Investigation of the two-dimensional O(3) model using the overrelaxation algorithm

We investigate the two-dimensional O(3) model with the standard action by Monte Carlo simulation at couplings $\beta$ up to 2.05. We measure the energy density, mass gap, and susceptibility of the model, and gather high statistics on lattices of size $L\le 1024$ using the floating point systems T-series vector hypercube. Asymptotic scaling does not appear to set in for this action, even at $\beta =2.05\mathrm{}$, where the correlation length is 304. We observe a 20% difference between our estimate $\frac{m}{{\Lambda }_{\overline{\mathrm{MS}}}}=3.52\mathrm{}\left(6\right)\mathrm{}$ at this $\beta$ and the recent exact analytical result. We use the overrelaxation algorithm interleaved with Metropolis updates and show that the decorrelation time scales with the correlation length and the number of overrelaxation steps per sweep. We determine its effective dynamical critical exponent to be $z^{\prime }=1.079\mathrm{}\left(10\right)\mathrm{}$; thus critical slowing down is reduced significantly for this local algorithm that is vectorizable and parallelizable.