Lie-Schwinger block-diagonalization and gapped quantum chains

TL;DR

This paper proves that small perturbations of gapped quantum chain Hamiltonians preserve a uniform spectral gap, using a novel local Lie-Schwinger conjugation method.

Contribution

Introduces a new technique based on local Lie-Schwinger conjugations to analyze spectral gaps in perturbed quantum chains.

Findings

Spectral gap remains positive under small perturbations

Method applies uniformly to chains of arbitrary length

Establishes stability of the gap for a class of quantum Hamiltonians

Abstract

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.