Phase Transitions in Spin Glasses

Whereas there is a general consensus that spin glasses with discrete (i.e. Ising) symmetry have a transition at finite temperature in zero magnetic field in three dimensions, there has not been general agreement on two other important questions concerning phase transitions in spin glasses in three dimensions:

(i) is there a transition at finite temperature in isotropic vector spin glasses (such as Heisenberg) which have a continuous symmetry, and

(ii) is there a line of phase transitions (Almeida-Thouless line) in in a magnetic field in Ising spin glasses?

By using Monte Carlo simulations, analyzed with finite-size scaling, I will argue that the answer to the first question is "yes", and to the second question is "no".