TOPOLOGY SHARED BETWEEN CLASSICAL AND QUANTUM MATERIALS

113 pagesThe concept of topology has been widely used to classify materials. Majority of works are focused on quantum systems. Until recently, many advancements have also been made in the field of topological mechanics. However, the connections between them are still limited to the linear level of mechanical systems which are naturally nonlinear. In this thesis, we expose this field with different approaches by studying topology of nonlinear classical systems and possible connections to quantum systems. Firstly, we present a generic prescription of defining topological indices which accommodates nonlinear effects in mechanical systems without taking any approximation. Invoking the tools of differential geometry, a Z-valued quantity in terms of a topological index mu in differential geometry known as the Poincare-Hopf index, that features the topological invariant of nonlinear zero modes (ZMs), is predicted. We further identify one type of topologically protected solitons that are robust to disorders. Our prescription constitutes a new direction of searching for novel topologically protected nonlinear ZMs in the future. Secondly, we connect this topological index mu to the Witten index W in supersymmetric quantum systems.To establish the connection, we study two topological number in isostatic mechanical systems and supersymmetric quantum systems respectively. On one hand, we define Qnet for an isostatic mechanical system that counts the number of robust zero-energy configurations. On the other hand, we write a supersymmetric Hamiltonian that has a well-defined Witten index that tells us the number of robust zero-energy states. Finally, we show that Qnet=W under very general conditions. Our result suggests a direct connection between nonlinear mechanical systems and interacting quantum systems, and therefore points out an alternative way to understand the topology of quantum systems. Finally, we study topological frustration which is the existence of classical zero modes that are robust to many but not all distortions of the Hamiltonian. For a magnet whose classical limit exhibits topological frustration, an important question is what happens to this topology when the degrees of freedom are quantized and whether such frustration could lead to exotic quantum phases of matter like a spin liquid. In quantum spin ladders, we find low-energy eigenstates corresponding to known symmetry protected topological (SPT) ground states, and a special role of SU(2) symmetry that demands the existence of extra dimensions of classical zero modes---the phenomena we call symmetry-enriched topological frustration (SETF).These results suggest that in the absence of magnetic order, classical topological frustration manifests at finite spin as asymptotically low energy modes with support for exotic quantum phenomena