Since its introduction by Hastings (Phys Rev B 69:104431, 2004), the technique of quasi-adiabatic continuation has become a central tool in the discussion and classification of ground-state phases. It connects the ground states of self-adjoint Hamiltonians in the same phase by a unitary quasi-local transformation. This paper takes a step towards extending this result to non-self-adjoint perturbations, though, for technical reason, we restrict ourselves here to weak perturbations of non-interacting spins. The extension to non-self-adjoint perturbation is important for potential applications to Glauber dynamics (and its quantum analogues). In contrast to the standard quasi-adiabatic transformation, the transformation constructed here is exponentially local. Our scheme is inspired by KAM theory, with frustration-free operators playing the role of integrable Hamiltonians.
Besides the restriction to frustration-free systems the result @cite_20 @cite_32 rests on assumptions concerning the presence of a local gap and topological order in the unperturbed ground state subspace, which are trivially satisfied in our setting of independent spins and unique ground state.
@cite_32 @cite_20
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We prove stability of the spectral gap for gapped, frustration-free Hamiltonians under general, quasi-local perturbations. We present a necessary and sufficient condition for stability, which we call Local Topological Quantum Order and show that this condition implies an area law for the entanglement entropy of the groundstate subspace. This result extends previous work by on the stability of topological quantum order for Hamiltonians composed of commuting projections with a common zero-energy subspace. We conclude with a list of open problems relevant to spectral gaps and topological quantum order. We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming outgoing tails. An important application is wave decay on Kerr and Kerr-de Sitter black holes. We recover the optimal size of the strip and give an @math resolvent bound there. We next show existence of deeper resonance free strips under the @math -normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the ne-sided cutoff resolvent on the real line. We consider the relation of symmetries and subspace controllability for spin-1 2 networks with XXZ couplings, subject to perturbation of a single node by a local potential (Z-control). The Hamiltonians for such networks decompose into excitation subspaces. Focusing on the single excitation subspace, it is shown for single-node Z-controls that external symmetries are characterized by eigenstates of the system Hamiltonian which have zero overlap with the control node, and there are no internal symmetries. It is further shown that there are symmetries which persist even in the presence of random perturbations. For XXZ chains with uniform coupling strengths, a characterization of all possible symmetries is given which shows a strong dependence on the position of the node we control. We then show for Heisenberg and XX chains with uniform coupling strength subject to single-node Z-control that the lack of symmetry is both necessary and sufficient for subspace controllability. Finally, the latter approach is generalized to establish controllability results for simple branched networks. We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and threfore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with a number of examples.