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arXiv · 2212.11300

Finite-size Topology

Ashley M. Cook·Anne E. B. Nielsen
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Abstract

We show that topological characterization and classification in $D$-dimensional systems, which are thermodynamically large in only $D-\delta$ dimensions and finite in size in $\delta$ dimensions, is fundamentally different from that of systems thermodynamically large in all $D$-dimensions: as $(D-\delta)$-dimensional topological boundary states permeate into a system's $D$ dimensional bulk with decreasing system size, they hybridize to create novel topological phases characterized by a set of $\delta+1$ topological invariants, ranging from the $D$-dimensional topological invariant to the $(D-\delta)$-dimensional topological invariant. The system exhibits topological response signatures and bulk-boundary correspondences governed by combinations of these topological invariants taking non-trivial values, with lower-dimensional topological invariants characterizing fragmentation of the underlying topological phase of the system thermodynamically large in all $D$-dimensions. We demonstrate this physics for the paradigmatic Chern insulator phase, but show its requirements for realization are satisfied by a much broader set of topological systems.

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Ashley M. Cook, Anne E. B. Nielsen. 2022-12-21. Finite-size Topology. https://doi.org/10.1103/physrevb.108.045144

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