Zero-temperature Monte Carlo study of the non-coplanar phase of the classical bilinear-biquadratic Heisenberg model on the triangular lattice

TL;DR

This study uses large-scale Monte Carlo simulations to analyze the non-coplanar phase of a classical Heisenberg model on a triangular lattice, revealing long-range order in scalar chirality and the necessity of finite-temperature phase transition.

Contribution

Introduces an Ising pseudospin representation and employs advanced cluster algorithms to uncover the true asymptotic behavior of the model at large system sizes.

Findings

Long-range order in scalar chirality at zero temperature

System consists of equivalent ordered sublattices

Finite-temperature phase transition implied by long-range chirality order

Abstract

We investigate the ground-state properties of the highly degenerate non-coplanar phase of the classical bilinear-biquadratic Heisenberg model on the triangular lattice with Monte Carlo simulations. For that purpose, we introduce an Ising pseudospin representation of the ground states, and we use a simple Metropolis algorithm with local updates, as well as a powerful cluster algorithm. At sizes that can be sampled with local updates, the presence of long-range order is surprisingly combined with an algebraic decay of correlations and the complete disordering of the chirality. It is only thanks to the investigation of unusually large systems (containing $\sim 10^8$ spins) with cluster updates that the true asymptotic regime can be reached and that the system can be proven to consist of equivalent (i.e., equally ordered) sublattices. These large-scale simulations also demonstrate that the scalar chirality exhibits long-range order at zero temperature, implying that the system has to undergo a finite-temperature phase transition. Finally, we show that the average distance in the order parameter space, which has the structure of an infinite Cayley tree, remains remarkably small between any pair of points, even in the limit when the real space distance between them tends to infinity.