Finite-size Topology

The bulk-boundary correspondence is a corner stone of topological physics and yields phenomena of great interest, such as the quasiparticles of topologically-protected quantum computation schemes and the boundary metals of spintronics platforms. Here, we show that bulk-boundary correspondence in finite-size systems is fundamentally different from that of infinite-size systems: as D-1 dimensional topological boundary states permeate into a system's D dimensional bulk with decreasing system size, they interfere to create novel topological phases, including some with an additional bulk-boundary correspondence: the value of a topological invariant computed for the quasi-(D-1) dimensional bulk (D-dimensional system with open boundary conditions and finite size in one direction) determines the presence or absence of quasi-(D-2) dimensional boundary modes. This additional bulk-boundary correspondence occurs while the system exhibits topological responses associated with a non-trivial D-dimensional topological invariant. 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.