A theorem in statistical mechanics stating that there cannot be spontaneously broken symmetry in two-dimensional systems at non-zero temperatures. It was proved by David Mermin (1935– ) and Herbert Wagner (1935– ) in 1967–68. Although the theorem rules out the possibility of phase transitions associated with spontaneously broken symmetries in two dimensions, it is still possible to have phase transitions with discrete symmetry, as in the Ising model, or changes in topology, as in a Kosterlitz–Thouless transition.