Bulkboundary

Bulkboundary is a concept in physics and mathematics describing the relationship between a system's bulk properties and the behavior that occurs at its boundaries. In particular, it refers to how nontrivial topological properties of the bulk give rise to edge, surface, or boundary states that are robust against local perturbations.

The idea is formalized in the bulk-boundary correspondence, a principle within topological phases of matter. When a gapped bulk has a nonzero topological invariant, such as a Chern number in two dimensions or a Z2 invariant in certain symmetry classes, gapless modes appear at the boundary. These boundary states are protected by symmetries and the bulk gap, making them resistant to perturbations that do not close the bulk gap.

Examples include the quantum Hall effect, where a two-dimensional electron gas with a nonzero Chern number supports chiral edge modes, and topological insulators and superconductors, which host edge or surface states with specific spin or Majorana properties. In these systems, the boundary phenomena provide direct signatures of the bulk topology.

The bulkboundary concept extends beyond electronic materials to photonic, phononic, and cold-atom systems, where robust boundary transport and waveguiding can be engineered. Mathematically, the bulk-boundary correspondence connects bulk invariants to boundary spectra, and has intersections with index theorems and K-theory in topology.

See also: bulk-boundary correspondence, topological insulator, edge state, surface state, Chern number, Majorana mode.