Finite-size topology
Ashley M. Cook, Anne E. B. Nielsen
Abstract
We show that topological characterization and classification in $D$-dimensional systems, which are thermodynamically large in only $D\ensuremath{-}\ensuremath{\delta}$ dimensions and finite in size in $\ensuremath{\delta}$ dimensions, is fundamentally different from that of systems thermodynamically large in all $D$ dimensions: As $(D\ensuremath{-}\ensuremath{\delta})$-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 $\ensuremath{\delta}+1$ topological invariants, ranging from the $D$-dimensional topological invariant to the $(D\ensuremath{-}\ensuremath{\delta})$-dimensional topological invariant. The system exhibits topological response signatures and bulk-boundary correspondences governed by combinations of these topological invariants taking nontrivial 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.