Exploring the 3D Ising gauge-Higgs model in exact Coulomb gauge and with a gauge-invariant substitute for Landau gauge

The Z2 gauge-Higgs model in three dimensions has two different types of phase transition, confinement-deconfinement and Higgs magnetization. Here they are explored through two order parameters, the Coulomb magnetization which is a local order parameter for confinement, and a replica-based gauge-invariant order parameter which tracks the Higgs transition in a way similar to Landau-gauge magnetization. Minimal Coulomb gauge is set exactly, using the polynomial-time minimum-weight matching algorithm of Edmonds. This is a tremendous speed improvement over relaxation/annealing methods and completely eliminates the systematic error. The replica-overlap is also an improvement over relaxation methods for setting Landau gauge, in that it has an easily controllable and measurable systematic error. These simulations show the phase transitions not ending at the first-order endpoint but bifurcating near there and continuing even through the expected analyticity region. The specific heat critical exponents, $\alpha$, are highly negative, explaining the lack of strong signals in the energy. Nevertheless, energy moments are seen to follow curves consistent with the predictions from the order parameters, showing these to be true thermal transitions, albeit weak ones. These results cast further doubt on Fradkin-Shenker analyticity. They also suggest one or more new yet to be explored phases in gauge-Higgs models in the bifurcated region.