Nonlinear quadrupole topological insulators

TL;DR

The work addresses extending higher-order topological insulators to nonlinear regimes by introducing nonlinear quadrupole topological insulators (NLQTIs) implemented in a 2D electric-circuit lattice. It maps a tight-binding model with amplitude-dependent onsite energies, realized via nonlinear circuit elements, onto a topological quadrupole phase that is quantized in the linear limit with $q_{xy}=1/2$ for $\gamma<\lambda$. Through quench dynamics, it demonstrates nonlinear corner states in the weakly nonlinear regime and corner solitons in the strongly nonlinear regime, along with two distinct bulk-soliton branches: one in the middle finite gap (weak nonlinearity) and one in the semi-infinite gap (strong nonlinearity). The results expand the nonlinear HOTI family, showing a nonlinear platform for soliton control and suggesting pathways to higher multipole nonlinear topological phases and broad soliton phenomena in topological lattices.

Abstract

Higher-order topological insulators (HOTIs) represent a family of topological phases that go beyond the conventional bulkboundary correspondence. d-dimensional n-th order HOTIs maintain (d - n)-dimensional gapless boundary states (in particular, zero-dimensional corner states in the case of d = n = 2). HOTIs of the Wannier type cam be extended into the nonlinear regime. Another prominent class of HOTIs, in the form of multipole insulators, was investigated only in the linear regime, due to the challenge of simultaneously achieving both negative hopping and strong nonlinearity. Here we propose the concept of nonlinear quadrupole topological insulators (NLQTIs) and report their experimental realization in an electric circuit lattice. Quench-initiated dynamics gives rise to nonlinear topological corner states and topologically trivial corner solitons, in weakly and strongly nonlinear regimes, respectively. Furthermore, we reveal the formation of two distinct types of bulk solitons, one existing in the middle finite gap under the action of weak nonlinearity, and another one found in the semi-infinite gap under strong nonlinearity. This work realizes another member of the nonlinear HOTI family, suggesting directions for exploring novel solitons across a broad range of topological insulators.

Figures (18)

• Figure 1: Schematic representation of the nonlinear quadrupole topological insulator (NLQTI) and its topological properties in the linear limit. (a) The tight-binding lattice model of the NLQTI, where $\gamma$ and $\lambda$ represent the intracell and intercell hopping strengths, respectively. Solid and dashed lines designate positive and negative hoppings, respectively. Parameters $\delta _{m,n}^{(1,2,3,4)}$ denote the amplitude-dependent onsite energies in the $(m,n)$-th unit cell. (b) The unit cell of the electric-circuit lattice realizing the NLQTI. Blue and red circuit elements realize the intracell and intercell hoppings, respectively, while yellow-green grounded circuit elements determine the onsite energies. (c) The dependence of the quadrupole moment $q_{x,y}$ on the dimerization ratio $\gamma /\lambda$. (d) The frequency spectrum of the lattice with open boundary conditions in the $x$ and $y$ directions, plotted as a function of $\gamma /\lambda$. The blue curve indicates the corner states, while the gray curves denote the bulk and edge ones. The red lines in (c) and (d) mark the phase-transition boundary.
• Figure 2: Nonlinear topological corner states and topologically trivial corner solitons in the NLQTI. (a) The dependence of the eigenfrequency on the maximum voltage in the lattice. The solid and dashed blue curves represent localized and delocalized nonlinear corner states, respectively, while the red curve pertains to the corner solitons. The frequencies corresponding to the linear corner, edge, and bulk states are indicated by the green regions. (b) Absolute values of the voltage distributions for the states labeled in (a), normalized to their respective maximum values. (c) The side view of the fabricated PCB, with the inset exhibiting the unit cell of the circuit lattice. The circuit elements are labeled in the inset. (d,e) Experimental and theoretical IPRs of the voltage distributions (see Eq. (\ref{['IPR']})) at $t=9~\mathrm{\mu s}$ for different initial voltages. Plots marked as ⑤ through ⑧ display the voltage distributions for the respective values of $\psi _{0}$ in panels (d) and (e).
• Figure 3: Bulk solitons in the NLQTI. (a) The eigenfrequency versus the maximum voltage in the lattice. The solid blue and red curves represent two distinct types of bulk solitons, both of which are localized states. The dashed blue curve indicates the delocalized states that arise from one type of the bulk soliton. For comparison, the frequencies corresponding to the linear corner, edge, and bulk states are indicated by the green regions. (b) Absolute values of the normalized voltage distributions for the quasi-antisymmetric triangular solitons labeled in (a). (c,d) Experimentally measured and theoretically predicted IPRs of the voltage distributions at $t=8~\mathrm{\mu }\text{s}$ for different initial voltages, with insets ⑤ -- ⑧ showing the voltage distributions at representative values of $\psi _{0}$.
• Figure 4: The capacitance-voltage relation of two parallel-connected common-cathode diodes. The curve represents theoretical results, while the circular points correspond to data of the experimental measurements.
• Figure 5: A schematic of the NQTI realized in the circuit lattice. For the clarity's sake, only the unit cell is shown.