Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems

We unveil the geometrical meaning of winding number and utilize it to characterize the topological phases in one-dimensional chiral non-Hermitian systems. While chiral symmetry ensures the winding number of Hermitian systems are integers, it can take half integers for non-Hermitian systems. We give a geometrical interpretation of the half integers by demonstrating that the winding number $\nu$ of a non-Hermitian system is equal to half of the summation of two winding numbers ${\nu }_1$ and ${\nu }_2$ associated with two exceptional points, respectively. The winding numbers ${\nu }_1$ and ${\nu }_2$ represent the times of the real part of the Hamiltonian in momentum space encircling the exceptional points and can only take integers. We further find that the difference of ${\nu }_1$ and ${\nu }_2$ is related to the second winding number or energy vorticity. By applying our scheme to a non-Hermitian Su-Schrieffer-Heeger model and an extended version of it, we show that the topologically different phases can be well characterized by winding numbers. Furthermore, we demonstrate that the existence of left and right zero-mode edge states is closely related to the winding number ${\nu }_1$ and ${\nu }_2$.

Figure 1

Schematic diagram of three typical winding cases by calculating the model with $h_x=t+t^{\prime }cos\left(k\right)$ and $h_y=t^{\prime }sin\left(k\right)-i\delta$. In (a)–(c) $t=0.5,t^{\prime }=1$, and $\delta$ takes $1.75,1$, and 0.2, respectively. Times for the trajectory surrounding the EP marked by the asterisk gives ${\nu }_1$ and the one surrounding the pentagram gives ${\nu }_2$. It is straightforward to get (a) ${\nu }_1=0$ and ${\nu }_2=0$, (b) ${\nu }_1=0$ and ${\nu }_2=1$, and (c) ${\nu }_1=1$ and ${\nu }_2=1$.

Figure 2

Schematic diagram of non-Hermitian SSH model with different nearest-neighbor hopping strengths inside a cell.

Figure 3

(a) Phase diagram of the non-Hermitian SSH model. The colors from shallow to deep represent $\nu =0,1/2$, and 1 as marked. (b)–(h) Winding diagrams with different parameters. They are all on the dashed line in (a) with $t=0.5$ and different values of $\delta$. In each subfigure, (b) $\delta =-1.75$, (c) $\delta =-1$, (d) $\delta =-0.2$, (e) $\delta =0$, (f) $\delta =0.2$, (g) $\delta =1$, and ($\mathrm{h})\delta =1.75$. The red asterisk and blue pentagram represent two different EPs.

Figure 4

(a) Phase diagram of non-Hermitian SSH model characterized by the number of zero-mode edge states. The red shallow represents the existence of a left zero-mode edge state, while the blue shallow represents the right edge state. The number represents the quantities of edge states in its mark area. Subgraphs (b)–(d) are distributions of moduli of the zero-mode edge states of the three marks (ring, star, and square) at $t=0.5$ and $\delta =-1,0,1$ in subgraph (a), respectively. (e) The dispersion of $H^{†}\times H$ with parameters $t=0.5$ and $\delta$ changing from $-2$ to 2 for the lattice size with 200 sites.

Figure 5

(a) The phase diagram of the non-Hermitian extended SSH model with $\mathrm{\Delta }=1$. The colors from shallow to deep represent $\nu =0,1/2,1,3/2$, and 2 as marked. (b)–(h) Winding diagrams with different parameters. They are all on the dashed line in (a) with $t=0.3$ and different values of $\delta$. In each subfigure, (b) $\delta =-2.6$, (c) $\delta =-1.5$, (d) $\delta =-0.5$, (e) $\delta =0$, (f) $\delta =0.5$, (g) $\delta =1.5$, and (h) $\delta =2.6$. The red asterisk and blue pentagram represent two different EPs.

Figure 6

(a) In this figure, we distinguish different regions by the number of left zero-mode edge states for the semi-infinite system with the end at the left side. In the dark area, there are two left zero-mode edge states corresponding to the region with both eigenvalues less than unity, while in the shallow area, there is only one zero-mode edge state. In the blank area, there is no zero-mode state. Subfigure (b) shows the case with the end at the right side. Similarly, blank, shallow, and dark areas correspond to regions with no, one, and two right zero-mode edge states.

Figure 7

(a)–(e) Distribution of moduli of the zero-mode edge-state wave functions under open boundary conditions, which correspond to the situation in subgraphs (c)–(g) in Fig. 5 with $\mathrm{\Delta }=1,t=0.3$, and $\delta =-1.5,-0.5,0,0.5,1.5$, respectively. Red solid lines and black dash-dot lines represent left-side edge states, while blue dashed lines and purple dotted lines represent right-side edge states.