Topological states in generalized electric quadrupole insulators

The modern theory of electric polarization has recently been extended to higher multipole moments, such as quadrupole and octupole moments. The higher electric multipole insulators are essentially topological crystalline phases protected by underlying crystalline symmetries. Henceforth, it is natural to ask what are the consequences of symmetry breaking in these higher multipole insulators. In this work, we investigate topological phases and the consequences of symmetry breaking in generalized electric quadrupole insulators. Explicitly, we generalize the Benalcazar-Bernevig-Hughes model by adding specific terms in order to break the crystalline and nonspatial symmetries. Our results show that chiral-symmetry breaking induces an indirect gap phase which hides corner modes in bulk bands, ruining the topological quadrupole phase. We also demonstrate that quadrupole moments can remain quantized even when mirror symmetries are absent in a generalized model. Furthermore, it is shown that topological quadrupole phase is robust against a unique type of disorder presented in the system.

Figure 1

(a) Lattice structure of extended BBH model $H_1$. The dashed dark blue lines represent hopping at equivalent sites with strength $t$ between nearest unit cells. For simplicity, here we only sketch parts of this kind of hopping. (b) Energy bands corresponding to (a) with parameters fixed at ${\gamma }_x={\gamma }_y=0.5$ and $t=0.5$. Here, the system has an indirect gap.

Figure 2

Indirect gap phase induced by chiral-symmetry breaking. (a) Energy spectrum of $H_1$ as functions of parameter $t$. (b) Wannier bands for fixed $t=0.5$. (c) Calculated quadrupole moments when varying parameter $t$ (red line is calculated analytically with the nested Wilson loop approach, blue line is calculated numerically under periodic boundary condition with size $L_x\times L_y=40\times 40$). (d) Electron charge density in the nontrivial phase with $t=0.2$. Other parameters are ${\gamma }_x={\gamma }_y=0.5$.

Figure 3

Signatures of the indirect gap phase of Hamiltonian $H_1$. (a) Energy spectrum of $H_1$ at a ribbon with width $L_y=20$ for $t=0.35$. Red lines denote edge states. (b), (d) Plot the edge polarization and Wannier centers of states for $t=0.1$ and 0.8, respectively. (c) Phase diagram modified by the chiral breaking term in $H_1$ with fixed parameter $t=0.38$. Here, $q_{xy}$ is calculated numerically under periodic boundary condition with size $L_x\times L_y=30\times 30$. Red color region denotes indirect gap phases. Other parameters are ${\gamma }_x={\gamma }_y=0.5$.

Figure 4

(a) Lattice structure of extended BBH model $H_2$. The dashed dark blue lines represent hopping within and between unit cells. These hopping with imaginary amplitudes will break time-reversal symmetry. (b) Energy bands corresponding to (a) with parameters fixed at ${\gamma }_x=0.8,{\gamma }_y=0.5,$ and $J_2=0.3.$ The band degeneracy is lifted.

Figure 5

Edge spectrum and corresponding Wannier bands for extended model $H_2$. (a) $J_2=0.2$. (b) $J_2=0.5$. Gap closure happens both at edge spectrum and Wannier bands. (c) $J_2=0.6$. Edge spectrum opens a gap while Wannier bands are still gapless. Other parameters are fixed at ${\gamma }_x=0.8,{\gamma }_y=0.5$ for all.

Figure 6

The topological phase transition induced by $i{J}_2$ hopping terms. (a) Energy spectrum as function of $J_2$. We find that the zero-energy mode splits around $J_2=0.5$ due to topological phase transition. (b) Calculated quadrupole moments $q_{xy}$ for fixed parameter $J_2=0.5$. The nontrivial regions shrink due to topological phase transitions. (c) Numerically calculated $q_{xy}$ corresponding to (a). Here, $q_{xy}$ jumps from one-half to zero around $J_2=0.5$. (d) Electron charge density in the nontrivial phase for $J_2=0.2$. Other parameters are fixed at ${\gamma }_x=0.8,{\gamma }_y=0.5$ for all. Here, we set system at size $L_x\times L_y=40\times 40$ with periodic boundary for the calculation of $q_{xy}$.

Figure 7

Wannier values and edge polarization before ($J_2=0.2$) and after ($J_2=0.6$) the topological phase transition. (a) ${\nu }_x$ for different eigenstates, (b) edge polarization $p_x$ along $R_y$, (c) ${\nu }_y$ for different eigenstates, (d) edge polarization $p_y$ along $R_x$. Other parameters are ${\gamma }_x=0.8,{\gamma }_y=0.5.$ Here, the red circles in the upper panel of (a) and (c) are used to stress the two isolated topological modes with quantized Wannier center one-half.

Figure 8

(a) Lattice structure of extended BBH model $H_3$. The dashed dark blue lines represent hopping within and between unit cells. Here, the artificial asymmetric hopping breaks mirror symmetry. (b) Energy bands corresponding to (a) with parameters fixed at ${\gamma }_x={\gamma }_y=0.5$ and $J_3=0.3$. The band degeneracy is totally lifted.

Figure 9

(a) Energy spectrum as function of parameter $J_3$. (b) Edge polarization $p_x$ along $R_y$ (the upper panel) and Wannier center ${\nu }_x$ for different eigenstates (the lower panel). (c) Numerically calculated $q_{xy}$ corresponding to (a). (d) Edge polarization $p_y$ along $R_x$ (the upper panel) and Wannier center ${\nu }_y$ for different eigenstates (the lower panel). Other parameters are fixed at ${\gamma }_x={\gamma }_y=0.5$ and $J_3=0.2$. Here, we set system size $L_x\times L_y=40\times 40$ with periodic boundary for the calculations of $q_{xy}$. Here, the red circles in the lower panel of (b) and (d) are used to stress the two isolated topological modes with quantized Wannier center one-half.

Figure 10

Quadrupole moments in the presence of disorder. Upper panels (a) and (b) plot quadrupole moments $q_{xy}$ as functions of disorder strength $U_0$. In topological nontrivial case ${\gamma }_x=0.8$, ${\gamma }_y=0.5$ and trivial case with ${\gamma }_x=1.5$, ${\gamma }_y=0.5$. In (a) and (b) we use periodic boundary condition. (c) Eigenstates and (d) electron charge density in the presence of ${\mathrm{\Gamma }}_4$-type disorder with strength $U_0=3.0$ for the nontrivial case ${\gamma }_x=0.8$, ${\gamma }_y=0.5$. In (c) and (d) we use open boundary condition. The system size is $L_x\times L_y=50\times 50$, and we only take one disorder configuration for all.