 Topological quantum matter

Topological quantum matter has robust low-energy properties controlled by global topology rather than local order parameters.

The central question is which quantum states can be joined by a smooth change of a local Hamiltonian while keeping a nonzero bulk spectral gap. A Chern insulator makes this question concrete: two band insulators can have different integer Chern numbers, forcing a gap closure between them. The Haldane phase supplies a one-dimensional example in which a protecting symmetry matters. These are distinct from intrinsic topological order, with its anyons and topology-dependent ground-state degeneracy. The Berezinskii–Kosterlitz–Thouless transition concerns a separate mechanism involving vortices; topology organizes more than one kind of physical phenomenon.