Reconsidering gauge-Higgs continuity

The 3-d Z(2) lattice gauge-Higgs theory is cast in a partial axial gauge leaving a residual Z(2) symmetry, global in two directions and local in one. It is shown both analytically and numerically that this symmetry breaks spontaneously in the Higgs phase and is unbroken in the confinement phase. Therefore they must be separated everywhere by a phase transition, in contradiction to a theorem by Fradkin and Shenker. It relied on a fully fixed unitary gauge, which prohibits this phase transition explicitly. Thus the unfixed gauge theory is not, in this case, equivalent to the unitary-gauge version. This letter presents evidence for a hidden symmetry breaking in 3-d Z(2) lattice gauge-Higgs theory, made visible by a partial axial gauge fixing. The symmetry which breaks is simply a subgroup of the original gauge symmetry, which is fixed enough to invalidate Elitzur’s theorem [1] but leaves unbroken N “layered” Z(2) symmetries on the N lattice, one for each plane perpendicular to the third axis. In this gauge the average Higgs field on each layer and the average third-dimension-pointing link (link-magnetization) on each layer become order parameters for this residual exact Z(2) symmetry. It is found that in the Coulomb phase, the link magnetization takes on a non-zero expectation value, but the Higgs field does not. In the Higgs phase they both have nonzero expectation values, which can be shown analytically. In the confinement phase it can be shown analytically that both expectation values vanish and the symmetry is unbroken there. Therefore, due to the symmetry difference, these phases must be separated by a phase transition. This is a surprising result because a well-known theorem by Fradkin and Shenker (FS)[2] states that the Higgs and confinement phases are analytically connected. This phase-continuity has played an important role in attempts to understand the confinement mechanism. The FS proof relies crucially on using a completely-fixed unitary gauge, in which all of the Higgs fields are set to unity. Normally, any fixed gauge is considered equivalent to the unfixed gauge theory, with the partition function differing only by an infinite constant. However, if a portion of the gauge symmetry itself breaks spontaneously (e.g. global gauge symmetry or the layered symmetry considered above) then they will not be equivalent, because of the loss of ergodicity at a phase transition. The unfixed theory will differ from the unitary-gauge theory by a different infinite constant in the symmetry-broken phase than in the unbroken phase. Therefore, the FS proof, valid in the unitary gauge, is not necessarily valid for the unfixed or partially-fixed theories. In fact the presence of the phase transition mentioned above shows it is invalid. The change in ergodicity at the phase transition produces a sudden change in entropy which can be seen as the source of non-analyticity. The plan of this letter is as follows. First, the previously-known phase structure of the theory is reviewed including the FS analyticity region and the Monte-Carlo results of Jongeward, Stack, and Jayaprakash[3]. Then it is shown that in partial axial gauge along the β = 0 line, the theory is equivalent to the one-dimensional Ising model (here β is the gauge coupling parameter). This places a point of non-analyticity at (β = 0, βH = ∞), well within the FS analyticity region. The full theory is self-dual, so if there is only one phase transition it must occur along the self-dual line. Thus the FS analyticity arguments can be safely used away from it. They are used to show that the order parameters are identically zero in the strong coupling region (confined phase) and non-zero in the dual region (Higgs phase). This is sufficient to prove the two regions are everywhere separated by a phase boundary. Monte Carlo simulations are then used to explore the low-β region where no phase transition was seen in Ref. [3]. Using the new order parameters a clear transition is seen near the self-dual line, which is most likely weakly first-order. Finally the situation in the Coulomb phase, where the symmetry is also broken is discussed. The fact that the gauge