Spontaneous breaking of remnant gauge symmetries in zero-temperature SU(2) lattice gauge theory

The 4-d SU(2) lattice gauge theory is simulated in the minimal Coulomb gauge which aims to maximize the traces of all links in three directions. Fourth-direction links are interpreted as spins in a Heisenberg-like model with varying interactions. These spins magnetize in 3-d hyperlayers at weak coupling, breaking a remnant gauge symmetry, as well as the Polyakov-loop symmetry. They demagnetize at a phase transition around β = 2.5 on the infinite lattice, as determined by Binder cumulant crossings. Because N symmetries are breaking on an N lattice, the transition is unusually broad, encompassing most of the crossover region on typical lattices. It is well known that Elitzur’s theorem[1] prevents the spontaneous breaking of local gauge symmetries. However, once the gauge is fixed with a suitable gauge-fixing term, the remaining global or partially-global remnant gauge symmetry can be spontaneously broken, such as by the Higgs field in the standard model. Here, the minimal Coulomb gauge is used, which can be thought of as a “spin-like” gauge as detailed below. It attempts to transform all links lying in the first three lattice directions so that their traces are as large as possible. This makes them as close to the identity matrix as is possible through a gauge transformation. Interactions of the remaining links pointing in the fourth direction are through plaquettes involving, in each case, two gauge-fixed links. To the extent these are close to the identity, the interaction is spin-like in the following sense. If the sideways links were exactly the identity, then the four-link gauge interactions on plaquettes collapse to two-link spin interactions between adjacent fourth-direction pointing links. The lattice gauge theory would become exactly a set of uncoupled 3-d O(4) Heisenberg spin models, one for each hyperlayer perpendicular to the fourth direction. Of course the sideways links are not exactly the unit matrix, but if the gauge condition can drive them close, and at weak coupling it demonstrably can, then the fourth-direction links may still act more or less like spins in the Heisenberg model. In the following it is found that indeed, if these links are used to define a magnetization, the system is magnetized at weak coupling and demagnetizes at strong coupling. These phases are separated by a phase transition occurring around β = 1/g = 2.5. Finite size scaling indicates it is most likely a weak first-order transition. The Binder cumulant for different lattice sizes shows a definite crossing at the critical point and the magnetization actually strengthens with increasing lattice size in the magnetized phase. This seems to leave little doubt that the result also applies to the infinite 4-d lattice. This magnetic order parameter, which breaks a remnant gauge symmetry, was previously introduced by Greensite et. al.[2]. They showed that, due to its connection to the instantaneous Coulomb potential, realization of this symmetry was a necessary condition for absolute confinement. They also gave numerical evidence that the symmetry was realized in the confining phase of SU(2) lattice gauge theory and also, somewhat surprisingly, in the deconfined phase of an N × 2 lattice. Nakamura and Saito drew a similar conclusion for the SU(3) case[3]. However, Greensite et. al. also found that in gauge-Higgs models, the symmetry did appear to break in the Higgs phase[2]. In the present paper, the same analysis is applied to large symmetric lattices in the pure SU(2) theory, where it is found that the symmetry is broken at couplings β > 2.6. The transition is somewhat unconventional in that each lattice hyperlayer takes on its own magnetization. The symmetry broken is an N-fold SU(2) remnant gauge symmetry, global in three directions, but local in the fourth. The magnetization also breaks the Polyakov loop symmetry. A single constraint from the gauge-invariant Polyakov loop relates the different layered magnetizations, but otherwise they are independent. The transition is considerably more spread out in β on the finite lattice than a conventional phase transition would be for two reasons. First, the effective volume for the order parameter is only N due to the symmetry breakings occurring separately in layers, and second, an extra entropy factor from the binomial distribution favors states in which only some of the hyperlayers are magnetized over states with no magnetization or full magnetization. However as far as the energy is concerned, this is still a four-dimensional transition because the layers interact indirectly through multiple plaquettes. Lattice gauge theory is usually studied without fixing the gauge, in which case one is limited to gauge-invariant observables such as the Wilson and Polyakov loops. The nonlocal nature of these observables make them more difficult to work with than the local magnetization of a spin theory. In recent years it has been recognized that gauge fixing may be necessary or at least convenient to uncover hidden features of gauge configurations. Here