Topological equivalence of gapped Hamiltonians

Two local Hamiltonian operators describe the same gapped phase when an allowed continuous path $H(s)$, $0\le s\le 1$, connects them while retaining a nonzero bulk spectral gap. The path must keep locality bounds under control. For a family of finite systems with an isolated low-energy sector, a standard sufficient condition is a lower boundindependent of system size $L$ and path parameter $s$. A positive gap in each small finite system alone does not establish a gapped thermodynamic limit.

For sufficiently regular paths of local interactions, quasi-adiabatic continuation transports the low-energy subspace by a quasi-local evolution. Lieb-Robinson bounds control its spatial tails. The resulting equivalence concerns locality-preserving transport, not arbitrary unitary operators on a finite Hilbert space. ^[ref]

The permitted paths define what is being classified. In a symmetry-protected topological phase, they must preserve the specified symmetry. Allowing symmetry-breaking terms can join phases that were distinct under that restriction; the Haldane phase is an important example of why the symmetry condition matters.

Quasi-adiabatic continuation transports a separated low-energy subspace along a sufficiently regular path of local Hamiltonian operators whose spectral gap stays open. In finite volume, a suitable unitary operator $U(s)$ relates the spectral projectors by $P(s)=U(s)P(0)U(s{)}^{†}$.

The generator can be built from a filtered time evolution of ${\partial }_{s}H$. Lieb-Robinson bounds then control how far a transformed local observable spreads, providing a quasi-local evolution rather than a strictly local one. Exact spectral-flow constructions can have tails decaying faster than every power and almost exponentially; an unconditional pure exponential claim is too strong. The method compares gapped phases and transports their low-energy structure. Its parameter evolution need not equal the physical real-time evolution generated by $H(s)$.^[ref]

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.