Non-Hermitian Topological Invariants in Real Space

The topology of non-Hermitian systems is drastically shaped by the non-Hermitian skin effect, which leads to the generalized bulk-boundary correspondence and non-Bloch band theory. The essential part in formulations of bulk-boundary correspondence is a general and computable definition of topological invariants. In this Letter, we introduce a construction of non-Hermitian topological invariants based directly on real-space wave functions, which provides a general and straightforward approach for determining non-Hermitian topology. As an illustration, we apply this formulation to several representative models of non-Hermitian systems, efficiently obtaining their topological invariants in the presence of non-Hermitian skin effect. Our formulation also provides a dual picture of the non-Bloch band theory based on the generalized Brillouin zone, offering a unique perspective of bulk-boundary correspondence.

Figure 1

(a) SSH model with alternating $t_1$, $t_2$ hoppings and a non-Hermitian $t_3\pm \gamma /2$ hopping. (b),(c),(d) The OBC energy spectrum (real and imaginary part, and modulus) with length $L=80$ (in unit cell), with varying $t_1$ and fixed $t_2=1$, $t_3=0.2$, $\gamma =0.1$. (e) Open-bulk winding number calculated in real space [red solid dots, by Eq. (5)] and non-Bloch winding number calculated in GBZ [black hollow dots, by Eq. (3)]. For the open-bulk winding number, the chain length $L=100$; the boundary cutoff $l=15$.

Figure 2

(a) Profiles (modulus square) of all bulk energy eigenstates plotted as a function of eigenenergies; $x$ refers to the coordinate of open chain; $E$ refers to the eigenenergy. Eigenstate profiles are normalized to have maximum value 1. Extended eigenstates exist only at the discrete Bloch points $E\approx \pm 0.8$; all other eigenstates are localized by NHSE. $t_1=t_2=1$,$t_3=0.2,\gamma =0.1$. (b) The GBZ of (a). The dashed line is the unit circle. (c) Energy spectrum for varied $t_1$; other parameters the same as (a)(b). Yellow and blue color indicates eigenstate localization at the right and left end, respectively.

Figure 3

Open-bulk Chern number $C$ for Eq. (12). Fixed parameters: $t=1$, ${\gamma }_x={\gamma }_y=0.3$, ${\gamma }_z=0$. The size is $L_x=L_y=30$ and the truncation is $l_x=l_y=14$. The jump of $C$ occurs near the critical value $m_c\approx 2.09$ found in previous numerical energy spectrums [20]. The blue area $m\in [m_{-},m_{+}]$ stands for the gapless phase of the Bloch Hamiltonian $H(k_x,k_y)$, which is not seen in the open-boundary spectrums.