Robust Symmetry Breaking in Gapless Quantum Magnets

We prove the existence of spontaneous symmetry breaking in suitably low-energy eigenstates of certain gapless and frustrated many-body quantum systems, namely symmetric quantum perturbations to classical models that exhibit spontaneous symmetry breaking of a finite group at some positive temperature. Additionally, the classical model need not be local in space, as long as it satisfies a quantum analog of the Peierls condition. As an example of our technique, we establish robust ferromagnetism in random-bond Ising models in d=2 dimensions with sufficiently biased random couplings, with weak transverse field. Our mathematical technique is based on establishing quantum bottlenecks, similar to a "many-body Wentzel-Kramers-Brillouin" method for evaluating tunneling rates. Using these same methods, we provide new proofs of metastability and the slow decay of the false vacuum, applicable to gapless metastable states. Our Letter represents a first step toward a rigorous classification of stable gapless quantum phases.