Confinement-Higgs and deconfinement-Higgs transitions in three-dimensional $Z(2)$ LGT

We re-examine by numerical simulation the phase structure of the three-dimensional Abelian lattice gauge theory (LGT) with $Z(2)$ gauge fields coupled to $Z(2)$-valued Higgs fields. Concretely, we explore two different order parameters which are able to distinguish the three phases of the theory: (i) the Fredenhagen-Marcu operator used to discriminate between deconfinement and confinement/Higgs phases and (ii) the Greensite-Matsuyama overlap operator proposed recently to distinguish confinement and Higgs phases. The latter operator is an analog of the overlap Edwards-Anderson order parameter for spin-glasses. According to it, the Higgs phase is realized as a glassy phase of the gauge system. For this reason standard tricks for simulations of spin-glass phases are utilized in this work, namely tempered Monte Carlo and averaging over replicas. In addition, we also present results for a certain definition of distance between Higgs field configurations. Finally, we calculate various gauge-invariant correlation functions in order to extract the corresponding masses.