High-Temperature Series Analyses of the Classical Heisenberg and XY Model

TL;DR

This paper uses extended high-temperature series expansions and advanced analysis methods to accurately determine critical temperatures and exponents for 3D classical Heisenberg and XY models, resolving previous discrepancies with Monte Carlo results.

Contribution

It provides a refined analysis of high-temperature series for $O(n)$ models up to 14th order, achieving better agreement with Monte Carlo estimates for critical parameters.

Findings

Good agreement with Monte Carlo estimates for critical temperatures.

Reanalysis of series for Heisenberg model on BCC and FCC lattices.

Validation of field theory exponent estimates.

Abstract

Although there is now a good measure of agreement between Monte Carlo and high-temperature series expansion estimates for Ising ($n=1$) models, published results for the critical temperature from series expansions up to 12{\em th} order for the three-dimensional classical Heisenberg ($n=3$) and XY ($n=2$) model do not agree very well with recent high-precision Monte Carlo estimates. In order to clarify this discrepancy we have analyzed extended high-temperature series expansions of the susceptibility, the second correlation moment, and the second field derivative of the susceptibility, which have been derived a few years ago by L\"uscher and Weisz for general $O(n)$ vector spin models on $D$-dimensional hypercubic lattices up to 14{\em th} order in $K \equiv J/k_B T$. By analyzing these series expansions in three dimensions with two different methods that allow for confluent correction terms, we obtain good agreement with the standard field theory exponent estimates and with the critical temperature estimates from the new high-precision MC simulations. Furthermore, for the Heisenberg model we reanalyze existing series for the susceptibility on the BCC lattice up to 11{\em th} order and on the FCC lattice up to 12{\em th} order.