Note on the Abelian Higgs-Kibble model on a lattice: Absence of spontaneous magnetization

Abstract
We study the magnetization (expectation value of the scalar field in the presence of a symmetry-breaking term) for the Abelian Higgs-Kibble model quantized on a Euclidean space-time lattice according to Wilson's procedure and show that it vanishes when the external field goes to zero. More explicitly, we show that the gauge invariance of the infinite-volume system is recovered when the symmetry-breaking term is removed.