Klein-bottle quadrupole insulators and Dirac semimetals

The Benalcazar-Bernevig-Hughes (BBH) quadrupole insulator model is a cornerstone model for higher-order topological phases. It requires $\pi$-flux threading through each plaquette of the two-dimensional Su-Schrieffer-Heeger model. Recent studies showed that particular $\pi$-flux patterns can modify the fundamental domain of momentum space from the shape of a torus to a Klein bottle with emerging topological phases. By designing different $\pi$-flux patterns, we propose two types of Klein-bottle BBH models. These models show rich topological phases, including Klein-bottle quadrupole insulators and Dirac semimetals. The phase with nontrivial Klein-bottle topology shows twined edge modes at open boundaries. These edge modes can further support second-order topology, yielding a quadrupole insulator. Remarkably, both models are robust against flux perturbations. Moreover, we show that different $\pi$-flux patterns dramatically affect the phase diagram of the Klein-bottle BBH models. Going beyond the original BBH model, Dirac semimetal phases emerge in Klein-bottle BBH models featured by the coexistence of twined edge modes and bulk Dirac points.

Figure 1

(a) Sketch of the lattice for the first Klein-bottle BBH model with a specific $\pi$-flux pattern. The dashed lines indicate a negative sign to account for the $\pi$ fluxes. (b) Phase diagram in the parameter space $(t_x,t_y)$. The light blue region indicates the nontrivial Klein-bottle quadrupole insulators (KBQI); the light green region indicates the Dirac semimetal (DSM) phase. The region between the two dashed lines represents phases with nontrivial Klein-bottle topology. Other regions are normal insulators (NI). (c) Fundamental domain of momentum space in the BZ (blue). The boundaries marked with the same colored arrows should be identified in that sense and thus a Klein bottle. (d) Band structure for the first Klein-bottle BBH model in the insulating phase. There are four nondegenerate bands. The parameters are taken as $t_x=0.6\mathrm{and}{t}_y=0.3$ in units of $t$.

Figure 2

(a) Spectra of a ribbon along the $x$ direction. The blue and red lines indicate the twined edge modes. (b) Wave function distribution of the twined edge modes along the $y$ direction of the ribbon. (c) Similar to (a), but in a topological trivial case without edge modes. (d) The Wannier spectrum ${\nu }_y\left(k_x\right)$ of the lowest energy band. The parameters are taken as $t_x=0.6,t_y=0.3$ for (a), (b), and (d) and $t_x=1.2,t_y=2$ for (c) in units of $t$.

Figure 3

(a) Edge polarization $p_x$ along $y$. (b) Wannier center ${\nu }_x$ for different eigenstates. (c) Edge polarization $p_y$ along $x$. (d) Wannier center ${\nu }_y$ for different eigenstates. The parameters are taken as $t_x=0.6,t_y=0.3$ in units of $t$.

Figure 4

(a) Energy spectrum of the model with open boundaries in both direction. There are four zero-energy corner modes at zero energy. (b) Electron charge density distribution on the lattice. It gives fractional corner charges $\pm \frac{e}{2}$. The parameters are taken as $t_x=0.6,t_y=0.3$ in units of $t$.

Figure 5

(a) Four Dirac points in the band structure. (b) Coexistence of edge Dirac points and bulk edge Dirac points in the spectra of a ribbon along the $x$ direction. The parameters are taken as $t_x=1.2,t_y=0.6$ in units of $t$.

Figure 6

(a) Energy spectrum on a ribbon along the $x$ direction when $\varphi =0.95\pi$. (b) Quadrupole moment $q_{xy}$ as a function of flux deviation $\mathrm{\Delta }\varphi =\pi -\varphi$. (c) Eigenstates around zero energy for different $\mathrm{\Delta }\varphi$. (d) Quadrupole moment $q_{xy}$ as a function of random flux strength $U_d$. The parameters are taken as $t_x=0.6, t_y=0.3$ in units of $t$.

Figure 7

(a) Sketch of the lattice for the second Klein-bottle BBH model with a different $\pi$-flux pattern. The dashed lines indicate a negative sign to account for the $\pi$ fluxes. (b) Phase diagram in the parameter space $(t_x,t_y)$. The light blue region indicates the Dirac semimetal (DSM) phase; the light green region indicates the trivial quadrupole insulators (TQI). The region between the two dashed lines represent phases with nontrivial Klein-bottle topology. Other regions are the normal insulators (NI).

Figure 8

(a) Edge polarization $p_y$ along $x$ (left panel) and the Wannier center ${\nu }_y$ for different eigenstates (right panel) in the second Klein-bottle BBH model. (b) Electron density distribution in the lattice. The parameters are taken as $t_x=0.8,t_y=0.2$ in units of $t$ for all plots.

Figure 9

(a) Energy spectrum of the second Klein-bottle BBH model on a ribbon along the $x$ direction. The edge and bulk Dirac points coexist. (b) Same as (a), but with flux $\varphi =0.9\pi$. The parameters are taken as $t_x=t_y=0.2$ in units of $t$.

Figure 10

(a)–(e) and (g) Energy spectra of the model specified in Eq. (A1) on a ribbon along the $y$ direction. The red lines indicate the twined edge modes. (f) Wannier bands of the lowest energy band (blue) and the second energy band (red) corresponding to (c). (h) and (i) correspond to (g), but with the flux deviation from $\pi$. The other parameters are $t_{11}^x=t_{22}^x=1, t_{12}^x=t_{21}^x=3.5, t_1^y=2$, and $t_2^y=1.5$, the same as in Ref. [45]. The values of $\epsilon$ and $\lambda$ are labeled on each plot.

Figure 11

(a) Energy spectra for the BBH model with a $\pi$-flux defect. There are six zero-energy modes in the gap in total in the topological nontrivial phase. The $\pi$-flux defect bounds two extra zero-energy modes. (b) The wave function distribution corresponds to the bound state in (a). Here, $t_x=t_y=0.5$ for (a) and (b). (c) The same as (a), but the two bound states are away from zero energy. (d) The wave function distribution corresponds to the bound state in (c). Here, $t_x=0.3,t_y=0.6$ for (c) and (d).