# Thermodynamics of O(N) Antiferromagnets in 2+1 Dimensions

## TL;DR

This paper uses effective Lagrangian methods to compute the thermodynamics of O(N) antiferromagnets in 2+1 dimensions, revealing weak magnon interactions and universal low-temperature behavior.

## Contribution

It provides a systematic, symmetry-based calculation of thermodynamic quantities up to three-loop order, demonstrating the universality and model-independence of the low-temperature series in 2+1 dimensions.

## Key findings

- Magnon-magnon interaction in O(3) antiferromagnet is weak and repulsive.
- The free energy density's temperature dependence is determined by a coefficient from the leading-order Lagrangian.
- The low-temperature series are universal and do not depend on higher-order lattice anisotropies.

## Abstract

Within the framework of effective Lagrangians we calculate the free energy density for an O($N$) antiferromagnet in 2+1 dimensions up to three-loop order in the perturbative expansion and derive the low-temperature series for various thermodynamic quantities. In particular, we show that the magnon-magnon interaction in the O(3) antiferromagnet in $d$=2+1 -- the O(3)-invariant quantum Heisenberg antiferromagnet on a square or a honeycomb lattice -- is very weak and repulsive and manifests itself through a term proportional to five powers of the temperature in the free energy density. Remarkably, the corresponding coefficient is fully determined by the leading-order effective Lagrangian ${\cal L}^2_{eff}$ and does not involve any higher order effective constants from ${\cal L}^4_{eff}$ related to the anisotropies of the lattice -- the symmetries are thus very restrictive in $d$=2+1. We also compare our results that apply to O($N$) antiferromagnets in 2+1 dimensions with the those for O($N$) antiferromagnets in 3+1 dimensions. The present work demonstrates the efficiency of the fully systematic effective Lagrangian method in the condensed matter domain, which clearly proves to be superior to spin-wave theory. We would like to emphasize that the structure of the low-temperature series derived in the present work is model-independent and universal as it only relies on symmetry considerations.