Criticality in 1-dimensional field theories with mesoscopic, infinite range 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 mesoscopic 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. The framework introduced here holds particular relevance for monolayer spintronics research, where the ultimate goal is to achieve a strong ferromagnetic order at room temperature.

💡 Research Summary

The paper introduces a novel class of one‑dimensional statistical‑field theories in which the interaction range becomes effectively infinite through a mesoscopic feedback mechanism. In this scheme the coupling constants or temperature‑like parameters are not fixed a priori but are dynamically adjusted according to global observables such as the total magnetisation or the action density. Consequently, every spin feels the state of the whole system, while the underlying one‑dimensional lattice geometry, boundary conditions and topology are preserved.

The authors first review the classic results that forbid spontaneous symmetry breaking in 1 D systems with short‑range interactions – the Hohenberg‑Mermin‑Wagner theorem for continuous symmetries and the absence of phase transitions for the nearest‑neighbour Ising chain. They then recall Dyson’s and Fröhlich‑Spencer’s work on algebraically decaying long‑range couplings, emphasizing that the range and functional form of the interaction, rather than dimensionality alone, determine the possibility of ordering.

Two concrete models are studied.

1. Ising‑type model with a non‑local S² term
   The standard nearest‑neighbour Ising action (S=\sum_{\langle xy\rangle}z_xz_y) is used to define a density of states (\rho(E)) that can be computed exactly. The authors then consider a modified action (S_2 = \frac{1}{L-1}\sum_{\langle xy\rangle,\langle uv\rangle}z_xz_yz_uz_v), i.e. the square of the nearest‑neighbour sum. The effective coupling is taken to be (\beta(E)=\kappa^2E/(L-1)), so the partition function becomes
   \