= Topological equivalence of gapped Hamiltonians
= Topologically equivalent
{synonym}
Two local <Hamiltonian operators> describe the same gapped phase when an allowed continuous path $H(s)$, $0\le s\le1$, 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 bound
$$
\Delta_L(s)\ge\Delta_*>0
$$
independent 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>. <Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems by Sven Bachmann and others>[Assumptions 4.2–4.4 and Section 6, printed p. 29, uniform finite-volume gap]{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.
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