Dual topological characterization of non-Hermitian Floquet phases

Non-Hermiticity is expected to add far more physical features to the already rich Floquet topological phases of matter. Nevertheless, a systematic approach to characterize non-Hermitian Floquet topological matter is still lacking. In this work we introduce a dual scheme to characterize the topology of non-Hermitian Floquet systems in momentum space and in real space using a piecewise quenched nonreciprocal Su-Schrieffer-Heeger model for our case studies. Under the periodic boundary condition, topological phases are characterized by a pair of experimentally accessible winding numbers that make jumps between integers and half integers. Under the open boundary condition, a Floquet version of the so-called open boundary winding number is found to be integers and can predict the number of pairs of zero and $\pi$ Floquet edge modes coexisting with the non-Hermitian skin effect. Our results indicate that a dual characterization of non-Hermitian Floquet topological matter is necessary and also feasible because the formidable task of constructing the celebrated generalized Brillouin zone for non-Hermitian Floquet systems with multiple hopping length scales can be avoided. This work hence paves a way for further studies of non-Hermitian physics in nonequilibrium systems.

Figure 1

Schematic illustration of the periodically quenched NHSSH model. Each unit cell contains two sublattices, which are coupled by the intracell hopping amplitude $\mu$. The intercell hopping amplitudes in the first and second halves of the driving period are $\frac{J_1\pm \lambda }{2}$ and $\pm \frac{\lambda \pm J_2}{2}$. The lightning symbols denote quenches applied in the middle/end of each driving period, after which the lattice is switched from the configuration of the upper/lower to the lower/upper array of the figure.

Figure 2

Bulk topological phase diagram of the periodically quenched NHSSH model vs hopping parameters $\mu$ and $J_1$. Other system parameters are $(J_2,\lambda )=(0.5\pi ,0.25)$. Each region with a uniform color denotes a topological phase characterized by the winding numbers $(w_0,w_{\pi })$. The solid lines between different regions are boundaries between distinct non-Hermitian Floquet topological phases. The values of $w_0$ and $w_{\pi }$ for each phase are denoted explicitly in panels (a) and (b).

Figure 3

Spin textures and dynamic winding angles of the periodically quenched NHSSH model in time frames $\alpha =1$ [panels (a, c)] and $\alpha =2$ [panels (b, d)]. System parameters are $(J_1,J_2,\mu ,\lambda )=(2.4\pi ,0.5\pi ,0.4\pi ,0.25)$, and evolutions are averaged over 500 driving periods to generate the winding angles ${\theta }_{yx}^{1,2}$ in panels (c, d). In the panel (a) [(b)], the red (magenta) points denote $(\langle {\sigma }_x\rangle ,\langle {\sigma }_y\rangle )$ in the first (second) time frame. The gray solid line denotes the origin of $\langle {\sigma }_x\rangle \text{−}\langle {\sigma }_y\rangle$ plane. In panel (c) [(d)], the red (magenta) points correspond to the dynamic winding angles ${\theta }_{yx}^1$ (${\theta }_{yx}^2$) in time frame 1 (2) [19]. The values of $(w_1,w_2)$ are also shown in panels (c, d).

Figure 4

Gap functions ${\mathrm{\Delta }}_0$ (blue and gray solid lines), ${\mathrm{\Delta }}_{\pi }$ (red and green dotted lines), OBWNs ${\nu }_0$ (circles), ${\nu }_{\pi }$ (crosses), and PBC winding numbers $w_0$ (diamonds), $w_{\pi }$ (pentagrams) of the model. System parameters are $(\mu ,J_2,\lambda )=(0.4\pi ,0.5\pi ,0.25)$. Phase transitions under OBC happen at $J_1=(0.4\pi ,0.6\pi ,1.4\pi ,1.6\pi ,2.4\pi ,2.6\pi ,3.4\pi ,3.6\pi )$, denoted by the ticks along the horizontal axis. Only the first 20 smallest values of $({\mathrm{\Delta }}_0,{\mathrm{\Delta }}_{\pi })\mathrm{under}$ OBC are shown for clear illustrations. The numbers of zero and $\pi$ Floquet edge modes are denoted below the horizontal axis.