Edge state mimicking topological behavior in a one-dimensional electrical circuit

For one-dimensional (1D) topological insulators, the edge states always reside in the bulk bandgaps as isolated modes. The emergence and vanishing of these topological edge states are always associated with the closing/reopening of the bulk bandgap and changes in topological invariants. In this work, we discover a special kind of edge state in a 1D electrical circuit, which can appear not only inside the bandgap but also outside the bulk bands with the changing of bulk circuit parameters, resembling Tamm states or Shockley states. We prove analytically that the emergence/vanishing of this edge state and its position relative to the bulk bands depends on the intersections of certain critical frequencies. Specifically, the edge mode in the proposed circuit can be mathematically described by polynomials with roots equal to some critical frequencies in the bulk circuit. From this point of view, the transition of the edge state is uniquely determined by the order of the critical frequencies in the bulk circuit. Such topological behaviors shown by the edge state in the proposed electrical circuit may indicate, in a broader sense, the presence of certain type of topology.


Introduction
Edge states, or surface states, appearing at a material interface, have been observed in in various physical systems. They can be classified into trivial and nontrivial phases, depending on whether they possess certain nontrivial topologies in the bulk. For topological edge states, they always appear either as isolated modes inside the bandgap of one-dimensional (1D) topological systems, or as continuous lines/surfaces between the bulk bands in two-dimensional (2D)/three-dimensional (3D) topological systems. The emergence and vanishing of them are always accompanied by the closing and reopening of bulk bandgap and are related to changes in the topological invariant of the bulk system. This is known as the bulk-edge correspondence [1][2][3], which ensures a one-to-one correspondence between the number of topological edge states in an open system and the topological invariant in the bulk. Meanwhile, trivial edge states, commonly known as the Tamm states [4] or Shockley states [5], do not present such topological behavior. Their existence is highly related to the boundary properties and can appear almost everywhere in the band structure, even inside the bulk band as the bound states in the continuum [6,7].
In this work, we discover a special type of edge state in a 1D electrical circuit which displays certain topological behaviors. Distinct from conventional topological systems, we found that the edge state can traverse the bulk bands by varying only the bulk circuit parameters, i.e. they can appear not only inside the bulk bandgap, but also below or above the bulk bands. Most importantly, the emergence/vanishing of the edge states as well as their transition among the bulk bands is strictly determined by the competition of certain critical frequencies of the bulk circuit. That is to say, the transition of the surface state can only occur at the crossings of these critical bulk frequencies, making the circuit an analog of a topological insulator. The transport of edge state through the bulk bands is experimentally demonstrated through the measurement of the bulk and edge states in the circuit along a sweeping path in three different phase diagrams.
We note that the edge state observed in the proposed circuit belongs to the category of Tamm state. Although there have been literatures reported on the manipulation of the position of the Tamm states, or Shockley states, to be above or below the bulk bands [36][37][38], the boundary sites or the defect sites where the surface states appear have been intentionally modified. It is important to note that, in the circuit we proposed, the boundary sites are kept the same as the bulk sites, which forms a natural boundary truncation.

Circuit design and phase analysis
Each unit cell of the circuit comprises two resonators and two identical coupling links, as shown in figure 1(a). The grounded resonators are composed of a shunt-resonant circuit with inductors L a /L b and capacitors C a /C b , and the coupling links are made of identical shunt-resonant circuits with L 0 and C 0 . We remark that this circuit is different from the Su-Schrieffer-Heeger (SSH) type topological circuit reported previously [11,29,30], in that both the on-site and coupling terms in our circuit are made of parallel resonant LC tanks, while in previous designs, the coupling terms are made of a single inductor or capacitor.
For the above circuit unit cell in the periodic boundary condition, we have the following Kirchhoff's current law circuit equations at the two nodes, (2) and the Kirchhoff's voltage law circuit equations in the four loops as, in which V 1 /V 2 are the voltage at node 1/2, I 1 /I 2 are the current flowing through the coupling branches O 1 /O 2 , I a /I b are the current flowing through the resonator branches A/B, ω is the angular frequency, k is the quasi-wavevector. By defining E = ω 2 C 0 L 0 , C i = r i C 0 , L i = s i L 0 , t i = r i s i , we can rewrite equations (1)-(3) in a dimensionless form as (see supplementary materials note S1 (https://stacks.iop.org/NJP/23/103005/mmedia)), in which, Noted that by choosing a proper gauge V = R 1/2 V, we can have a Hermitian circuit Hamiltonian Figure 1(b) shows the bulk band structure of a typical circuit with parameter s a = 1, s b = 1, r a = 1.2, r b = 0.8. The six dashed lines in the plot mark two resonant frequencies of the A/B branches E r,a = 1/t a , E r,b = 1/t b , and four frequency limits of the bulk bands, E π,a = 1+2s a 2s a +t a , E π,b = 1+2s b , in which Δ 1 is a positive real number (see supplementary materials note S1). The upper/lower bounds of the bulk bands E π,a , E π,b depends only on the resonant frequencies of the A/B links, while the bounds of the bandgap E 0,+ , E 0,− depend on all circuit parameters. Crossing among the six critical frequencies can occur as the parameters s a , s b , r a , r b are tuned. We will see in the following demonstrations that the relative relations between these critical frequencies can determine the edge states of the system, that is, the number of edge modes and its position in the open circuit.
To investigate the edge state in such an electrical circuit, we consider a finite circuit chain with 50.5 unit cells (i.e. 101 circuit nodes), in which both boundaries are terminated with resonator B. We first consider the simple case with s a = s b = 1. Figure 2(a) shows the phase diagram of the circuit for [r a , r b ] ranging from 0 to 2. Two critical lines r b = r a , r b = 1, divide the phase diagram into three regions, each representing an individual state describing a distinct location of the edge state relative to the bulk bands. The square bracket representation [1/0, 1/0, 1/0] indicates the existence/absence of edge state below, inside, and above the bulk band. For example, the yellow region marked with [0,1,0] represents an in-gap edge state which is located between the lower and upper bands; while the [1,0,0] and [1,0,0] regions represent a bottom and top edge state which reside below or above the bulk bands. To observe how the edge state crosses through the two bulk bands, we provide in figures 2(b)-(e) the variation of the band structure along four straight lines in the phase diagram (figure 2(a)), respectively, at r a = 0.5, r a = 1.5, r b = 0.5, and r b = 1.5, as marked by gray-dashed lines. The black dots in figures 2(b)-(e) represent the bulk modes, while the cyan, magenta, and blue circles represent the bottom, in-gap, and top edge states. For r b = 0.5 ( figure 2(b)), the edge state locates inside the bulk band gap for r a < 0.5, and jumps above the upper band as r a > 0.5; while for r b = 1.5, the edge state remains below the lower band until the band gap closes at r a = 1.5, and appears inside the band gap as r a further increases. Note that the gap closing line r a = r b is not the only condition for phase transition; the edge state also experiences phase changes at r b = 1, when the resonant frequency of the resonator B equals that of the coupling link, which is normalized as 1. Note that this phase transition line would become r a = 1 if the boundary is terminated with resonator A. For example, when r b sweeps along the line r a = 0.5 (figure 2(d)), the edge state firstly transits from above the upper band to inside the band gap, and then crosses the lower band at r b = 1 to the lower side of the bulk bands. For r a = 1.5 (figure 2(e)), the edge state firstly passes through the upper band from above at r b = 1, and then transits to Note that one can find Tamm states transversing across the bulk bands in other systems [36][37][38]. For example, we can induce a similar surface state in the conventional SSH model by tuning the potential of the boundary sites (see supplementary materials note S4). However, the boundary sites in this case have been intentionally modified to be different from the bulk sites. While in the our circuit, both the boundary sites and bulk sites are kept identical. Most importantly, the phase transition of the edge state observed in the proposed circuit is strictly dependent on the bulk frequencies, as will be demonstrated in the following section.

Analytical solution of the edge state
To further gain a physical insight into the mechanism of the edge state in our electrical circuit, we next consider a more general case with an arbitrary parameter set s a , s b , r a , r b , and give analytical solutions to its edge mode. To obtain the analytical solution of the edge state, we consider a semi-infinite chain with one end terminated by resonator B. Using the transfer matrix method to obtain the exponential decay ratio of all modes in the bulk circuit and applying it to the boundary, we obtain the equation for solving the edge mode (see supplementary materials note S2), in a more general case with arbitrary boundary site L b0 , representing a mixed frequency term combining the resonant frequencies of both the bulk and boundary sites. However, for a circuit with natural boundary truncation in our cases, in which the boundary site needs to be kept identical to the bulk site, E r,b is exactly the same as the resonant frequency of resonator B. It is important to note that just because E r,a and E r,b are not relevant to the bulk bands, the transition of the edge state is not solely determined by the bandgap closing condition as in the conventional SSH-type model, as will be demonstrated in the following. They are also of vital importance for the unusual edge state to emerge outside the bulk bands.
It is intriguing to note that equation (6) takes the simple form of the equating between two polynomials with roots equal to five critical frequencies E π,a , E π,b , E 0,+ , E 0,− , E r,b of the bulk circuit. The status of the edge state, i.e. the existence of edge state and its position relative to the bulk bands, can only change at the crossings of these critical frequencies, that is, when any two elements each from one of the two critical frequency sets {E π,a , E b,r } and {E π,b , E 0,+ , E 0,− } are equal to each other. Note that for the case of boundary termination with resonator A, one should replace E b,r on the left side of equation (6) with E a,r , and swap E π,a , E π,b . By numerically analyzing the variations of the five critical frequencies in the phase diagram, one can find the following three equations that determine the boundaries of distinct phases in our circuit, The theoretical prediction of the above critical lines is verified from the phase diagram in figures 3(a) and (b), for two cases with s a = 0.  Figure 3(c) shows that as we scan θ = tan −1 t b /t a counterclockwise from θ = 0 • , the edge state first appears inside the band gap until θ reaches critical line 1 at 45 • . As θ resides between critical line 1 and line 3, the edge state appears both below and inside the bulk band. As θ further increases, the circuit enters phase [1,0,0], with the edge located below the bulk bands. The edge state then crosses from below the lower band into the band gap at t b = 1 (line 2), until it reaches again at line 3, at which the edge state is located both inside and above the bulk bands. As θ passes line 3, the circuit enters into phase [0,0,1], at which the edge state is located above the bulk bands. To investigate how the competition of the critical frequencies determines the phase of the circuit, we present in figures 3(e) and (f) the variation of the five critical frequencies as well as the bulk/edge modes as θ scans along the same circular path as in figures 3(c) and (d). It is observed that the crossings of the critical frequency curves correspond exactly to the changes of the edge state in either its existence or position. In  (6), including both boundary terminations with resonator A and B. All phases are shaded with the same color as in (a) and (b). Those edge states corresponding to boundary termination with site B are selected by using the corresponding decay direction (see supplementary materials note S3) and are marked by colored circles. The red pins mark the crossings of these critical frequencies that correspond to phase transitions. other words, the existence and position of the edge state remain unchanged as long as the critical frequency curves E π,a (θ) , E r,b (θ) do not cross with E π,b (θ) , E 0,+ (θ) , E 0,− (θ), as marked in figures 3(e) and (f) by small red pins. The continuous red dotted lines represent all the edge mode solutions solved from equation (6). Those red dots marked by circles correspond to the case with boundary terminated by resonator B, while the rest red dots without circles correspond to the case with boundary terminated by resonator A (see supplementary materials note S3). Note that the crossings between the curves E π,a (θ) and E r,b (θ) do not correspond to phase transitions because both frequencies are on the left-hand side of equation (6).

Experimental demonstration
To experimentally verify the existence of the edge state and its transition among the bulk bands, we construct a circuit with 10.5 unit cells (21 nodes), as shown in figure 1(c). C a /C b and L a /L b are designed as capacitor and inductor sets, which are composed of multiple capacitors/inductors with different capacitance/inductances being connected in parallel to a multiway switching. This configuration allows us to reach all the phases of the circuit as the capacitance C a /C b and L a /L b can be manually swept from 100 to 2900 pF and 11 to 22 uH, respectively. L 0 and C 0 are fixed at 22 μH and 1000 pF, respectively, during the measurement. This is equivalent to sweep r a /r b from 0.1 to 2.9, and s a /s b from 0.5 to 1.0. Three different cases, s a = 1.0, s b = 1.0, s a = 1.0, s b = 0.5 and s a = 0.5, s b = 1.0 are chosen for the experiment, which cover all possible types of phase diagrams. In the first case with L a = L b = 22 μH, figures 4(a) and (b) show the measured and simulated finite band structures as C a and C b sweep along a rectangular path as indicated in supplementary figure S1(a). The bulk states are obtained from the impedance spectrum (Z 11 ) measured at node 10 and 11 in the circuit, as indicated by the gray shading in figure 4(a). The red dots represent the edge state which are obtained as the frequency peaks of the impedance spectrum measured at node 1. More experimental details are given in the method. The first point starts from the bottom left corner of the path with C a /C b of 300/700 pF, which lies in the [0,1,0] phase with the ingap edge state. As C a increases, the circuit enters [0,0,1] phase, with the edge state remaining above the upper bulk band. The phase moves back to phase [0,1,0] as C a reaches 1800 pF and C b exceeds 1000 pF, as can be observed in figure 4(a) that and hence, leading to the absence of red dots in these regions. The frequency spectra of the edge state and bulk state for the other sweeping point follow a similar trend as those in the first case. One may notice that some edge states in case 2 are further away from the bulk states than those in case 1, due to the fact that those points have larger distances to the phase boundaries.
For case 3 with the value of L a and L b swapped, the phase diagram displays a new center at C a = 2000 pF and C b = 1000 pF, and two new phase regions [

Conclusion
In this work, we present a special kind of edge state in a 1D electrical circuit that mimics the behavior of the topological one. We found that the emergence/vanishing of the edge state and its position relative to the bulk bands in such a Hermitian electrical circuit is uniquely determined by the competition among certain critical frequencies of the bulk circuit, which can be strictly solved from an elegant equation formed by two polynomial functions with roots corresponding exactly to these critical bulk frequencies. Noted that although the trimer model having three sites per unit cell can exhibit similar Tamm state that are controlled by the bulk parameters [39,40], the circuit proposed in this work composed of only two sites per unit cell serves as the simplest model to exhibit such properties. In addition, the edge states in the proposed circuit can appear outside the bulk bands, while they are confined inside the bandgaps for the trimer model.
We emphasize that the unusual edge state found in our circuit is not the conventional Tamm state, because the emergence and vanishing of Tamm state in quantum electronic and photonic is simply a boundary effect and does not rely on the bulk parameters. Most importantly, there is no literatures reported on the observation of conventional Tamm state to exhibit such topological behaviors. It should also be noted that the unusual edge state shown in this work is neither the SSH-type topological edge state. Because the Chiral symmetry is broken by the unbalanced grounded terms (resonator A and B), Zak phase does not take quantized value (0 and π) for the bulk circuit Hamiltonian in equation (4). However, the fact that the phase of the edge state is ambiguously determined by the order of the bulk critical frequencies can be reasonably viewed, in a broader sense, as the original definition of topology in mathematics, which describes certain quantities that are invariant under continuous deformations. We expect more intriguing physics of the extension of such unusual edge stage to higher-dimensional, and non-Hermitian circuits.

Experimental details
The impedance spectra for each circuit node are measured using a vector network analyser (VNA, Agilent 8753ES) in the S 11 (reflection coefficient) format with 2 KHz frequency resolution, and is transformed to Z 11 (input impedance) using Z 11 = Z 0 (1 + S 11 ) / (1 − S 11 ). It can be proved that the impedance Z 11 measured for a certain circuit node is strictly equivalent to the voltage response on it when excited by an ideal current source on the same node, which does not affect the circuit to be measured and can correctly reflect the permitted mode on the tested node.