Non-Hermitian non-Abelian topological insulators with $\mathcal{PT}$ symmetry

We study a non-Hermitian non-Abelian topological insulator preserving $\mathcal{PT}$ symmetry, where the non-Hermitian term represents nonreciprocal hoppings. As it increases, a spontaneous $\mathcal{PT}$ symmetry breaking transition occurs in the perfect-flat band model from a real-line-gap topological insulator into an imaginary-line-gap topological insulator. By introducing a band bending term, we realize two phase transitions, where a metallic phase emerges between the above two topological insulator phases. We discuss an electric-circuit realization of non-Hermitian non-Abelian topological insulators. We find that the spontaneous $\mathcal{PT}$ symmetry breaking as well as the edge states are well observed by the impedance resonance.

Figure 1

Illustration of the tight-binding Hamiltonian. (a) Hermitian and (b) non-Hermitian models. Interactions between the $\alpha$ and $\beta$ chains yield a non-Abelian topological number. All other chains shown in green act as spectators. Red arrows represent nonreciprocal hoppings.

Figure 2

Energy spectrum of the non-Hermitian Hamiltonian in nanoribbon geometry. Eigenvalues of (a1) the perfectly flat $\alpha$ and $\beta$ bands with $\xi =0$ and (a2) the bended bands with $\xi =0.5$ shown in blue. The red dots represent the topological edge states. The band structure as a function of $\xi$ with (b1) $\gamma =0$ and (b2) $\gamma =0.25$. The red lines represent the topological edge states. (c1, c2) Real part of the energy. (d1, d2) Imaginary part of the energy. We set $\xi =0$ for (c1) and (d1), while we set $\xi =0.5$ for (c2) and (d2). The bulk bands are colored in blue, while the edge states are colored in red. The spectator band is colored in green. When $\xi =0$, there are two phases, a real-line-gap topological insulator (real-TI) phase and an imaginary-line-gap topological insulator (im-TI) phase. When $\xi \ne 0$, a metallic phase emerges between these two topological insulator phases. We set ${\varepsilon }_{\alpha }=1$ and ${\varepsilon }_{\beta }=2$.

Figure 3

(a1)–(e1) Real part of the energy and (a2)–(e2) imaginary part of the energy. (a1) and (a2) for a real-line-gap topological insulator (real-TI) with $\gamma =0$; (b1) and (b2) for a phase transition point with $\gamma ={\gamma }_1=0.3$; (c1) and (c2) for a metal with $\gamma =0.4$; (d1) and (d2) for a phase transition point with $\gamma ={\gamma }_0=0.5$; (e1) and (e2) for an imaginary-line-gap topological insulator (im-TI) with $\gamma =0.6$. We set $\xi =0.2$ for all figures. See also the caption of Fig. 2.

Figure 4

Non-Abelian topological number marked in red as a function of $\gamma$ for various $\xi$. (a) $\xi =0$, (b) $\xi =-0.2$, (c) $\xi =-0.4$, (d) $\xi =0.2$, and (e) $\xi =0.4$. It is quantized at $1/2$ except for the metallic phase. In the figures, real-TI (im-TI) stands for real(imaginary)-line-gap topological insulator phase.

Figure 5

(a) Real and imaginary parts of the bulk-band energy in the $(\gamma ,\xi )$ plane. (b) Topological phase diagram in the $(\gamma ,\xi )$ plane. Metallic phase appears except for $\xi =0$. In the figure, real-TI (im-TI) stands for real(imaginary)-line-gap topological insulator phase.

Figure 6

(a) Illustration of the electric circuit corresponding to the lattice in Fig. 1. The hopping along the $\alpha$-chain ($\beta$-chain) is represented by the inductance $L$ (the capacitance $C$). (b) Negative impedance converter $R_X$ represents an imaginary hopping [40]. (c) Operational amplifier circuit $C_{\gamma }$ represents a nonreciprocal hopping [41].

Figure 7

Real and imaginary parts of impedance $Z_{aa}$ at the edge as a function of frequency $\omega$. (a1)–(a4) Hermitian model with $\gamma =0$. (b1)–(b4) Non-Hermitian model with $\gamma =0.025$, (c1)–(c4) with $\gamma =0.04$ (phase transition point ${\gamma }_1$), (d1)–(d5) with $\gamma =0.045$, (e1)–(e4) with $\gamma =0.05$ (phase transition point ${\gamma }_0$), and (f1)–(f4) with $\gamma =0.075$. We used a finite chain with open boundary condition (red) and periodic boundary condition (cyan). (a1)–(f2) $\xi =0$. (a3)–(f4) $\xi =0.2$. We set ${\varepsilon }_{\alpha }=1$ and ${\varepsilon }_{\beta }=1.1$. The length of the chain is 20.