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 ℤ2, with a ``helical" symmetry group ℤ2rtimes ℤ mixing flavor symmetry with spatial translation. For any 0<β<∞, we construct an uncountable infinity of DLR (Gibbs) states parameterized by ℝ2, with a free ℤ2 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 ℝ2 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.
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BibTeX
@misc{lucas2026spontaneous,
title = {{Spontaneous symmetry breaking at any temperature in a local one-dimensional model}},
author = {Andrew Lucas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07693},
eprint = {2610.07693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07693},
}
RIS
TY - UNPB
TI - Spontaneous symmetry breaking at any temperature in a local one-dimensional model
AU - Andrew Lucas
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07693
UR - https://arxiv.org/abs/2610.07693
AB - We present a one-dimensional classical statistical physics model on an infinite lattice with nearest-neighbor interactions and local state space ℤ^2, with a ``helical" symmetry group ℤ^2rtimes ℤ mixing flavor symmetry with spatial translation. For any 0<β<∞, we construct an uncountable infinity of DLR (Gibbs) states parameterized by ℝ^2, with a free 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 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.
ER -