Decay of Correlations in 2D Quantum Systems with Continuous Symmetry

We study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, and we prove a general McBryan-Spencer-Koma-Tasaki theorem concerning algebraic decay of correlations. We present applications of our main result to the Heisenberg, Hubbard, and t-J models, and to certain models of random loops.