Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization

A prominent feature of some one-dimensional non-Hermitian systems is that all right eigenstates of the non-Hermitian Hamiltonian are localized in one end of the chain. The topological and trivial phases are distinguished by the emergence of zero-energy modes within the skin states in the presence of chiral symmetry. Skin states are formed when the system is nonreciprocal, where it is said to be nonreciprocal if the absolute values of the right- and left-going hopping amplitudes are different. Indeed, zero-energy edge modes emerge at both edges in the topological phase of a reciprocal non-Hermitian system. Then, analyzing higher-order topological insulators in nonreciprocal systems, we find the emergence of topological zero-energy modes within the skin states formed in the vicinity of one corner. Explicitly, we explore the anisotropic honeycomb model in two dimensions and the diamond lattice model in three dimensions. We also study an electric-circuit realization of these systems. Electrical circuits with (without) diodes realize nonreciprocal (reciprocal) non-Hermitian topological systems. Topological phase transitions are observable by measuring the impedance resonance due to zero-admittance topological corner modes.

Figure 1

(a)–(e) Development of skin states (in gold) and topological corner modes (in red) in rhombus geometry of a nonreciprocal honeycomb system as nonreciprocity $\gamma$ increases. The size of a ball represents the magnitude of LDOS. We have set $t_A=0.4$ and $t_B=1$. (f)–(h) The corresponding ones in rhombohedron geometry of a diamond lattice system. We have set $t_A=0.5$ and $t_B=1$.

Figure 2

(a) Illustration of the SSH model. One unit cell contains two sites $A$ and $B$. Nonreciprocal links are shown by the symbol $⊳$. (b) Illustration of the SSH circuit. The reciprocal link is represented by a condenser and a resistance connected in series. The nonreciprocal link is obtained by replacing this resistance with a set of a diode and a resistance connected in parallel. (c) Illustration of the anisotropic honeycomb model. (d) Illustration of the electric-circuit realization. Each node is connected to the ground via inductance $L$ as in (b).

Figure 3

LDOS in the topological and trivial phases of the SSH model for (a1)–(d1) and (a2)–(d2), respectively. The horizontal axis denotes the lattice site number. Red curves represent the topological edge modes in (a1)–(d1). (a1) LDOS of the Hermitian system, where the topological edge modes are prominent at both edges. (b1) LDOS of the reciprocal non-Hermitian system, which is quite similar to that of the Hermitian system. (c1)–(d1) LDOS of the nonreciprocal non-Hermitian system, where skin states are formed. (a3)–(d3) Impedance (in units of $\mathrm{\Omega }$) in the corresponding SSH circuits. We have taken $t_A=0.25, t_B=1$, and $\gamma =0$ for (a), $t_A=0.25+0.5i, t_B=1+0.5i$, and $\gamma =0$ for (b), while $t_A=0.25+0.5i, t_B=1+0.5i$, and $\gamma =0.1$ for (c), and $t_A=0.25+0.5i, t_B=1+0.5i$, and $\gamma =0.4$ for (d).

Figure 4

Energy spectra and topological numbers of the rhombus made of the anisotropic honeycomb lattice. The horizontal axis is $t_A$, while the vertical axis is the real part of the energy for (a1)–(d1), the imaginary part for (a2)–(d2), the absolute value for (a3)–(d3), and the topological number $W$ for (a4)–(d4). The horizontal red lines represent the topological corner modes, where the system is a second-order topological insulator. (a1)–(a4) for the Hermitian model, (b1)–(b4) for the reciprocal non-Hermitian model, and (c1)–(c4) and (d1)–(d4) for the nonreciprocal non-Hermitian model with increasing $\gamma$. The vertical lines represent the phase transition points at the value $t_A^{+}$ (magenta), $t_A^{-}$ (violet), and $\gamma /2$ (green). We have taken $t_B=1$ for (a), $t_B=1+i$ for (b), $t_B=1$ for (c), and $t_B=1$ for (d).

Figure 5

Impedance peaks are shown in the $(C_A/C_B)-\omega$ plane. The phase transition occurs at $C_A=C_B$. (a) Many impedance peaks are generated as implied by Eq. (8) both in the topological and trivial phases of the Hermitian SSH model ($R_A=R_B=0$). White curves represent the analytical result (8). (b) All these peaks are suppressed drastically, except for the topological peak in the topological phase of the reciprocal non-Hermitian SSH model ($R_A\ne 0,R_B\ne 0$).

Figure 6

Spatial distribution of impedance in the topological phase of the anisotropic honeycomb circuit. (a) In the Hermitian system, enhanced topological peaks emerge at both corners. (b) In the reciprocal non-Hermitian system topological peaks emerge at both corners, but they are not so prominent. (c) In the nonreciprocal system, an enhanced topological peak emerges only at corner $B$ corresponding to the topological skin corner states around corner $B$ as in Fig. 2.