Systematic design of topological metamaterials by symmetry relaxation

We present a method based on mathematical optimization to systematically design topological metamaterials. The method is combined with the adjoint gradient technique which allows us to design systems with a large number of design parameters, such as they appear in finite element simulations. 

 

The robustness of metamaterial properties that have a topological origin is both theoretically interesting as well as a promising feature for practical applications. However, it represents a challenge when looking for a new topological configuration of the metamaterial: small perturbations cannot be used to identify the direction of change of the topological indices. The method we present overcomes this challenge by breaking the symmetries which protect the topology. This causes the topological indices to become smooth, i.e. non-quantized and differentiable functions of the system parameters. Exploiting the differentiability, we compute gradients that indicate the direction of growth of the smoothed topological index. Following the gradients, we are able to identify the topological configuration. 

 

We successively apply the method to conventional and higher order topological systems as well as to tight binding and continuously parametrized models.