O(n) Spin Systems- Some General Properties: A Generalized Mermin-Wagner-Coleman Theorem, Ground States, Peierls Bounds, and Dynamics

TL;DR

This paper explores the properties of O(n) spin systems with arbitrary interactions, establishing ground states, bounds, fluctuation effects, and a generalized Mermin-Wagner-Coleman theorem, revealing novel effects and thermodynamic behaviors.

Contribution

It provides a comprehensive framework for analyzing O(n) systems, including ground states, bounds, fluctuation spectra, and a generalized theorem applicable to various dimensions and interaction ranges.

Findings

Derived ground states for translationally invariant O(n) systems

Established Peierls bounds for long-range Ising interactions

Discovered an odd-even n effect in thermal fluctuations

Abstract

Here we examine O(n) systems with arbitrary two spin interactions (of unspecified range) within a general framework. We shall focus on translationally invariant interactions. In the this case, we determine the ground states of the $O(n \ge 2)$ systems. We further illustrate how one may establish Peierls bounds for many Ising systems with long range interactions. We study the effect of thermal fluctuations on the ground states and derive the corresponding fluctuation integrals. The study of the thermal fluctuation spectra will lead us to discover a very interesting odd-even $n$ (coupling-decoupling) effect. We will prove a generalized Mermin-Wagner-Coleman theorem for all two dimensional systems (of arbitrary range) with analytic kernels in $k$ space. We will show that many three dimensional systems have smectic like thermodynamics. We will examine the topology of the ground state manifolds for both translationally invariant and spin glass systems. We conclude with a discussion of O(n) spin dynamics in the general case.