# Numerical Simulations of a Spin Dynamics Model Based on a Path Integral Approach

## TL;DR

This paper develops a path integral-based classical spin model to efficiently simulate quantum spin systems at finite temperatures, capturing quantum effects as effective anisotropies and reproducing thermal expectation values.

## Contribution

It introduces a novel path integral formulation for spin systems that simplifies quantum simulations by mapping them onto effective classical models.

## Key findings

- Effective classical Hamiltonians describe quantum fluctuations.
- The model reproduces quantum thermal expectation values.
- Facilitates analysis of magnetic ordering at finite temperatures.

## Abstract

Inspired by path integral molecular dynamics, we build a spin model, in terms of spin coherent states, from which we can compute the quantum expectation values of a spin in a constant magnetic field, at finite temperature. This formulation facilitates the description of a discrete quantum spin system in terms of a continuous classical model and recasts the quantum spin effects within the framework of path integrals in a double $1/s$ and $\hbar s$ expansion, where $s$ is the magnitude of the spin. In particular, it allows for a much more direct path to the low- and high-temperature limits of the quantum system and to the definition of effective classical Hamiltonians that describe both thermal and quantum fluctuations. In this formalism, the quantum properties of the spins emerge as an effective anisotropy. We use atomistic spin dynamics to sample the path integral, calculate thermodynamic observables and show that our effective classical models can reproduce the thermal expectation values of the quantum system within temperature ranges relevant for studying magnetic ordering.