Spontaneous symmetry breaking at any temperature in a local one-dimensional model

Abstract

We present a one-dimensional classical statistical physics model on an infinite lattice with nearest-neighbor interactions and local state space $\mathbb{Z}^2$, with a ``helical" symmetry group $\mathbb{Z}^2\rtimes \mathbb{Z}$ mixing flavor symmetry with spatial translation. For any $0<β<\infty$, we construct an uncountable infinity of DLR (Gibbs) states parameterized by $\mathbb{R}^2$, with a free $\mathbb{Z}^2$ flavor symmetry group action; as such, we interpret the model as exhibiting spontaneous symmetry breaking at any temperature. The most general DLR state we construct is interpreted as an infinite sequence of localized domain walls separating a pair of asymptotic ``symmetry sectors". Although any normalizable Gibbs state necessarily breaks a countably infinite freely-acting symmetry group, a natural regularization of the problem to (arbitrarily) large but finite systems with periodic boundary conditions has measure concentration in disjoint and far-separated symmetry-related clusters in the (now unique) Gibbs state, which justifies our interpretation of spontaneous symmetry breaking on the infinite line. A modification of the model with local state space $\mathbb{R}^2$ must break a continuous helical symmetry at any temperature. This is not a contradiction with the Mermin-Wagner Theorem, because this continuous helical symmetry is non-compact and does not commute with translation.