# Monte Carlo determination of the low-energy constants for a two-dimensional spin-1 Heisenberg model with spatial anisotropy

## TL;DR

This study uses Monte Carlo simulations combined with magnon chiral perturbation theory to accurately determine low-energy constants of a 2D spin-1 Heisenberg model with spatial anisotropy, providing new benchmark data.

## Contribution

First-principles Monte Carlo calculations of low-energy constants for anisotropic 2D spin-1 Heisenberg models, including new values for ${\

## Key findings

- Quantitative agreement of $
ho_{s1}$ with series expansion results.
- New benchmark values for ${\
- The approach validates the use of magnon chiral perturbation theory in anisotropic systems.

## Abstract

The low-energy constants, namely the spin stiffness $\rho_s$, the staggered magnetization density ${\cal M}_s$ per area, and the spinwave velocity $c$ of the two-dimensional (2D) spin-1 Heisenberg model on the square and rectangular lattices are determined using the first principles Monte Carlo method. In particular, the studied models have antiferromagnetic couplings $J_1$ and $J_2$ in the spatial 1- and 2-directions, respectively. For each considered $J_2/J_1$, the aspect ratio of the corresponding linear box sizes $L_2/L_1$ used in the simulations is adjusted so that the squares of the two spatial winding numbers take the same values. In addition, the relevant finite-volume and -temperature predictions from magnon chiral perturbation theory are employed in extracting the numerical values of these low-energy constants. Our results of $\rho_{s1}$ are in quantitative agreement with those obtained by the series expansion method over a broad range of $J_2/J_1$. This in turn provides convincing numerical evidence for the quantitative correctness of our approach. The ${\cal M}_s$ and $c$ presented here for the spatially anisotropic models are new and can be used as benchmarks for future related studies.