Topological Euler insulators and their electric circuit realization

The Euler number is a topological number recently debuted in the topological physics. It is defined for a set of bands unlike the Chern number defined for a single band. We propose a simple model realizing the topological Euler insulator. We utilize the fact that the Euler number in a three-band model in two dimensions is reduced to the Pontryagin number. A skyrmion structure appears in momentum phase, yielding a nontrivial Euler number. Topological edge states emerge when the Euler number is nonzero. We discuss how to realize this model in electric circuits. We show that topological edge states are well signaled by impedance resonances.

Figure 1

Band structures of a nanoribbon with 40-site width. There are three bulk bands, one of which is perfectly flat at zero energy. The horizontal axis is the momentum $k$, and the vertical axis is the energy in unit of $t$. (a) Topological insulator phase (TEI-1) with $\mu =-t$, (b) metal phase with $\mu =0$, (c) topological insulator phase (TEI-2) with $\mu =t$, (d) metal phase with $\mu =2t$, and (e) trivial insulator phase with $\mu =4t$. We have set $\lambda =t$. (*1) Bird's eye view of the bulk band. (*2) Projection of the bulk bands. (*3) Band structure of a nanoribbon. The edge states are marked in magenta. (*4) Enlarged band structure of (*3) near the zero energy. The gap closing points of topological edge states are marked in green circles. We have set $c_3=1$ and $c_1=0.25$. The topological numbers for the insulator phases are illustrated in Fig. 2.

Figure 2

Euler numbers as a function of $\mu /t$ for (a) $Q_{31}$ and (b) $Q_{12}$. We note that $Q_{23}=0$ for all $\mu$. There are two topological insulator phases denoted by TEI-1 and TEI-2. The horizontal axis is $\mu /t$.

Figure 3

Spin textures in the $k_x\text{−}{k}_y$ plane for (*1) ${\mathit{n}}_1$, (*2) ${\mathit{n}}_2$, and (*3) ${\mathit{n}}_3$, yielding the Pontryagin numbers $Q_1, Q_2$, and $Q_3$. (a*), (c*), and (e*) are for the topological insulator phase (TEI-1) with $\mu =-t$, for the topological insulator phase (TEI-2) with $\mu =t$, and for the trivial insulator phase with $\mu =4t$. They correspond to the band structures (a*), (c*), and (e*) in Fig. 1. The corresponding Euler numbers are shown in Fig. 2.

Figure 4

Local Pontraygin density in the $k_x\text{−}{k}_y$ plane for (a) metal phase with $\mu =0$, (b) topological insulator phase with $\mu =t$, and (c) metal phase with $\mu =2t$.

Figure 5

Absolute value of the eigenfunction for localized edge states in a nanoribbon with 40-site width. Panels (a), (b), and (c) are for the bands $|u_1\rangle , |u_2\rangle$ and $|u_3\rangle$. They are well localized at the edges.

Figure 6

Impedance in the $k\text{−}\omega$ plane. (a*) Topological phase with $\mu =-t$ and (b*) trivial phase with $\mu =-4t$. (*1) Top view and (*2) bird's eye's view. We have set $c_3=1$ and $c_1=0.25$. The vertical axis is the impedance in unit of $\mathrm{k}\mathrm{\Omega }$. We have used a nanoribbon with width 6. We see clearly six curves in the bulk band in both the topological and trivial phases, and the edge band only in the topological phase. The impedance peak at ${\omega }_0$ is due to the perfect-flat bulk band in Fig. 1.