Higher-order topological electric circuits and topological corner resonance on the breathing kagome and pyrochlore lattices

Electric circuits are known to realize topological quadrupole insulators. We explore electric circuits made of capacitors and inductors forming the breathing kagome and pyrochlore lattices. They are known to possess three phases (trivial insulator, higher-order topological insulator, and metallic phases) in the tight-binding model, where the topological phase is characterized by the emergence of zero-energy corner states. A topological phase transition is induced by tuning continuously the capacitance, which is possible by using variable capacitors. It is found that the two-point impedance yields huge resonance peaks when one node is taken at a corner in the topological phase. It is a good signal to detect a topological phase transition. We also show that the topological corner resonance is robust against the randomness of capacitance and inductance. Furthermore, the size of the electric circuits can be quite small to realize the topological phase together with topological phase transitions.

Figure 1

Illustration of the breathing kagome circuit composed of two types of capacitors (with capacitance $C_A$ and $C_B$) and inductors (with inductance $L$). Adjacent nodes are connected by capacitors and each node is grounded by an inductor. The size of the triangle is $\ell =6$.

Figure 2

Two-point impedance for the breathing kagome circuit. (a) The maximum value of the two-point impedance as a function of $C_A/C_B$. (b) The admittance spectrum at the resonant frequency $\omega ={\omega }_{\text{c}}$ as a function of $C_A/C_B$. (c) The numerator of the impedance ${\sum }_{n=1}^3{|{\psi }_{n,a}-{\psi }_{n,b}|}^2$. Spatial distribution of two-point impedance (d) in the trivial phase, (e) in the topological phase, and (f) in the metallic phase. One node is fixed in the vicinity of the triangle center. The absolute value of the impedance is represented by the length of the tubes. We have taken $C_B=1\mu \mathrm{F}$ and $L=1\mu \mathrm{H}$. We use a triangle with $\ell =6$.

Figure 3

Corner impedance $\left|Z\right|$ as a function of $\omega /{\omega }_{\text{c}}$ of the triangle made of the breathing kagome circuit (a) in the trivial phase ($C_A/C_B=-1.5$), (b) in the topological phase ($C_A/C_B=0.25$), and (c) in the metallic phase ($C_A/C_B=1.5$). Red circles indicate the resonance peak arising from the topological corner modes. The height of the peak is as huge as ${10}^9 \mathrm{\Omega }$ in the case of $\ell =9$. The size $\ell$ is shown in the figure. (a1)–(${\mathrm{c}1}^{\prime }$) Corner impedance $\left|Z\right|$ without randomness. (a2)–(${\mathrm{c}2}^{\prime }$) Corresponding impedance $\left|Z\right|$ in the presence of $5\%$ randomness. The inset of (${\mathrm{c}1}^{\prime }$) illustrates the breathing kagome circuit with the size $\ell =2$.

Figure 4

Two-point impedance for the breathing pyrochlore circuit. (a) The maximum value of the two-point impedance as a function of $C_A/C_B$. (b) The admittance structure at the resonant frequency $\omega ={\omega }_{\text{c}}$ as a function of $C_A/C_B$. (c) The numerator of the impedance ${\sum }_{n=1}^4{|{\psi }_{n,a}-{\psi }_{n,b}|}^2$. Spatial distribution of two-point impedance (d) in the trivial phase, (e) in the topological phase, and (f) in the metallic phase. One node is fixed in the vicinity of the tetrahedron center. Absolute value of the impedance is represented by the size of a ball. Huge balls are found at four corners in the topological phase. We have taken $C_B=1\mu \mathrm{F}$ and $L=1\mu \mathrm{H}$.