Mirror skin effect and its electric circuit simulation

We analyze the impacts of crystalline symmetry on non-Hermitian skin effects. Focusing on mirror symmetry, we propose another type of skin effect, a mirror skin effect, which results in a significant dependence of the energy spectrum on the boundary condition only for the mirror invariant line in the two-dimensional Brillouin zone. This effect arises from the topological properties characterized by a mirror winding number. We further reveal that the mirror skin effect can be observed for an electric circuit composed of negative impedance converters with a current inversion where switching the boundary condition significantly changes the admittance eigenvalues only along the mirror invariant lines. Furthermore, we demonstrate that extensive localization of the eigenstates for each mirror sector results in an anomalous voltage response.

Figure 1

Sketch of the electric circuit showing the mirror skin effect. The blue (cyan) symbols represent negative impedance converters with current inversion connecting intralayer (interlayer) nodes, respectively. We refer to the upper (lower) layer as A (B), respectively. The black symbols denote capacitors. Dots represent nodes. This system preserves the mirror symmetry whose mirror plane is illustrated with a vertical yellow plane.

Figure 2

Energy spectrum of the Hamiltonian (4) for $(t,\mu ,\mathrm{\Delta })=(1,2,1.8)$. Blue (orange) dots are data obtained under the (open) periodic boundary condition for the $y$ direction. For the $x$ direction, the periodic boundary condition is imposed. The number of sites for the $y$ direction is set to $L_y=20$.

Figure 3

(a) [(b)] Amplitude of the right eigenvectors of the Hamiltonian (4) for $(t,\mu ,\mathrm{\Delta })=(1,2,1.8)$ at $k_x=0$ ($\pi /6$), respectively. The data are obtained under the periodic (open) boundary condition for the $x$ ($y$) direction, respectively. The number of the sites along the $y$ direction is set to $L_y=20$.

Figure 4

(a) [(b)] Admittance spectrum for each case of the boundary condition for $k_x=0$ ($k_x=\pi /6$), respectively. Blue (orange) dots denote the admittance eigenvalue $j_n$ under the periodic (open) boundary condition for the $y$ direction with $L_y=14$. The periodic boundary condition is imposed for the $x$ direction. The data are obtained for $L_0=120\mu \mathrm{H}$, $C_0=47\mathrm{n}\mathrm{F}$, $C_1=33\mathrm{n}\mathrm{F}$, and $f=117.4\mathrm{kHz}$, where $f$ denotes frequency $\omega =2\pi f$. (c) [(d)] Voltage profile of layer A for the case where the current with the plus (minus) sector of reflection is fed. (e) Illustration of setup to observe the anomalous voltage response with $I_{\mathrm{in}}=I_{+\mathrm{L}}$. For the above parameter sets, the fed current is $I_{\mathrm{s}}=0.0001\mathrm{A}$ with a fed voltage $V_{\mathrm{s}}=0.002\mathrm{V}$ for $C_{\mathrm{s}}=0.068\mu \mathrm{F}$, and $L_{\mathrm{s}}=27\mu \mathrm{H}$.