Criticality in 1-dimensional field theories with mesoscopic, infinite range interactions

TL;DR

This work introduces mesoscopic feedback mechanisms in one-dimensional field theories to generate effective infinite-range interactions, enabling genuine phase transitions despite the usual 1D constraint. By coupling interaction strengths to global observables, the authors formulate nonlocal actions such as $S^2$ and $S^3$ for Ising-like spins and extend the analysis to a continuous $O(3)$ model with $S^2$-type feedback. They show a second-order transition at $κ_c=1$ for the $S^2$ Ising model with $ u\approx2$ and a first-order transition at $ω_c\approx2$ for the $S^3$ case, along with spontaneous symmetry breaking in the 1D $O(3)$ model at large coupling and a residual $Z_2$ symmetry. The approach leverages density-of-states methods, Langfeld–LLR coefficients, and finite-size scaling to establish new 1D universality classes, with potential realizations in polymer physics and holographic reductions.

Abstract

The study of critical phenomena in one-dimensional field theories with long-range interactions has garnered significant attention following the implications of the Hohenberg-Mermin-Wagner theorem, which precludes phase transitions in most low-dimensional theories with local interactions. This research investigates a novel class of one-dimensional theories characterised by a distinctly defined infinite interaction range. We propose that such theories emerge naturally through a meso- scopic feedback mechanism. In this proof-of-concept study, we examine Ising-type models and a model with continuous O(3) symmetry, and demonstrate that the natural emergence of phase transitions, criticality, spontaneous symmetry breaking and previously unidentified universality classes is evident.

Figures (8)

• Figure 1: Left and Middle: Effective statistical theory for polymer thermodynamics with mesoscopic feedback on model parameter. Right: Effective 1D field theory emerging at the 1-dimensional subdomain of a local 3D statistical field theory.
• Figure 2: Left: Marginal distribution for the nearest-neighbour interaction $E$ (non-local interaction) for several interactions strengths $\kappa$. Middle: Maximum marginal probability over that at $E=0$ for several extensions $L$. Right: Interface tension $\sigma$ as a function of the coupling strengths $\kappa$.
• Figure 3: Left: Distribution marginal distribution for the nearest neighbour interaction $E$. Middle: Suppression of the symmetric state $E=0$ in the broken phase. Right: Gap formation in the limit $L \to \infty$.
• Figure 4: Left: LLR coefficient at finite volume over its asymptotic value (\ref{['eq:2.14']}). Screening for phase transitions using the LLR coefficient for the the $S^2$ (middle) and $S^3$ (right) case.
• Figure 5: The internal energy density as a function of the couplings for the $S^2$ and $S^3$ theory.