Energy density of variational states

We show, in several important and general cases, that a low variational energy density of a trial state is possible even when the trial state represents a different phase from the ground state. Specifically, we ask whether the ground-state energy density of a Hamiltonian whose ground state is in phase A can be approximated to arbitrary accuracy by a wave function, which represents a different phase B. We show this is indeed the case when A has discrete symmetry breaking order in one dimension or topological order in two dimensions, while B is disordered. We argue that, if reasonable conditions of physicality are imposed upon the trial wave function, then this is not possible when A has discrete symmetry breaking in dimensions greater than one and B is symmetric. Some other situations are also discussed.