Adjoint computation of Berry phase gradients

Berry phases offer a geometric perspective on wave propagation and are key to designing materials with topological wave transport. However, controlling Berry phases is challenging due to their dependence on global integrals over the Brillouin zone, making differentiation difficult. We present an adjoint-based method for efficiently computing the gradient of the Berry phase with respect to system parameters. We introduce an adjoint-based algorithm that computes Berry-phase gradients via only one forward and one adjoint solve. Under reasonable assumptions the algorithm’s time complexity is $O\left(N^{1+1/D}\right)$, where $N$ is number of grid points in a numerical discretization scheme and $D$ is the space dimension. Thereby it outperforms numerical differentiation and perturbation theory for problems with a large number of design variables. This approach enables the use of advanced, gradient-based optimization techniques to design new continuously parameterized materials with tailored topological wave properties. Furthermore, via multi-objective optimizations this method allows to co-design the topological characteristics in tandem with other objectives. We apply the method to an elastic metamaterial rod.