Structural-Disorder-Induced Second-Order Topological Insulators in Three Dimensions

Higher-order topological insulators are established as topological crystalline insulators protected by crystalline symmetries. One celebrated example is the second-order topological insulator in three dimensions that hosts chiral hinge modes protected by crystalline symmetries. Since amorphous solids are ubiquitous, it is important to ask whether such a second-order topological insulator can exist in an amorphous system without any spatial order. Here, we predict the existence of a second-order topological insulating phase in an amorphous system without any crystalline symmetry. Such a topological phase manifests in the winding number of the quadrupole moment, the quantized longitudinal conductance, and the hinge states. Furthermore, in stark contrast to the viewpoint that structural disorder should be detrimental to the higher-order topological phase, we remarkably find that structural disorder can induce a second-order topological insulator from a topologically trivial phase in a regular geometry. We finally demonstrate the existence of a second-order topological phase in amorphous systems with time-reversal symmetry.

Figure 1

(a) In-gap hinge states with the size of each sphere indicating the sum of local densities of four eigenstates closest to zero energy at that site. (b) The quadrupole moment $Q_{xy}$ with respect to an inserted flux ${\mathrm{\Phi }}_z$, reflecting that the winding number $W_Q=1$ defined in (3). Here, we consider a typical sample of a SOTI on an amorphous lattice of size $L=20$ for $M=-3$ in Hamiltonian (1).

Figure 2

(a) Configuration averaged longitudinal conductances $\overline{G}$ along $z$ (blue and green lines) and winding numbers ${\overline{W}}_Q$ of the quadrupole moment (red line) versus the mass $M$ for Hamiltonian (1) on amorphous lattices in comparison with the conductance for a cubic system (black line). The blue, green, and red lines correspond to systems with size $L=30$, 40, 16, respectively. Three distinct phases including amorphous SOTIs, metals, and trivial insulators are identified for amorphous systems as separated by the light yellow lines. For the metal phase, the close-up view of the conductance is plotted in the inset. See the Supplemental Material [49] for standard deviations of the winding number. (b) Configuration averaged bulk energy gaps (left vertical axis) versus $M$ for amorphous lattices in comparison with the result for a cubic lattice and the density of states (DOS) at zero energy $\rho (E=0)$ (right vertical axis) versus $M$ for amorphous lattices with $L=30$ calculated by the kernel polynomial method with the expansion order $N_c=2^9$.

Figure 3

(a) Schematics of lattice structures for three structural disorder strengths $W$ added on a regular cubic lattice. (b) Configuration averaged longitudinal conductances along $z$, quadrupole moment winding numbers and bulk energy gaps versus $W$. (c) The finite-size scaling of the conductance versus the system size $L$ for $W=6$, which displays a power-law decay fitted by a black line. A top view of LDOS at zero energy for (d) $W=0$ and (e) $W=6$, obtained by averaging over different random configurations and summing over the coordinates along $z$. For [(b)–(e)], we take $M=-6.5$ in Hamiltonian (1).

Figure 4

Sample averaged longitudinal conductances $\overline{G}$ along $z$ (blue line) and ${\mathbb{Z}}_2$ invariants ${\overline{\nu }}_Q$ (red line) versus the mass $M$ for Hamiltonian (4) on amorphous lattices in comparison with the conductance (black line) and the ${\mathbb{Z}}_2$ invariant (green line) on a cubic lattice. The blue (red) line corresponds to a system with size $L=30$ ($L=12$).