It is shown, by means of Monte Carlo simulation and Finite Size Scaling analysis, that the Heisenberg spin glass undergoes a finite-temperature phase transition in three dimensions. There is a single critical temperature, at which both a spin glass and a chiral glass orderings develop. The Monte Carlo algorithm, adapted from lattice gauge theory simulations, makes possible to thermalize lattices of size L = 32, larger than in any previous spin glass simulation in three dimensions. High accuracy is reached thanks to the use of the Marenostrum supercomputer. The large range of system sizes studied allow us to consider scaling corrections. Palassini and Caracciolo [3]. These works demonstrated the applicability to spin-glasses of our approach [4,5] to Finite Size Scaling (FSS) at the critical temperature, as well as that of Caracciolo and coworkers [6] for the paramagnetic state (see also [7,8]).However, the situation is still confusing for Heisenberg spin glasses [9,10,11,12,13,14,15,16,17], which is the more experimentally interesting case (see e.g. [18,19]). Indeed, numerical studies are the only theoretical tool to achieve progress in three dimensions. Early simulations [9] could not reach low enough temperatures due to the dramatic dynamical arrest of the algorithms available at the time, and concluded that the critical temperature, T c , was strictly zero. Yet, recent studies at lower temperatures [16,17] found indirect indications of a spin glass transition for Heisenberg spin glasses. Matters are further complicated by the Villain et al. [12] suggestion of a low temperature chiral glass phase, with an Ising like ordering due to the handedness of the non-collinear spin structures (see definitions below). The simulations of Kawamura and coworkers [13,14] gave ample support to this spin-chirality decoupling scenario (i.e. T c = 0 for the spin glass, but T c > 0 for the chiral glass ordering).In order to clarify the situation for Heisenberg and XY spin glasses in D = 3, Young and Lee [15] have recently tried our FSS methods at T c [2,4]. Although parallel tempering only allowed them [15] to thermalize systems of size up to L = 12, very clear results were reached for the XY spin glasses. The finite-lattice correlation length [20], ξ L , was analyzed for several system sizes L. As expected [2,4,5], the dimensionless ratio ξ L /L crosses neatly at the same T c , for the chiral glass and the spin glass ordering, for XY spin glasses. In the more important case of Heisenberg spin glasses, their results, although inconclusive, were interpreted also as lack of spin-chirality decoupling. This conclusion has been criticized by Kawamura and Hukushima [14], that studied somehow larger systems on very few samples.Here we show that a finite-temperature spin glass transition occurs for the Heisenberg spin glass in D = 3. The critical temperature for the spin glass transition coincides with that of the chiral glass. Our results rely on Monte Carlo simulation and FSS analysis at T c [2,4,5]. We adapt a lattice gauge theory alg...
