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Enumerating Copies in the First Gribov Region on the Lattice in up to four Dimensions

The covariant gauges are known to suffer from the Gribov problem: even after fixing a gauge non-perturbatively, there may still exist residual copies which are physically equivalent to each other, called Gribov copies. While the influence of Gribov copies in the relevant quantities such as gluon propagators has been heavily debated in recent studies, the significance of the role they play in the Faddeev--Popov procedure is hardly doubted. We concentrate on Gribov copies in the first Gribov region, i.e., the space of Gribov copies at which the Faddeev--Popov operator is strictly positive (semi)definite. We investigate compact U($1$) as the prototypical model of the more complicated standard model group SU($N_{c}$). With our Graphical Processing Unit (GPU) implementation of the relaxation method we collect up to a few million Gribov copies per orbit. We show that the numbers of Gribov copies even in the first Gribov region increase exponentially in two, three and four dimensions. Furthermore, we provide strong indication that the number of Gribov copies is gauge orbit dependent.