Goldstone Boson Interaction in D=2+1 (Pseudo-)Lorentz-Invariant Systems with a Spontaneously Broken Internal Rotation Symmetry

TL;DR

This paper systematically analyzes Goldstone boson interactions in 2+1 dimensional systems with spontaneously broken internal rotation symmetry, revealing unique behaviors in quantum XY and Heisenberg antiferromagnets at low temperatures.

Contribution

It provides a three-loop order effective field theory calculation of Goldstone boson interactions in 2+1D systems, highlighting special cases of XY and Heisenberg models with distinct interaction signs.

Findings

Goldstone boson interactions are generally attractive or repulsive depending on the system.

XY model exhibits positive interaction contribution to magnetization at low T.

Heisenberg antiferromagnet shows negative interaction contribution to susceptibility.

Abstract

The low-temperature properties of systems characterized by a spontaneously broken internal rotation symmetry, O($N$) $\to$ O($N$-1), are governed by Goldstone bosons and can be derived systematically within effective Lagrangian field theory. In the present study we consider systems living in two spatial dimensions, and evaluate their partition function at low temperatures up to three-loop order. Although our results are valid for any such system, here we use magnetic terminology, i.e., we refer to quantum spin systems. We discuss the sign of the Goldstone boson interaction in the pressure, staggered magnetization, and susceptibility as a function of an external staggered field for general $N$. As it turns out, the $d$=2+1 quantum XY model ($N$=2) and the $d$=2+1 Heisenberg antiferromagnet ($N$=3), are rather special, as they represent the only cases where the spin-wave interaction in the pressure is repulsive in the whole parameter regime where the effective expansion applies. Remarkably, the $d$=2+1 XY model is the only system where the interaction contribution in the staggered magnetization (susceptibility) tends to positive (negative) values at low temperatures and weak external field.