Study of Critical Slowing-Down in SU(2) Landau Gauge Fixing

Here we report on the status of our study [1,2] of Landau gauge-fixing algorithms. The efficiency of these algorithms is of key importance when gauge-dependent quantities — such as gluon and quark propagators — are evaluated, especially if the effect of Gribov copies is considered. The main issue regarding the efficiency of these algorithms is the problem of critical slowing-down (CSD), which occurs when the relaxation time τ of an algorithm diverges as the lattice volume is increased (see for example [3]). Conventional local algorithms have dynamic critical exponent z ≈ 2, namely τ ∼ N, while global methods may succeed in eliminating CSD completely, i.e. z ≈ 0. We consider five different algorithms: the Los Alamos method (a conventional local algorithm), the Cornell method, which is generally believed to have z ≈ 2, the overrelaxation and stochastic overrelaxation methods (improved local algorithms, which are expected to show z ≈ 1), and the Fourier acceleration method, which is a global method. We confirm the predictions for z (in both two and four dimensions) with the exception of the Cornell method. For this case we obtain that the dynamic critical exponent is actually z ∼< 1, a result that can be understood by a comparative analysis between the Cornell and the overrelaxation methods. Besides the problem of CSD, we are also interested in understanding which quantities should be used to test the convergence of the gauge fixing, and in finding prescriptions for the tuning of parameters, when