Spontaneous breaking of U(1) symmetry at zero temperature in one dimension

The Hohenberg-Mermin-Wagner theorem is the no-go theorem for the spontaneous breaking of continuous symmetries in spatial dimensions d>1 at finite temperature. The classical/quantum mapping implies the absence of continuous symmetry breaking in one dimension at zero temperature, which is also known as Coleman's theorem in the context of relativistic quantum field theories. Except for the classic example of the Heisenberg ferromagnet and its variations, there has been no known counterexample to the theorem. In this work, we discuss new examples that display spontaneous breaking of a U(1) symmetry at zero temperature, although the order parameter does not commute with the Hamiltonian, unlike the Heisenberg ferromagnet. We argue that a more general condition for this behavior is the Hamiltonian being frustration-free.