Topological Phase Transitions in Disordered Electric Quadrupole Insulators

We investigate disorder-driven topological phase transitions in quantized electric quadrupole insulators in two dimensions. We show that chiral symmetry can protect the quantization of the quadrupole moment $q_{xy}$, such that the higher-order topological invariant is well defined even when disorder has broken all crystalline symmetries. Moreover, nonvanishing $q_{xy}$ and consequent corner modes can be induced from a trivial insulating phase by disorder that preserves chiral symmetry. The critical points of such topological phase transitions are marked by the occurrence of extended boundary states even in the presence of strong disorder. We provide a systematic characterization of these disorder-driven topological phase transitions from both bulk and boundary descriptions.

Figure 1

Phase diagrams of disordered electric quadrupole insulators. (a) Electric quadrupole moment $q_{xy}$ as a function of mass parameters $t_x$ and $t_y$ at a fixed disorder strength $W=2.5$. The dashed lines indicate the phase boundaries in the clean limit. (b) $q_{xy}$ as a function of $t_x$ and $W$ at fixed $t_y=0.8$. The dot-dashed lines are the phase boundaries obtained from an effective medium theory. In these phase diagrams, the disorder is of the $V(\mathbf{r}){\gamma }_4$ type, with $120/150$ random configurations averaged in (a)/(b). The system is of size $L_x\times L_y=30\times 30$ with periodical boundary conditions. The two distinct phases are higher-order topological insulator and normal insulator.

Figure 2

Signatures of delocalized states at the open boundaries: a finite-size scaling analysis of the localization length (a) ${\mathrm{\Lambda }}_x$ and (b) ${\mathrm{\Lambda }}_y$ as functions of $W$ with an open boundary condition in the transverse dimension; effective boundary mass terms (c) $m_x$ and (d) $m_y$ as functions of $W$; phase boundaries indicated by the divergence of (e) ${\mathrm{\Lambda }}_x$ and (f) ${\mathrm{\Lambda }}_y$ as analyzed in (a),(b). In (a)–(d), we have set $t_x=1.1$ and $t_y=0.8$.

Figure 3

Charge density distributions (a)–(c) that correspond to the three points, $A$, $B$, and $C$ (marked by crosses) in the phase diagram in Fig. 1. The points $A$ and $C$ sit in different phase boundaries; the point $B$ sits inside the nontrivial QEQI phase. The values of the disorder strength in (a)–(c) are $W=2.02$, $W=2.70$, and $W=3.10$, respectively. In all these calculations we have set $t_x=1.1$ and $t_y=0.8$, and taken an average of 10 disorder configurations.