Enumerating Gribov copies on the lattice

In the modern formulation of lattice gauge fixing, the gauge-fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions, called Gribov copies. The dependence of the number of Gribov copies, n[U]n[U], on the different gauge orbits plays an important role in constructing the Faddeev–Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n[U]n[U] for different orbits using three complimentary methods: (1) analytical results in lower dimensions, and some lower bounds on n[U]n[U] in higher dimensions, (2) the numerical polynomial homotopy continuation method, which numerically finds all   Gribov copies for a given orbit for small lattices, and (3) numerical minimisation (“brute force”), which finds many distinct Gribov copies, but not necessarily all. Because nn for the coset SU(Nc)/U(1) of an SU(Nc) theory is orbit independent, we concentrate on the residual compact U(1) case in this article, and establish that nn is orbit dependent for the minimal lattice Landau gauge and orbit independent for the absolute lattice Landau gauge. We also observe that, contrary to a previous claim, nn is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension.

► We compute the number of Gribov copies in three variants of lattice Landau gauge.
► The results are obtained using analytical, semi-analytical and numerical methods.
► In two dimensions, the number of copies depends on the gauge orbit in all three gauges.
► In the stereographic gauge, the number is far higher than in the naive gauge.