Low-Temperature Properties of Two-Dimensional Ideal Ferromagnets

TL;DR

This paper analyzes the low-temperature properties of two-dimensional ideal ferromagnets using effective Lagrangians, revealing that spin-wave interactions only appear at three-loop order and discussing implications of the Mermin-Wagner theorem.

Contribution

It provides a systematic, model-independent low-temperature expansion for 2D ferromagnets, including the effect of external magnetic fields, and clarifies the order at which spin-wave interactions emerge.

Findings

Spin-wave interaction appears only at three-loop order.

No interaction term of order T^3 in the free energy density.

Results are applicable to various lattice types and consider the Mermin-Wagner theorem.

Abstract

The manifestation of the spin-wave interaction in the low-temperature series of the partition function has been investigated extensively over more than seven decades in the case of the three-dimensional ferromagnet. Surprisingly, the same problem regarding ferromagnets in two spatial dimensions, to the best of our knowledge, has never been addressed in a systematic way so far. In the present paper the low-temperature properties of two-dimensional ideal ferromagnets are analyzed within the model-independent method of effective Lagrangians. The low-temperature expansion of the partition function is evaluated up to two-loop order and the general structure of this series is discussed, including the effect of a weak external magnetic field. Our results apply to two-dimensional ideal ferromagnets which exhibit a spontaneously broken spin rotation symmetry O(3) $\to $ O(2) and are defined on a square, honeycomb, triangular or Kagom\'e lattice. Remarkably, the spin-wave interaction only sets in at three-loop order. In particular, there is no interaction term of order $T^3$ in the low-temperature series for the free energy density. This is the analog of the statement that, in the case of three-dimensional ferromagnets, there is no interaction term of order $T^4$ in the free energy density. We also provide a careful discussion of the implications of the Mermin-Wagner theorem in the present context and thereby put our low-temperature expansions on safe grounds.