Topological Matter by Inverse Design
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Date
2024
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Doctoral Thesis
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Abstract
Topological states of matter exhibit a range of exceptional properties such as unidirectional wave propagation along topological boundaries and an unparalleled robustness to different types of perturbations. The primary aim of this thesis is to leverage gradient-based optimization techniques for the design of topological matter. Unlike other inverse design approaches for topological systems, this work targets the topological invariant directly. At first, this approach may seem paradoxical, as topological invariants are invariant under continuous deformation of materials, meaning that small perturbations cannot identify the direction of change of the invariant.
To solve the paradox, in my first study, I introduce the concept of symmetry relaxation. Since topological invariants are protected by symmetries, relaxing these symmetries converts the quantized topological invariant into a differentiable entity which I call ``topological indicator". This indicator matches the invariant when symmetries are intact and interpolates between various topological phases when symmetries are relaxed. By following the gradient of this indicator with respect to design or model parameters, a material can be guided through a topological phase transition. This technique is applied to both discrete and continuous, as well as higher-order, topological insulators.
In a subsequent project, I employ the adjoint method from PDE-constrained optimization to compute these gradients. This method enables the efficient and exact computation of gradients, particularly when faced with a vast design space. Collectively, I believe these studies lay the groundwork for the systematic design of topological information processing devices.
An emerging domain exploiting topological effects in the classical realm involves time-modulated metamaterials. Here, topological effects often draw parallels to adiabatic processes observed in quantum systems, such as those of Majorana fermions and braiding operations. I have dedicated a study to contrast classical and quantum adiabatic evolutions. It is conventionally assumed that they are largely equivalent, especially concerning the conditions for adiabaticity and the emergence of geometric entities. However, my results challenge this perception. I discern that mode coupling in classical systems diverges fundamentally from its quantum counterpart, implying certain quantum adiabatic processes are not replicable in classical settings. When addressing the general multiband setting, the manifestation of the non-Abelian gauge potential is also distinct, requiring a classical ``correction'' term. Finally, classical adiabatic evolution requires an extra condition compared to quantum adiabatic evolution. Only under this extra condition adiabaticity is ensured, and geometric quantities like Berry phases or the Wilczek-Zee matrix emerge.
The fourth study provides an overview of an ongoing collaborative project with the Bertoldi Lab at Harvard. This project aims to outsource the sensory system and control unit of a mobile robot into a mechanical metamaterial. To achieve this goal the metamaterial is mounted onto a wheeled mobile basis. The task of the robot is to navigate through a course of obstacles. As the robot collides with an obstacle the metamaterial deforms nonlinearly because of the contact. This deformation is exploited to compute the motor command voltages to free the robot from the obstacle using Physical Reservoir Computing.
In the fifth study, a framework is established for mapping a 1-D discretized non-homogeneous elastic wave equation to a Schrödinger equation which can be solved using gate-based quantum computers. We validate feasibility by solving a small-scale problem on the quantum computer IBM Brisbane and discuss the potential for exponential speedup in comparison to classical wave equation integrators.
Lastly, a work is presented that employs random field interferometry for ultrasound computed tomography for medical imaging.
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Examiner : Fichtner, Andreas
Examiner : Serra Garcia, Marc
Examiner : Tournat, Vincent
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ETH Zurich
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03971 - Fichtner, Andreas / Fichtner, Andreas