Critical Spontaneous Breaking of U(1) Symmetry at Zero Temperature in One Dimension

The Hohenberg-Mermin-Wagner theorem states that there is no spontaneous breaking of continuous internal symmetries in spatial dimensions d <= 2 at finite temperature. At zero temperature, the quantum-toclassical mapping further implies the absence of such symmetry breaking in one dimension, which is also known as Coleman's theorem in the context of relativistic quantum field theories. One route to violate this "folklore" is requiring an order parameter to commute with a Hamiltonian, as in the classic example of the Heisenberg ferromagnet and its variations. However, a systematic understanding has been lacking. In this Letter, we propose a family of one-dimensional models that display spontaneous breaking of a U(1) symmetry at zero temperature, where the order parameter does not commute with the Hamiltonian. While our models can be deformed continuously within the same phase, there exist symmetry-preserving perturbations that render the observed symmetry breaking fragile. We argue that a more general condition for this behavior is that the Hamiltonian is frustration-free.