Anomalous Floquet topological phase in a lattice of LC resonators

Abstract

Periodically driven systems provide a new platform for studying and realizing novel topological phases of matter that cannot be observed in static systems. These so-called anomalous Floquet topological insulators support topologically protected edge states, despite having zero Chern number bands. Here, we propose a circuit realization of an anomalous Floquet topological insulator. Based on a simple model, we designed a lattice of inductors and capacitors connected through electrical switches. We cast the governing equations of the circuit in the form of a Schrodinger-like equation and implement the Hamiltonian of an anomalous Floquet topological insulator by the circuit. Using a current source for excitation, the propagation of the topological edge state in the circuit is analyzed.

Graphical abstract

Appendix: energy loss in the switching process

As discussed in the main text, energy stored in each element (in the form of electric field in capacitors or magnetic field in inductors) is transferred to the nearby elements through a switching process. In the ideal case, there will not be any energy loss and the scheme discussed in Sect. 2 will work perfectly. Although introducing non-ideal switches will cause energy dissipation in the circuit, the desired behavior would be still observed if certain conditions are met. These conditions which are totally achievable with the current switching technology are high speed switching in comparison to the resonant frequency of the LC resonators and high ratio between the resistance in the on and off states of the switch.

To examine the effect of a non-ideal switch on the circuit, we considered the simple case of two LC resonators with a shared inductor which are controlled by two switches, illustrated in Fig. 8. The circuit can be readily simulated by an ODE solver. Here we used MATLAB function ode45 to solve the circuit for the voltages of capacitors and the current of the inductor. The switches are modeled as time varying resistors with their time dependence given by a hyperbolic tangent function, varying between two values ${R}_{\mathrm{on}}$ and ${R}_{\mathrm{off}}$.

$$R\left(t-{t}_{s}\right)={R}_{on}+\left({R}_{\mathrm{off}}-{R}_{on}\right)\left(1\pm \mathrm{tanh}w(t-{t}_{s})\right)$$

(20)

Here, ${t}_{s}$ is the switching time and $w$ controls the switching speed. We define the switching period $\Delta T$ as the time difference between the moments switch reaches 99 and 1 percent of its new resistance value.

Fig. 8

The Schematics of a simple LC circuit for the demonstration of the switching scheme

We solved the dynamical equations of the circuit for different ratios of switching period $\Delta t$ to the resonance period of the resonators T, $0.01<\frac{\Delta t}{T}<0.1$. Capacitor 1 is initialized with an initial voltage of 1 Volt which is supposed to transfer to the capacitor 2 after t = $T/2$ according to the underlying switching scheme. We set ${R}_{\mathrm{off}}/{R}_{\mathrm{on}}=100$ with ${R}_{on}=0.1{Z}_{0}=0.1\sqrt{L/C}$ and plot the voltage on each capacitor at t = $T/2$ after the initialization.

As illustrated in Fig. 9, faster switching times helps the energy transport. The highest voltage in the capacitor 2 is obtained for the fastest switching time $\Delta t=0.001T$, in which the voltage in the second capacitor is a little more than 0.7 V. To better show the dynamics of the circuit, we also illustrate the variations of the voltages and inductor’s current for two cases of $\Delta T=0.01T$ and $\Delta T=0.1T$ in Fig. 10.

Fig. 9

The voltage at the nodes of each capacitor after half of the resonators period

Fig. 10

The variations of the voltages of two capacitors, ${\mathrm{V}}_{1}$ and ${\mathrm{V}}_{2}$, and the middle inductor’s current ${\mathrm{Z}}_{0}\mathrm{I}$ as a function of time. ${\mathrm{R}}_{1}$ and ${\mathrm{R}}_{2}$ curves denote the on and off state of two switches. They are normalized to their maximum value here and have no dimension. Comparing two cases, the final voltage in the second capacitor ${\mathrm{V}}_{2}$ when $\mathrm{\Delta T}=0.1\mathrm{T}$ is half of the case with $\mathrm{\Delta T}=0.01\mathrm{T}$ which shows higher energy dissipation when the switching time is increased

Increasing the switching time increases the dissipation of inductor’s current and therefore less energy transport into the second capacitor, as illustrated in Fig. 10. Therefore to work close to the ideal case, the switching time should be as small as possible.

The current technology of MOSFET switches enables switching times of orders of a few nanoseconds with on-state resistance of a few Ohms (for instance, N-channel MOSFETs like 2N7002). CMOS switches are currently employed for the on/off resistance ratios of order of 10³ – 10⁵. [37] By choosing a high performance switch, the elements of the circuit can be chosen to provide the best possible performance.