Lie-Schwinger block-diagonalization and gapped quantum chains with unbounded interactions

TL;DR

This paper proves that quantum chains with short-range interactions maintain a positive spectral gap under small perturbations, using a novel Lie-Schwinger conjugation method to analyze the Hamiltonians.

Contribution

It introduces a new method based on local Lie-Schwinger conjugations to establish spectral gap stability in perturbed quantum chains.

Findings

Spectral gap remains positive under small perturbations.

Method applies to interactions form-bounded w.r.t. on-site Hamiltonians.

Results hold uniformly in chain length.

Abstract

We study quantum chains whose Hamiltonians are perturbations by 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. For interactions that are form-bounded w.r.t. the on-site Hamiltonian 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. In our proof we use a novel method introduced in [FP] and based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.