Enumerating Gribov copies on the lattice

Abstract

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] 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] for different orbits using three complimentary methods: 1. analytical results in lower dimensions, and some lower bounds on 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 n for the coset SU(N_c)/U(1) of an SU(N_c) theory is orbit-independent, we concentrate on the residual compact U(1) case in this article and establish that n 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, n is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension.Comment: 39 pages, 15 eps figures. Published version: minor changes onl

Potential energy landscapes for the 2D XY model: Minima, transition states, and pathways

“…As expected from symmetries of the Hamiltonian [25], for each N with PBC the global minimum is unique, the next minimum is 4-fold degenerate, then…”

“…[25] at smaller N with the help of the more powerful algorithms. Saddles of index one were only obtained from the OPTIM runs.…”

“…Saddles of index one were only obtained from the OPTIM runs. Since the 2D XY model possesses a number of discrete symmetries, as discussed in [25], we also tabulate the number of distinct minima and transition states in the table, i.e. solutions unrelated by symmetries of the Hamiltonian.…”

“…[25] application of the conjugate gradient method for small N suggested that the number of local minima for the 2D XY model would increase exponentially. One of the important results of the current paper is to verify this conjecture and make it more precise by characterizing the landscapes for larger N values, while improving on the earlier results for the number of minima.…”

“…This method was subsequently used to study the potential energy landscapes of various other models in statistical mechanics and particle physics [14][15][16][17][18][19][20][21][22][23]. In particular, it was employed to study the potential energy landscape of the two-dimensional (2D) XY model [24][25][26]. In the latter work, along with all the isolated solutions for a small 3×3 lattice, two types of singular solutions were characterized: (1) isolated singular solutions, where the Hessian matrix is singular (these solutions are in fact multiple solutions); and (2) a continuous family of singular solutions.…”