[Paper Review] Existence of Algebraic Decay in Nonabelian Ferromagnets

Adrian Patrascioiu|ArXiv.org|Feb 9, 2000

Theoretical and Computational Physics5 citations

TL;DR

This paper establishes that two-dimensional non-Abelian ferromagnets in the O(N) class exhibit algebraic decay of spin correlations at low temperatures, using a novel mapping to site-bond percolation and ergodicity arguments. By analyzing cluster structures in a 'cut' action model and applying Russo's theorem, it proves the existence of a massless phase for all N ≥ 2, contradicting conventional expectations of exponential decay.

ABSTRACT

The low temperature regime of nonabelian two-dimensional ferromagnets is investigated. The method involves mapping such models into certain site-bond peroclation processes and using ergodicity in a novel fashion. It is concluded that all ferromagnets possessing a continuous symmetry (abelian or not) exihibit algebraic decay of correlations at sufficiently low temperatures.

Motivation & Objective

• To resolve the long-standing question of whether two-dimensional non-Abelian O(N) ferromagnets exhibit algebraic or exponential correlation decay at low temperatures.
• To extend the percolation approach used for O(2) models to O(N) models with N ≥ 3, where topological structure complicates the analysis.
• To establish the existence of a massless phase in all 2D O(N) models by proving divergent mean cluster sizes in a mapped percolation process.
• To challenge the conventional wisdom that such models should exhibit exponential decay due to instanton effects, by showing entropic suppression of topological defects.

Proposed method

• Maps the O(N) spin model into an Ising spin configuration by dividing the (N−1)-sphere into hemispheres and assigning σ = ±1 based on spin direction.
• Applies the Fortuin-Kasteleyn transformation to the Ising model, relating the magnetic susceptibility to the mean size of FK-clusters.
• Introduces a 'cut' action model where nearest-neighbor interactions are suppressed unless |s_i − s_j| < ε, enabling control over cluster formation.
• Uses conjectures on ergodicity (C2), percolation uniqueness (C3), and symmetry invariance (C1) to rule out percolation of H-clusters and D-clusters.
• Applies Russo’s theorem to show that if neither σ = +1 nor σ = −1 clusters percolate, then the mean H-cluster size must diverge.
• Employs a surface-area-based conjecture (C6) to argue that the mean cluster size of the equatorial strip (D̄) cannot diverge, implying no percolation.

Experimental results

Research questions

• RQ1 Do two-dimensional O(N) ferromagnets with N ≥ 3 exhibit algebraic decay of correlations at low temperatures?
• RQ2 Can the percolation mapping approach used for O(2) be extended to non-Abelian models with N ≥ 3?
• RQ3 Is the absence of percolation in the D̄-clusters (s_∥ < d) sufficient to imply divergent mean FK-cluster size and thus algebraic decay?
• RQ4 Can topological defects such as instantons be entropically suppressed in 2D O(N) models, leading to a massless phase?
• RQ5 Does the Callan-Symanzik β-function vanish in the low-temperature limit, suggesting asymptotic freedom in the continuum limit?

Key findings

• All two-dimensional O(N) ferromagnets with N ≥ 2 exhibit algebraic decay of spin correlations at sufficiently low temperatures.
• The mean FK-cluster size diverges due to the non-percolation of both σ = +1 and σ = −1 H-clusters, as guaranteed by Russo’s theorem.
• In the 'cut' model, D-clusters (s_∥ > d) cannot percolate because they consist of two disconnected components within non-percolating H-clusters.
• The equatorial strip D̄ (s_∥ < d) cannot percolate for sufficiently small ε, as shown by the surface-area conjecture C6, which implies finite mean cluster size.
• The divergent mean FK-cluster size implies that the magnetic susceptibility diverges, confirming the existence of a massless phase.
• The results contradict the common belief that 2D non-Abelian models should exhibit exponential decay, suggesting that spin waves dominate over instantons due to entropic suppression.