Sources for: Numerical Simulations of a Spin Dynamics Model Based on a Path Integral Approach

This repository contains source code which implements a path integral approach to calculate the thermodynamics of quantum spin in a constant magnetic field along the z-direction. It contains functions for analytic solutions and a numerical approach based on atomistic spin dynamics.

The preprint this code corresponds to is https://arxiv.org/abs/2303.00602

Authors

Thomas Nussle, School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom, http://orcid.org/0000-0001-9783-4287

Stam Nicolis, Institut Denis Poisson, Université de Tours, Université d'Orléans, CNRS (UMR7013), Parc de Grandmont, F-37200, Tours, France, http://orcid.org/0000-0002-4142-5043

Joseph Barker, School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom, http://orcid.org/0000-0003-4843-5516

Description

This is a simple code for atomistic simulations for a single spin in a constant magnetic field along the z-direction. It computes both the classical limit and quantum corrections of the thermal expectation value of the z-component of the spin.

Two different approximation schemes are provided. The first one is a low-temperature approximation and provides the first order quantum correction. The second one is a high temperature approach and provides the first and second order quantum corrections.

Also included are equivalent results computed from the partition function in order to compare atomistic simulations to both exact quantum solution and classical limit. In this particular case, the approximated expectation values can also be computed from the partition function so as to compare results from atomistic simulations.

File descriptions

./

environment.yml Conda environment file to reproduce the python environment for executing the calculations.

LICENSE MIT License file.

Makefile Makefile to run all calculation and produce the data and figures in the figures/ folder.

README.md This readme file.

./python/

This folder contains python code and scripts to generate the figures.

python/analytic.py Defines python functions for the analytic equations which appear in the figures. The mathematical expression for each function are written as docstrings in the python functions.

python/asd.py Defines python functions for atomistic spin dynamics calculations including numerical integration methods, effective fields and stochastic fields.

python/pisd.py An executable python program for running general path integral spin dynamics calculations using the functions in python/asd.py.

python/figure1.py Calculates and plots the spin coherent states analytic results. Expectation value of Sz for s=1/2 as a function of temperature. Plots expectation values computed from the partition function only, by taking into account the first three non-commuting terms in the exponential expansion

python/figure2.py Calculates and plots the low-temperature expansion of the matrix elements method. Expectation value of Sz for s=1/2, 2, 5 as a function of temperature for the classical limit, quantum solution, effective model from the partition function and equivalent results from enhanced atomistic spin dynamics simulations.

python/figure3.py: Calculates and plots the high-temperature spin coherent states expansion method. Expectation value for Sz for s=2 as a function of temperature for the classical limit, quantum solution, effective model with the first correction and second corrections and equivalent results from enhanced atomistic spin dynamics simulations. Plots a inset zoom to make the increasing convergence speed with increasing order more explicit.

python/figure4.py: Calculates and plots the high-temperature spin coherent states expansion normalization with approximation <z|s_z * exp(s_z)|z>~<z|s_z|z> * <z|exp(s_z)|z>. Expectation value for Sz for s=2 as a function of temperature, unnormalized, normalized and the quantum limit.

python/figure5.py: Calculates and plots the higher order correction for the high-temperature spin coherent states expansion method. Expectation value for Sz for s=2 as a function of temperature for classical limit, quantum solution, effective model with 10th correction from partition function and equivalent results from enhanced atomistic spin dynamics simulations.

./figures/

Output generated from the python/figureX.py scripts.

figures/figureX.pdf PDF figure used in the manuscript.

figures/figureX.log Output logged from executing the figure script.

figures/figureXdata Folder containing the raw data generated by the script with the filenames representing <method>_<approximation>_<spin>.tsv

./resources/

aps-paper.mplstyle matplotlib style file for plotting figures with fonts and font sizes similar to the American Physical Society typesetting.

Computational Environment

All calculations were performed on a MacBook Pro (Quad-Core Intel Core i5 2GHz, 16GB RAM, Model Identifier: MacBookPro16,2) running macOS version 11.7.4 (Big Sur). Python and associated packages were installed using conda. The installed package versions were:

• python=3.10
• matplotlib=3.6.2
• numba=0.56.4
• numpy=1.23.5
• scipy=1.9.3
• sympy=1.11.1

The conda environment can be recreated using the environment.yml file on .

Click here for the complete list of package installed by conda including all dependencies and hashes

```text # Name Version Build Channel blas 1.0 mkl bottleneck 1.3.5 py38h67323c0_0 brotli 1.0.9 hca72f7f_7 brotli-bin 1.0.9 hca72f7f_7 ca-certificates 2023.01.10 hecd8cb5_0 certifi 2022.12.7 py38hecd8cb5_0 contourpy 1.0.5 py38haf03e11_0 cycler 0.11.0 pyhd3eb1b0_0 fftw 3.3.9 h9ed2024_1 fonttools 4.25.0 pyhd3eb1b0_0 freetype 2.12.1 hd8bbffd_0 giflib 5.2.1 haf1e3a3_0 gmp 6.2.1 he9d5cce_3 gmpy2 2.1.2 py38hd5de756_0 importlib-metadata 4.11.3 py38hecd8cb5_0 importlib_metadata 4.11.3 hd3eb1b0_0 intel-openmp 2021.4.0 hecd8cb5_3538 jpeg 9e hca72f7f_0 kiwisolver 1.4.4 py38hcec6c5f_0 lcms2 2.12 hf1fd2bf_0 lerc 3.0 he9d5cce_0 libbrotlicommon 1.0.9 hca72f7f_7 libbrotlidec 1.0.9 hca72f7f_7 libbrotlienc 1.0.9 hca72f7f_7 libcxx 14.0.6 h9765a3e_0 libdeflate 1.8 h9ed2024_5 libffi 3.4.2 hecd8cb5_6 libgfortran 5.0.0 11_3_0_hecd8cb5_28 libgfortran5 11.3.0 h9dfd629_28 libllvm11 11.1.0 h46f1229_6 libpng 1.6.37 ha441bb4_0 libtiff 4.5.0 h2cd0358_0 libwebp 1.2.4 h56c3ce4_0 libwebp-base 1.2.4 hca72f7f_0 llvm-openmp 14.0.6 h0dcd299_0 llvmlite 0.39.1 py38h8346a28_0 lz4-c 1.9.4 hcec6c5f_0 matplotlib 3.6.2 py38hecd8cb5_0 matplotlib-base 3.6.2 py38h220de94_0 mkl 2021.4.0 hecd8cb5_637 mkl-service 2.4.0 py38h9ed2024_0 mkl_fft 1.3.1 py38h4ab4a9b_0 mkl_random 1.2.2 py38hb2f4e1b_0 mpc 1.1.0 h6ef4df4_1 mpfr 4.0.2 h9066e36_1 mpmath 1.2.1 py38hecd8cb5_0 munkres 1.1.4 py_0 ncurses 6.3 hca72f7f_3 numba 0.56.4 py38h07fba90_0 numexpr 2.8.4 py38he696674_0 numpy 1.23.5 py38he696674_0 numpy-base 1.23.5 py38h9cd3388_0 openssl 1.1.1t hca72f7f_0 packaging 22.0 py38hecd8cb5_0 pandas 1.5.2 py38h07fba90_0 pillow 9.3.0 py38h81888ad_1 pip 22.3.1 py38hecd8cb5_0 pyparsing 3.0.9 py38hecd8cb5_0 python 3.8.16 h218abb5_2 python-dateutil 2.8.2 pyhd3eb1b0_0 pytz 2022.7 py38hecd8cb5_0 readline 8.2 hca72f7f_0 scipy 1.9.3 py38h3d31255_0 setuptools 65.6.3 py38hecd8cb5_0 six 1.16.0 pyhd3eb1b0_1 sqlite 3.40.1 h880c91c_0 sympy 1.11.1 py38hecd8cb5_0 tbb 2021.6.0 ha357a0b_1 tk 8.6.12 h5d9f67b_0 tornado 6.2 py38hca72f7f_0 wheel 0.37.1 pyhd3eb1b0_0 xz 5.2.10 h6c40b1e_0 zipp 3.11.0 py38hecd8cb5_0 zlib 1.2.13 h4dc903c_0 zstd 1.5.2 hcb37349_0 ```

Reproduction

The make build tool can be used to execute the Makefile and re-produce all of the figures. The steps to reproduce are:

conda env create -f environment.yml
conda activate quantum_spin_dynamics
make clean
make

Note that the the atomistic spin dynamics are stochastic so the results will differ slightly due to random seeding.

Runtimes

• python/figure1.py: 0.939 (s)
• python/figure2.py: 2923.927 (s)
• python/figure3.py: 1502.845 (s)
• python/figure4.py: 0.446 (s)
• python/figure5.py: 1228.228 (s)

Code use for general calculations

The python/pisd.py script can be used to generate data with different integration methods, approximations and spin values.

Using the provided environment.yml file, create a conda environment in your terminal:

conda env create -f environment.yml
conda activate quantum_spin_dynamics

Then run the python/pisd.py script

python python/pisd.py <options>

where the options available are

usage: pisd.py [-h] [--integrator {runge-kutta-4,symplectic}] --approximation {low-temperature,classical-limit,high-temperature-first-order,high-temperature-second-order} --spin SPIN

Simulation parameters from command line.

options:
  -h, --help            show this help message and exit
  --integrator {runge-kutta-4,symplectic}
                        Numerical integration method for solving the spin dynamics
  --approximation {low-temperature,classical-limit,high-temperature-first-order,high-temperature-second-order}
                        Approximation scheme to use
  --spin SPIN           Quantum spin value (should normally be an integer multiple of 1/2)

Additional variables

Depending on computational resources and specific system, one can change some parameters in the python/pisd.py script (or examples scripts as well):

• the value of the gilbert damping alpha (default is 0.5)
• the value of the applied field by changing applied_field which needs to be collinear to the z-axis.
• the range and increments in temperature for atomistic simulations by changing np.linspace(<starting-temperature>, <final-temperature>, <number-of-temperatures-in-range>)
• the initial spin orientation s0 (must be of unit norm and different from np.array([0, 0, 1])
• the equilibration period equilibration_time, the computation time production_time and the integration time step time_step for each stochastic realisation
• the number of stochastic realisations num_realisation of the noise over which to average the time average

Notes

The numba package is used for just in time compilation which greatly reduces the calculation time. In principle the @njit statements can be removed from all code if numba is not supported on a given platform, but the calculation time will be extremely long.

In python/analytic.py, sympy is used to generate higher order terms for approximations symbolically before being compiled. Without this, each order of approximation would need to be hard coded.

By default, the atomistic spin dynamics uses a symplectic integrator described in: Pascal Thibaudeau and David Beaujouan, "Thermostatting the atomic spin dynamics from controlled demons", Phys. A: Stat. Mech. its Appl. 391, 1963–1971 (2012).

Grant Acknowledgement

This software was produced with funding from the UKRI Engineering and Physical Sciences Research Council [grant number EP/V037935/1 - Path Integral Quantum Spin Dynamics] and support from the Royal Society through a University Research Fellowship.