Nonlinear quadrupole topological insulators

The quadrupole topological insulator (QTI) has attracted intense studies as a prototype of symmetry-protected higher-order topological phases of matter with a quantized quadrupole moment. The realization of QTIs has been reported in various static settings with periodic structures. Here, we theoretically investigate topological phase transitions and establish the QTI phase in a periodically driven system with disorder. In the clean limit, the Floquet QTI phase emerges from a topologically trivial band structure driven by elliptically polarized irradiation. More strikingly, starting from a pure and static system with trivial topology, we unveil an intriguing QTI phase which necessitates the simultaneous presence of disorder and periodic driving. Furthermore, we reveal that particle-hole symmetry is sufficient to protect the QTI. Our work not only establishes a new strategy to design QTIs but also enriches the symmetry-protected mechanism of higher-order topology.

A two-dimensional quadrupole topological insulator on a square lattice is a typical example of a higher-order topological insulator. It hosts an edge state localized near each of its $90^{\circ}$ corners at an energy $E$ inside the band gap, where $E$ is set equal to zero for simplicity. Although the appearance of an edge state has been shown in simple systems with only $90^{\circ}$ corners, it is uncertain whether a similar localized state can appear at $E = 0$ near a complicated edge consisting of multiple $90^{\circ}$ and $270^{\circ}$ corners. Here, we present a numerical method to determine the wavefunction of a zero-energy state localized near an arbitrary edge. This method enables us to show that one localized state appears at $E = 0$ if the edge consists of an odd number of corners. In contrast, the energy of localized states inevitably deviates from $E = 0$ if the edge includes an even number of corners.

Higher order topological tnsulators are $d$-spatial dimensional systems featuring topologically protected gapless states at their $(d\ensuremath{-}n)$-dimensional boundaries. With the help of ab initio calculations and tight-binding models along with symmetry considerations, we show that monolayer buckled honeycomb structures of group-V elements (Sb, As), which have already been synthesized, belong in this category and have a spinless charge fractionalization of $\frac{e}{2}$ at the corner states as well as weak topological edge states, protected by the ${S}_{6}$ symmetry operation, which classify this system as a quadrupole topological insulator. The robustness of these edge and corner states to perturbations is explicitly demonstrated.

Two-dimensional (2D) quadrupole topological insulators, featured by topologically protected 0D corner modes, have recently attracted tremendous interest in condensed matter and materials physics. Herein, we construct a specific electric circuit made of capacitors and inductors forming a 2D Kekulé hexagonal lattice for quadrupole topological phase and corner modes. Trivial–nontrivial topological phase transition can be controlled by varying capacitance in the circuit, so that distinct topological edge states appear in 1D ribbons and corner states emerge in 0D flakes. We explore the field strength distribution and two-point impedance with respect to excitation frequency, and reveal that the topological corner resonance is robust against size of the LC network and randomness of the capacitors/inductors, a great benefit for experimental detection. Our results enrich the family of designer topoelectrical circuit as a flexible and tunable platform to achieve exotic quantum phases, which may have potential for future telecommunications, signal processing and quantum computing.

Author Correction: Realization of quasicrystalline quadrupole topological insulators in electrical circuits