Topological equivalence of gapped Hamiltonians

Two local gapped Hamiltonians are topologically equivalent when they are connected by a continuous path of local Hamiltonians with a nonzero bulk gap, equivalently when their ground spaces are related by quasi-local evolution.

Quasi-adiabatic continuation implements a gapped path of local Hamiltonians by a quasi-local unitary. It dresses local operators with exponentially decaying tails while preserving their action within the corresponding low-energy spaces.

For a two-component gapped BdG Hamiltonian constrained to a plane, the winding number counts how many times its coefficient vector encircles the origin as crystal momentum traverses the Brillouin zone.