Sources for: Path integral spin dynamics for quantum paramagnets

This repository contains source code which implements a path integral approach to calculate the thermodynamics of quantum spin for a quantum paramagnet with easy axis along the applied field direction and with compressive/tensile stress along the same direction. It contains functions for analytic solutions and a numerical approach based on atomistic spin dynamics.

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

Pascal Thibaudeau, CEA/DAM le Ripault, F-37260, Monts, France, https://orcid.org/0000-0002-0374-5038

Description

This is a simple code for atomistic simulations for a single spin for a quantum paramagnet with easy axis along the applied field direction (arbitrary choice of z-axis here) and with compressive/tensile stress along the same direction. It computes both the classical limit and quantum corrections of the thermal expectation value of the z-component of the spin.

There are two approximation schemes included: The first one is a "high" temperature approximation, and the second one is a exact method which directly takes a log function as the Hamiltonian.

Also included are exact results computed from the partition function for the quantum case.

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/figure_{a,b,c,d}.py Calculates and plots the quantum analytic results and approximate results for the classical limit and first correction using the high temperature approximation of the atomistic approximation method. Expectation value of Sz for s={1/2, 1, 3/2, 2} as a function of temperature.

python/figure2_{a,b,c,d}.py Calculates and plots the quantum analytic results and approximate results for the "exact" quantum correction using the logarithm form of the Hamiltonian of the atomistic approximation method. Expectation value of Sz for s={1/2, 1, 3/2, 2} as a function of temperature.

python/figure3_{a,b,c,d}.py Calculates and plots the quantum analytic results and approximate results for the "exact" quantum correction using the logarithm form of the Hamiltonian of the atomistic approximation method. Expectation value of Sz for s=1 for K={10, 0, -1, -10}g * mu_0 * mu_B * H_z as a function of temperature.

./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 Mac Studio (Apple M2 Ultra, 64GB RAM, Model Identifier: Mac14,14) running macOS version 14.4.1 (Sonoma). Python and associated packages were installed using conda. The installed package versions were:

• python=3.10.12
• matplotlib=3.6.2
• numba=0.59.0
• numpy=1.23.5
• scipy=1.9.3
• sympy=1.12.1rc1

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 brotli 1.1.0 hb547adb_0 conda-forge brotli-bin 1.1.0 hb547adb_0 conda-forge bzip2 1.0.8 h3422bc3_4 conda-forge ca-certificates 2023.7.22 hf0a4a13_0 conda-forge certifi 2023.7.22 pyhd8ed1ab_0 conda-forge contourpy 1.1.1 py310h38f39d4_0 conda-forge cycler 0.11.0 pyhd8ed1ab_0 conda-forge fonttools 4.42.1 py310h2aa6e3c_0 conda-forge freetype 2.12.1 hadb7bae_2 conda-forge gmp 6.2.1 h9f76cd9_0 conda-forge gmpy2 2.1.2 py310h2e6cad2_1 conda-forge kiwisolver 1.4.5 py310h38f39d4_1 conda-forge lcms2 2.15 h40e5a24_2 conda-forge lerc 4.0.0 h9a09cb3_0 conda-forge libblas 3.9.0 18_osxarm64_openblas conda-forge libbrotlicommon 1.1.0 hb547adb_0 conda-forge libbrotlidec 1.1.0 hb547adb_0 conda-forge libbrotlienc 1.1.0 hb547adb_0 conda-forge libcblas 3.9.0 18_osxarm64_openblas conda-forge libcxx 16.0.6 h4653b0c_0 conda-forge libdeflate 1.19 hb547adb_0 conda-forge libffi 3.4.2 h3422bc3_5 conda-forge libgfortran 5.0.0 13_2_0_hd922786_1 conda-forge libgfortran5 13.2.0 hf226fd6_1 conda-forge libjpeg-turbo 2.1.5.1 hb547adb_1 conda-forge liblapack 3.9.0 18_osxarm64_openblas conda-forge libllvm11 11.1.0 hfa12f05_5 conda-forge libopenblas 0.3.24 openmp_hd76b1f2_0 conda-forge libpng 1.6.39 h76d750c_0 conda-forge libsqlite 3.43.0 hb31c410_0 conda-forge libtiff 4.6.0 h77c4dce_1 conda-forge libwebp-base 1.3.2 hb547adb_0 conda-forge libxcb 1.15 hf346824_0 conda-forge libzlib 1.2.13 h53f4e23_5 conda-forge llvm-openmp 16.0.6 h1c12783_0 conda-forge llvmlite 0.42.0 pypi_0 pypi matplotlib 3.6.2 py310hb6292c7_0 conda-forge matplotlib-base 3.6.2 py310h78c5c2f_0 conda-forge mpc 1.3.1 h91ba8db_0 conda-forge mpfr 4.2.0 he09a6ba_0 conda-forge mpmath 1.3.0 pyhd8ed1ab_0 conda-forge munkres 1.1.4 pyh9f0ad1d_0 conda-forge ncurses 6.4 h7ea286d_0 conda-forge numba 0.59.0 pypi_0 pypi numpy 1.23.5 py310h5d7c261_0 conda-forge openjpeg 2.5.0 h4c1507b_3 conda-forge openssl 3.1.3 h53f4e23_0 conda-forge packaging 23.1 pyhd8ed1ab_0 conda-forge pillow 10.0.1 py310hadb9e77_1 conda-forge pip 23.2.1 pyhd8ed1ab_0 conda-forge pthread-stubs 0.4 h27ca646_1001 conda-forge pyparsing 3.1.1 pyhd8ed1ab_0 conda-forge python 3.10.12 h01493a6_0_cpython conda-forge python-dateutil 2.8.2 pyhd8ed1ab_0 conda-forge python_abi 3.10 4_cp310 conda-forge readline 8.2 h92ec313_1 conda-forge scipy 1.9.3 py310ha0d8a01_2 conda-forge setuptools 68.2.2 pyhd8ed1ab_0 conda-forge six 1.16.0 pyh6c4a22f_0 conda-forge sympy 1.12.1rc1 pypi_0 pypi tk 8.6.12 he1e0b03_0 conda-forge tornado 6.3.3 py310h2aa6e3c_1 conda-forge tzdata 2023c h71feb2d_0 conda-forge unicodedata2 15.0.0 py310h8e9501a_0 conda-forge wheel 0.41.2 pyhd8ed1ab_0 conda-forge xorg-libxau 1.0.11 hb547adb_0 conda-forge xorg-libxdmcp 1.1.3 h27ca646_0 conda-forge xz 5.2.6 h57fd34a_0 conda-forge zstd 1.5.5 h4f39d0f_0 conda-forge ```

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/figure_a.py: 1529.213 (s)
• python/figure_b.py: 1401.299 (s)
• python/figure_c.py: 1511.157 (s)
• python/figure_d.py: 1520.961 (s)
• python/figure2_a.py: 702.361 (s)
• python/figure2_b.py: 783.914 (s)
• python/figure2_c.py: 750.803 (s)
• python/figure2_d.py: 793.125 (s)
• python/figure3_a.py: 702.815 (s)
• python/figure3_b.py: 655.724 (s)
• python/figure3_c.py: 692.955 (s)
• python/figure3_d.py: 780.047 (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 {classical-limit, quantum-approximation, quantum-exact} --spin SPIN --field FIELD --stress STRESS --anisotropy ANISOTROPY

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 {classical-limit,quantum-approximation, quantum-exact}
                            Approximation scheme to use
  --order ORDER             Order of approximation. These are the quantum correction terms to be taken into account for the approximation (3 is the first quantum correction and this is used neither for the classical limit where the order is 2 nor for the "exact" quantum field method)
  --spin SPIN               Quantum spin value (should normally be an integer multiple of 1/2)
  --field FIELD             Z-component of magnetic field (in Tesla)
  --stress STRESS           Applied stress energy (ƛσ in Joules)
  --anisotropy ANISOTROPY   Uniaxial anisotropy energy (K in Joules)

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 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.

This work is an extension of the method for a single spin in a constant magnetic field from: Thomas Nussle, Stam Nicolis and Joseph Barker, "Numerical simulations of a spin dynamics model based on a path integral approach", Phys. Rev. Research 5, 043075 (2023).

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.