Aspects of spontaneous symmetry breaking in Rindler and anti-de Sitter spacetimes for the O(N) linear sigma model

We investigate aspects of spontaneous breakdown of symmetry for the O(N) symmetric linear sigma model in the background of Rindler and anti-de Sitter spacetimes respectively. In the large N limit, by computing the one-loop effective action, we report that in three dimensional Rindler space, there is a phase transition from the disordered phase to an ordered phase past a certain critical Rindler acceleration parameter "a." Connections with finite temperature field theory results are established, thereby further reinforcing the idea that Rindler space can indeed be a proxy for Minkowski spacetime with finite temperature. We extend our calculations to anti-de Sitter space in various dimensions and observe that symmetry is broken in three dimensions, but not in four dimensions. We discuss the implications of our results.