The phase diagram of the Abelian lattice Higgs model. A review of rigorous results

We give a review (and some improvements) of the rigorous results on the phase structure of Abelian lattice Higgs models with gauge groups U(1) and . Emphasis is put on the relation between the Higgs mechanism (gauge-independent) and spontaneous symmetry breaking (gauge-dependent). We also discuss some nonperturbative effects due to Gribov copies in this context.This paper is based on an invited talk given by one of the authors (C.B.) at the conference on the Statistical Mechanics of Phase Transitions—Mathematical and Physical Aspects, Tebo, Czechoslovakia, September 1–6, 1986.