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Adjoint computation of Berry phase gradients

Cyrill Bösch, Marc Serra-Garcia, Christian Böhm, Andreas Fichtner

Abstract

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, involving only one forward and one adjoint calculation. This approach enables the use of advanced optimization techniques, such as topology optimization, to design new materials with tailored topological wave properties.

Adjoint computation of Berry phase gradients

Abstract

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, involving only one forward and one adjoint calculation. This approach enables the use of advanced optimization techniques, such as topology optimization, to design new materials with tailored topological wave properties.

Paper Structure

This paper contains 11 sections, 39 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries: quantum and classical Berry phases
  3. Adjoint method for geometric phase derivatives
  4. From constrained to unconstrained
  5. The adjoint problem and the gradient
  6. The adjoint source
  7. The algorithm
  8. Application to a continuous 1D elastic topological metamaterial: Hockey-Stick-Test
  9. Algorithm complexity
  10. Generalization to Multiband Berry Phases
  11. Conclusions

Figures (1)

  • Figure 1: (a) Elastic modulus profile across the unit cell in red and the gradient of the Berry phase of the 1st band in blue. From the profile, it is evident that altering the Zak phase necessitates breaking its inversion symmetry. (b) Bloch band structure over the Brillouin Zone. (c) Hockey-Stick-Test: The y-axis depicts the relative disparity between the adjoint and first-order finite differentiation gradients across varying design parameter perturbations (x-axis). The distinctive "hockey stick" curvature of the plot robustly underscores the accuracy of the adjoint gradient.