Lie–Schwinger Block-Diagonalization and Gapped Quantum Chains with Unbounded Interactions

We study quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian consisting of a sum of on-site terms that do not couple the degrees of freedom located at different sites of the chain and have a strictly positive energy gap above their ground-state energy. For interactions that are form-bounded w.r.t. the on-site terms, we prove that 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, for small values of a coupling constant. Our proof is based on an extension of a novel method introduced in [FP] involving local Lie–Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.