Discovery of topological metamaterials by symmetry relaxation and smooth topological indicators

Physical properties of a topological origin are known to be robust against small perturbations. This robustness is both a source of theoretical interest and a driver for technological applications, but presents a challenge when looking for new topological systems: Small perturbations cannot be used to identify the global direction of change in the topological indices. Here, we overcome this limitation by breaking the symmetries protecting the topology. The introduction of symmetry-breaking terms causes the topological indices to become smooth, nonquantized functions of the system parameters, which are amenable to efficient design algorithms based on gradient methods. We demonstrate this capability by designing discrete and continuous phononic systems realizing conventional and higher-order topological insulators.

Figure 1

Symmetry relaxation in the SSH model. (a) Example of an SSH chain. The rectangle highlights the unit cell. The model consists of a dimerized chain with alternating hopping potentials of strength $t$ and $s$. (b) Berry phase (left) and band structure (right) for an SSH chain with intact inversion symmetry, as the hopping $s$ is varied. (c) Berry phase (left) and unit cell and band structure (right) for an SSH chain with relaxed inversion symmetry, as the hopping $s$ is varied. The arrows indicate the evolution of a gradient-following optimization of the system parameters.

Figure 2

Design of a phononic topological insulator using smooth topological indicators. (a) Material design made by repeating a three-material unit cell. (b) Dispersion relation for the system in (a). The red region denotes the gap selected for optimization. (c) Multiband Berry phase as a function of the unit cell parameters $E_B$ and $E_C$, while $E_A$ and $\rho$ are kept constant. The red line shows a gradient-ascending optimization trajectory starting from a symmetric, trivial phase and ending in a symmetric, topological configuration. (d) Finite element method simulation (COMSOL®Multiphysics) of a localized mode at the interface between two sandwich metamaterials, one topological and one trivial [30], corresponding respectively to the start point and end point in (c). The domain wall is placed at $x=0$, and $W_A=0.6759W$.

Figure 3

Relaxation of topological indices based on symmetry eigenvalues at inversion-symmetric momenta. (a) Double-SSH chain with couplings $t_1$, $t_2$, $s_1$, $s_2$, $c_v$, and $c_x$. (b) Topological index in the presence of inversion symmetry. (c) Topological index after the symmetry has been lifted by perturbations ${\epsilon }_i\ne 0$. The red line represents a gradient-ascending optimization trajectory. (d) Bulk and boundary responsivity ${\left|\mathrm{\Psi }\right|}^2\left(\omega \right)$ of a finite sample, as the hopping strengths are being optimized from a trivial to a topological configuration following the red line in (c).

Figure 4

Symmetry relaxation in higher-order topological insulators. (a) Square lattice with four sites per unit cell (inset). Each unit cell interacts only with nearest-neighbor unit cells. (b) Evolution of the topological invariants ${\theta }_{\pm }\left(k\right)$ during the design process. (c) Evolution of the frequency spectrum for a finite, classical system consisting of $20\times 20$ unit cells, evaluated at the bulk, edges, and corners. (d) Bulk, edge, and corner responsivities ${\left|\mathrm{\Psi }\right|}^2\left(\omega \right)$ of the finite classical sample in (c), at the end of the design process. (e) Example of a corner-localized eigenmode of the finite system in (c).