Mermin-Wagner at the Crossover Temperature

TL;DR

This paper revisits the Mermin-Wagner theorem using effective field theory, estimating the crossover temperature in low dimensions and deriving bounds for magnetization, highlighting the theorem's implications in weak external fields.

Contribution

It provides an effective field theory perspective on the Mermin-Wagner theorem and estimates the crossover temperature behavior in weak external fields for different dimensions.

Findings

Crossover temperature $T_c$ tends to zero as external field $H$ approaches zero in 1D and 2D.

$T_c$ scales as $ oot{2}{H}$ in 1D and as $1/| ext{ln} H|$ in 2D.

Derived upper bounds for (staggered) magnetization, but these are not restrictive.

Abstract

Mermin-Wagner excludes spontaneous (staggered) magnetization in isotropic ferromagnetic (antiferromagnetic) Heisenberg models at finite temperature in spatial dimensions $d \le 2$. While the proof relies on the Bogoliubov inequality, here we illuminate the theorem from an effective field theory point of view. We estimate the crossover temperature $T_c$ and show that, in weak external fields $H$, it tends to zero: $T_c \propto \sqrt{H}$ ($d=1$) and $T_c \propto 1/|\ln H|$ ($d=2$). Including the case $d$=3, we derive upper bounds for the (staggered) magnetization by combining microscopic and effective perspectives -- unfortunately, these bounds are not restrictive.