Non-Bloch Band Theory of Non-Hermitian Systems

In spatially periodic Hermitian systems, such as electronic systems in crystals, the band structure is described by the band theory in terms of the Bloch wave functions, which reproduce energy levels for large systems with open boundaries. In this paper, we establish a generalized Bloch band theory in one-dimensional spatially periodic tight-binding models. We show how to define the Brillouin zone in non-Hermitian systems. From this Brillouin zone, one can calculate continuum bands, which reproduce the band structure in an open chain. As an example, we apply our theory to the non-Hermitian Su-Schrieffer-Heeger model. We also show the bulk-edge correspondence between the winding number and existence of the topological edge states.

Figure 1

Schematic figure of the band structure (a) in a finite open chain with various system sizes $L$, and (b) in the generalized Bloch Hamiltonian. The vertical axis represents the distribution of the complex energy $E$.

Figure 2

(a) Non-Hermitian SSH model. The dotted boxes indicate the unit cell. (b)–(d) Generalized Brillouin zone $C_{\beta }$ of this model. The values of the parameters are (b) $t_2=1$, $t_3=1/5$, ${\gamma }_1=4/3$, and ${\gamma }_2=0$, with (b-1) $t_1=1.1$ and (b-2) $t_1=-1.1$; (c) $t_1=0.3$, $t_2=1.1$, $t_3=1/5$, and ${\gamma }_1=0$, with (c-1) ${\gamma }_2=4/3$ and (c-2) ${\gamma }_2=-4/3$; and (d) $t_2=0.5$, $t_3=1/5$, ${\gamma }_1=5/3$, and ${\gamma }_2=1/3$, with (d-1) $t_1=0.3$ and (d-2) $t_1=-0.3$.

Figure 3

Phase diagram and bulk-edge correspondence with $t_3=1/5,{\gamma }_1=5/3$, and ${\gamma }_2=1/3$. (a) Phase diagram on the $t_1-t_2$ plane. The blue region represents that the winding number is 1, and the orange region represents that the system has exceptional points. Along the black arrow in (a) with $t_2=1.4$, we show the results for (b) the winding number, (d) energy bands in a finite open chain, and (e) the continuum bands from the generalized Brillouin zone $C_{\beta }$. The edge states are shown in red in (d). (c) shows ${\ell }_{+}$ (red) and ${\ell }_{-}$ (blue) on the $\mathit{R}$ plane with $t_1=1$ and $t_2=1.4$.

Figure 4

Phase diagram and bulk-edge correspondence with $t_1=0$, $t_2=1$, and $t_3=1/5$. (a) Phase diagram on the ${\gamma }_1-{\gamma }_2$ plane. The blue region represents that the winding number is 1, and the orange region represents that the system has exceptional points. (b) Energy bands calculated along the green arrow in (a) with ${\gamma }_2=1.4$. Note that ${\gamma }_c\simeq 1.89$. The edge states are shown in red. (c),(d) Loops ${\ell }_{+}$ (red) and ${\ell }_{-}$ (blue) on the $\mathit{R}$ plane. The values of the parameters are (c) ${\gamma }_1=-1$ and ${\gamma }_2=1.4$, and (d) ${\gamma }_1=2.1$ and ${\gamma }_2=1.4$. Note that ${\ell }_{-}$ passes the origin in (d), which corresponds to exceptional points.