equation (10)), but rather contributes to the offset 'potential' term, which may also be adjusted by tuning the common inductor L.
Thus far in our analytical derivation, we have considered an infinite LC circuit in all three directions. For the numerical verification of our analytical predictions, we consider a nanoribbon geometry in a LC circuit model with a finite width of l y =30 unit cells in the y-direction, under open boundary conditions. Here, multiple subbands are present due to the quantum confinement along the y direction. We consider the evolution of the admittance dispersion through the different WSM phases as the B-B coupling parameter along the z-direction, C Bz , is varied at a fixed k x =π and C Az , as shown in figures 3(a)-(c). With an inductive coupling between two neighbouring B sites along the z direction, we have ) . Any non-zero value of L B will result in a dispersion tilt such that The resulting Type-I WSM admittance dispersion is shown in figure 3(a). The dispersion relation consists of two symmetric bands touching each other at p =  k 2 z . The DOS at the WPs is zero and both signs of the admittance gradient are present in the vicinity of each WP. When the inductor L B is replaced with a capacitor C Bz , the B-B coupling C Bz is in-phase with the corresponding A-A coupling, C Az . In this case, both would have the same sign, so that > t v z z . In other words, for any positive value of C Bz , )holds and so the Type-II WSM would result. This is reflected by the tipping over of the admittance bands into two asymmetric branches with respect to k z (shown in figure 3(c)). The positions of the WPs are not shifted but the dispersion acquires a finite DOS at the nodal admittance-the line E=0 cuts across multiple states in the E-k z plot, spanning across the entire range of p p -< < k z , in contrast to the Type-I WSM in figure 2(a) where the E=0 line cuts across only the WP states. Finally, for the case of zero B-B coupling (C Bz = 0), one of the admittance bands exhibits a flat zero admittance state between the two WPs

Impedance spectrum analysis
In the previous section, we presented the admittance dispersions for the different WSM phases of a TE model. In this section, we consider the impedance of a TE circuit and show that it offers an experimentally convenient way to distinguish the different topological phase of the TE circuit. The impedance readout of a TE circuit can be obtained simply by connecting a fixed current source to two arbitrary nodes in the circuit and measuring the potential difference between the nodes. This constitutes a more convenient measurement than a direct determination of the admittance dispersion relation of the circuit. Moreover, the comparatively large impedance readout (in the range of few Ω to kΩ) compared to the admittance readout (in the range of a few Wm 1 ) provides a better measurement accuracy. We begin by analysing the mathematical significance of the impedance between any two lattice sites in a TE circuit. Consider a 2D TE circuit shown in figure 4 consisting of a finite number of nodes along the y direction, and having an infinite number of nodes along both the positive and negative x directions. Each node is capacitively coupled to its left and right neighbours by C x , its upper and lower neighbours by C y , and to the ground by a common grounding capacitance C. We also connect to every node an additional wire through which current may flow between the node and an external current supply. This wire is denoted as the dotted line with an arrow flowing into and out of the node, as shown in figure 4(b), and shall be referred to as the 'current wire' subsequently. We label the nodes by their (x, y) coordinates so that V x y , is the voltage at the node located at (x, y). The KCL at each node (except for those at the top and bottom rows) reads as Let us first consider the case where = I 0 x y , everywhere. Physically, this corresponds to leaving the current wire at each node unconnected to any current source or sink. Equation then has a similar form to the Schrödinger equation for a quantum mechanical TB Hamiltonian [36], where the common grounding capacitance C is the analogue of the eigenenergy. More generally, the KCL for a TE circuit which has voltage nodes connected to common grounding capacitances C and current flowing through the current wires, can be written as where i is the vector of currents flowing through the current wires at every node and v is the vector of voltages at every node. H is the matrix relating the voltages at the different nodes to one another, obtained by applying KCL at every node but excluding the contribution of the common grounding capacitance C, which is moved to left hand side of the equation (14). If I x y , is set to 0 everywhere, equation (14) becomes an eigenvalue equation in C and v, as mentioned earlier. In particular, if H is a finite-sized matrix, then the system becomes equivalent to an infinite potential well system with a discrete eigenspectrum of C consistent with having x y x y y y y y x y x , 1 1 ( ) ( ) ( ( ) ) ( ) and the system is mathematically equivalent to a one-dimensional system with a k x dependent 'on-site potential' -C k 2 cos 1 x x ( ( ) ). Equation (16) can be schematically written as where the 1 ( ) subscript denotes a quasi one-dimensional system, where the effects of the infinite-length x direction have been incorporated as a k x dependent on-site potential. In particular, for the nodes at x=0, we have We then consider the situation depicted in figure 4(b) where two nodes at y=a and y=b are connected to a current source supplying a current as I S (the subscript 'S' denotes source), while the other nodes are left unconnected. This corresponds to setting = - and substituting the above into equation (21), we have where h j a ; is the ath element of h j . Equations (21) and (25) are the key formulae to evaluate the impedance between any two arbitrary points. When C is one of the eigenvalues of H 1 ( ) and provided that h h j a j b ; ; | | is not simultaneously zero, the current flowing through the current wires attached to the nodes will be nearly zero, leading to a sharp spike in the impedance Z ab . This phenomenon is similar to that in quantum-dot (QD) systems [54,55], in which a resonance occurs in the transmission across the system when the Fermi energy in the leads coincides with one of the discrete energy levels in the QD. Here C plays the role of the lead Fermi energy and c j the discrete energy levels in the QD.
The impedance spectra across the whole circuit, i.e. between the first and last nodes ( = N n 2 ) are plotted as a function of k z at the resonant frequency for various WSM phases (see figure 5). Here, we consider n=30 unit cells so that there are N=60 alternating A and B lattice points. As can be seen in figure 5, the impedance distribution is symmetric about k z =0 for all types of WSM phases but assume different profiles for different phases. Figure 5(a) shows the impedance readout of a Type-I WSM TE circuit. The notable characteristic of the impedance spectrum is the presence of two clear peaks at the k points corresponding to the WPs in the k z direction. These may be explained by considering equation (25) and figure 3(a). Figure 3(a) shows the hole and particle-like bands almost touching only at the WPs, where the admittance matrix is nearly zero. (The hole and particle bands do not actually touch due to the small band gap as a result of the finite width.) Equation (25) shows that the impedance between two points is dominated by the eigenvalues of the Laplacian matrix closest to C=0, and inversely proportional to these eigenvalues. Thus, large impedance peaks are observed in the vicinity of WPs of Type-I circuit model. At k values away from the WPs, the eigenvalues of electron and hole bands have relatively large (non-zero) magnitudes, resulting in the decay of impedance readout ( figure 5(a)). The impedance spectrum along k z is plotted in figure 5(b) for Critical-type WSMs. As we have seen from figure 3(b), the electron and hole bands in the spectrum exhibit a flat band dispersion. The corresponding WPs are marked by two impedance minima, but the corresponding impedance value is large and of the same order of magnitude as that of the impedance peaks in Type-I WPs. At other k values between the WPs, the impedance exhibits much larger values by several orders of magnitude. The existence of large impedance over the whole of the Brillouin zone apart from the WPs is the direct consequence of the zero-admittance flat dispersion in the admittance spectrum. Finally, the resonant impedance characteristics of a Type-II WSM is illustrated in figure 5(c), which shows an Figure 5. Impedance spectra for various Weyl phases at resonant frequency between the first and last nodes (N = 60) with C 1 =0.72 mF, C 2 =0.95 mF, C y =0.167 mF, C n =0 mF, L=1 mH and p = k x . (a) Resonant impedance behaviour for Type-I WSM network that shows large impedance peaks at the WPs. (b) In the impedance spectrum for Critical-type WSM system, the two WPs are marked by deep troughs even though their magnitude is comparable to the peak impedance value of Type-I WSM. However, the other points in k space show even larger impedance due to flat band nature at the transition phase. (c) Impedance spectrum for Type-II WSM system shows two primary peaks at the WPs and multiple secondary impedance peaks around the WPs due to the non-zero DOS in the vicinity of WPs arising from the electron and hole pockets. oscillatory response along the momentum direction that is parallel to the tilt direction (i.e. the k z direction). The primary impedance peaks occur at p =  k 2 z , just as in the Type-I WSM system, but in addition there are multiple secondary peaks at other values of k z . The emergence of these secondary peaks is due to the presence of eigenvalues with small magnitude of admittance, thus indicating the existence of finite DOS in the proximity of Type-II WPs. This is in line with the presence of hole and electron pockets as can be seen in the admittance dispersion for Type-II WSM shown in figure 3(c).

Conclusion
In conclusion, we have realized and characterized various WSM phases in the admittance dispersion of threedimensional LC topoelectric circuit. The characteristics of such LC circuits are described by the circuit admittance matrices, which are analogous to the Hamiltonians of the quantum tight-binding model. The different phases of the circuit can be switched between one another by adjusting the magnitude and sign of the capacitive/inductive coupling in the circuit lattice. An intermediate Critical-type WSM phase with a flat admittance band emerges at the transition between the Type-I and Type-II phases. In practice, the impedance readout of the circuit can be used to classify its topological WSM phases. To show this, we numerically calculated the impedance between the terminal nodes using the Green's function analogy. The impedance spectra of the different WSM phases reveal different characteristics. The impedance spectrum of Type-I WSM shows peaks in the vicinity of WPs whereas a Type-II WSM exhibits multiple secondary peaks in addition to the main peaks at the WPs. This impedance behaviour reflects the vanishing and non-vanishing DOS at the WPs in the Type-I and Type-II WSM phases, respectively. The LC circuit model allows ready implementation of WSM phases using basic circuit elements. The accessibility and ease of fabrication of the LC circuits make them an ideal platform for the design and characterization of topological WSM states and their transport properties.