Floquet generation of hybrid-order topology and $\mathbb{Z}_2$-like bipolar localization

Abstract

Higher order topology, in the form of the emergence of corner modes, is observed in two dimensions when crystalline symmetries are superposed on the Altland-Zirnbauer classification of topological insulators. It occurs in Benalcazar-Bernevig-Hughes (BBH) model on a 2D square lattice, which owing to an embedded $\mathbb{Z}_2$ gauge field, features a bulk quadrupole moment with localized zero-energy corner states. Further, as a dividend, the BBH model transmutes the general notion of the space-time inversion ($\mathcal{PT}$) symmetry and behaves as a spinful system, without having to invoke `real' spin degrees of freedom. A two-fold engineering of the model, namely a periodic drive, followed by a non-reciprocal hopping render intriguing consequences. As a first, the drive activates first-order topology, and the resulting Floquet phase hosts a coexistence of first-order conducting edges at both zero and $π$ quasienergies along with higher order corner states, which qualifies the coexisting state to be denoted as a hybrid-order topological phase. Further, inclusion of non-reciprocal couplings features a $\mathbb{Z}_2$-like skin effect which demonstrates a drive-induced transition from a unipolar to a bipolar localized phase, and is evidenced via the generalized Brillouin zone (GBZ) theory. While depiction of a GBZ in 2D is challenging, a corresponding 1D map is still possible and can be implemented by exploiting the mirror symmetry. We further uncover conditions under which the skin effect is completely suppressed in our system. Putting together, our results manifest an efficient technique to dynamically engender and control Hermitian and non-Hermitian topological features, that remain otherwise masked in a static scenario.

Figures (8)

• Figure 1: Schematic illustration of an $N \times N$ square lattice with intercell hopping amplitude $\lambda$ and nonreciprocal intracell hopping amplitudes $v \pm \gamma$. The dashed bonds indicate hopping processes that differ by a $\pi$ phase relative to the solid bonds. This staggered phase pattern effectively introduces a $\mathbb{Z}_2$ gauge structure, which modifies the symmetry algebra and consequently alters the topological classification of the system.
• Figure 2: Quasienergy spectra under open boundary conditions along the $y$-direction in panel (a) and along both $x$- and $y$-directions in panel (b). The red branches in panels (a) and (b) correspond to gapless first-order boundary states, while the blue dots denote localized second-order corner states. Notably, the edge modes emerge within the quasienergy gap at zero quasienergy, whereas the corner states remain pinned at $\pi/T$ quasienergy. The corresponding spatial probability distributions shown in panels (c) and (d), for the $\pi$ quasienergy corner and zero-quasienergy edge states, respectively, confirm realization of a hybrid-order topological phase characterized by the coexistence of conducting edge modes and localized corner states. The parameters used are $T_1=1$, $T_2=4$, $\lambda=0.5$, $v=2.5$, and $N_x=N_y=30$.
• Figure 3: Quasienergy spectra under open boundary conditions along the $y$-direction in panel (a) and along both $x$- and $y$-directions in panel (b). The red branches in panels (a) and (b) correspond to gapless first-order boundary states, while the blue dots denote localized second-order corner states. Notably, the edge modes emerge within the quasienergy gap at $\pi$, whereas the corner states remain pinned at zero quasienergy. The corresponding spatial probability distributions shown in panels (c) and (d), for the $\pi$-quasienergy edge states and zero-quasienergy corner states respectively, confirm realization of a hybrid-order topological phase characterized by the coexistence of conducting edge modes and localized corner states. The parameters used are $T_1=1$, $T_2=4$, $\lambda=0.5$, $v=0.5$, and $N_x=N_y=30$.
• Figure 4: Unipolar-bipolar localization crossover followed by a complete suppression of the skin effect is shown. Panels (a)-(d) show the spatial distribution of all eigenstates in the presence of non-reciprocity, $\gamma$. In panel (a), corresponding to $\sqrt{2} v=0.1\pi$, all eigenstates accumulate at the left corner, indicating a unipolar localization regime. In panel (b), corresponding to $\sqrt{2} v=0.5\pi$, the states are distributed equally between two of the diagonal corners; specifically, eigenstates with the positive quasienergies localize at the left corner while the negative quasienergy states accumulate at the right corner, giving rise to a characteristic $\mathbb{Z}_2$-like skin effect occurring at the bidirectional points, given by $\sqrt{2} v=(2n+1)\pi/(T/2)$. In panel (c), for $\sqrt{2} v=0.9\pi$, unipolar localization reappears with reversed polarization relative to panel (a). Finally, panel (d), corresponding to $\sqrt{2} v=\pi$, shows a complete disappearance of boundary accumulation, signaling total suppression of the skin effect occurring at values, $\sqrt{2} v=2n\pi/(T/2)$. The system parameters are chosen as $T=2$, $N_x=N_y=30$, $\gamma=0.3$, and $\lambda=0.5$.
• Figure 5: Closed contours of the 1D mapped Floquet GBZ along the symmetry line $k_x = k_y$, plotted in the complex plane $\mathrm{Re}(\beta)$ versus $\mathrm{Im}(\beta)$. Panel (a), corresponding to $v=0.1\pi$, lies in the unipolar regime where all the eigenstates accumulate at the left boundary; the associated GBZ forms a closed loop with radius $|\beta|<1$. In contrast, panel (b), for $v=0.9\pi$, corresponds to right-boundary accumulation, and the GBZ contour has radius, $|\beta|>1$. In both the cases, the contours exhibit cusp-like behavior, reflecting the multiple solutions (more than two) of $\beta$ having similar amplitude. This turns out to be a direct consequence of the second-order Taylor expansion of the characteristic polynomial governing the Floquet operator. The system parameters are chosen as $T=2$, $\lambda=0.5$, and $\gamma=0.3$.