 Quasi-adiabatic continuation

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]