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Communications in Mathematical Physics

Lie–Schwinger Block-Diagonalization and Gapped Quantum Chains

Jürg Martin Fröhlich, Alessandro Pizzo

Published Mar 20, 2020• 20 citations
Publisher
DOI
10.1007/s00220-019-03613-2

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We study quantum chains whose Hamiltonians are perturbations by bounded interactions of short range of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. We prove that, for small values of a coupling constant, the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.
In this paper we study the low-lying spectrum of the AKLT model perturbed by small, finite-range potentials and with open boundary conditions imposed at the edges of the chain. Our analysis is based on the \emph{local, iterative Lie Schwinger block-diagonalization method} which allows us to control small interaction terms localized near the boundary of the chain that are responsible for the possible splitting of the ground-state energy of the AKLT Hamiltonian into energy levels separated by small gaps. This improves earlier results concerning the persistence of the so called \emph{bulk} gap in these models, besides illustrating the power of our general methods in a non-trivial application.