Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

In this paper we extend the local iterative Lie-Schwinger block-diagonalization method - introduced in [8] for quantum lattice systems with bounded interactions in arbitrary dimension- to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t| independent of the size of the lattice, we prove that the spectral gap above the ground-state energy of such Hamiltonians remains strictly positive. As in [8], we iteratively construct a sequence of local blockdiagonalization steps based on unitary conjugations of the original Hamiltonian and inspired by the Lie-Schwinger procedure. To control the supports of the effective potentials generated in the course of our block-diagonalization steps, we use methods introduced in [8] for Hamiltonians with bounded interactions potentials. However, due to the unboundedness of the interaction potentials, weighted operator norms must be introduced, and some of the steps of the inductive proof by which we control the weighted norms of the effective potentials require special care to cope with matrix elements of unbounded operators. We stress that no large-field problems appear in our con-struction. In this respect our operator methods turn out to be an efficient tool to separate the low-energy spectral region of the Hamiltonian from other spectral regions, where the un-bounded nature of the interaction potentials would become manifest. (c) 2022 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we extend the local iterative Lie-Schwinger block-diagonalization method to bosonic systems with unbounded interactions. We study Hamiltonians consisting of a gapped operator and a perturbation given by unbounded interaction potentials. For small values of the coupling constant, we prove the existence of a positive spectral gap above the ground-state energy. Using unitary conjugations and inspired by the Lie-Schwinger procedure, we iteratively construct a sequence of local block-diagonalization steps. Weighted operator norms are introduced to control the supports of the effective potentials generated during the block-diagonalization process.