Sources for: Path integral spin dynamics with exchange and external field

This repository contains source code which implements a path integral approach to calculate the thermodynamics of two quantum spins coupled by exchange interaction in a magnetic field. It contains a numerical approach based on atomistic spin dynamics, and in the special case where the magnetic field is along the z-direction, exact diagonalisation results are provided for the quantum system.

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

Iason Sofos, School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom, http://orcid.org/0009-0006-3666-0416

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 two spins coupled by exchange interaction in a magnetic field. It computes both the classical limit and quantum corrections of the thermal expectation value of the z-component of the total spin.

The approximation scheme is a high temperature approach, either explicitly by doing a Taylor series in the inverse temperature for the computed effective Hamiltonian or hidden in always required exponential series of the quantum partition function. The effective Hamiltonian is computed either directly from this series or as a difference to the classical limit of the system, which is slightly more accurate in the case of ferromagnetically coupled spins.

In the special case where the magnetic field is along the z-direction, exact diagonalisation results are used to compute thermal expectation values for the whole temperature range.

Exact results from exact diagonalisation serve as a reference where they can be computed.

File descriptions

./

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

LICENSE MIT License file.

README.md This readme file.

./python/

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

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

python/effective_field_computation.py Defines python functions for symbolical calculations leading to the effective field (numerical) used in the atomistic spin dynamics calculations.

python/exact_diagonalisation.py Defines python functions for obtaining the diagonalised Hamiltonian with eigenvalues and eigenvectors used in the atomistic spin dynamics code, when appropriate, or as exact results for comparison.

python/figure1.py Calculates and plots the quantum analytic result for the thermal expectation value of two spins s=1/2 with ferromagnetic exchange J=1 in units of g$\mu_B$, using results from exact diagonalisation. Expectation value of Sz for s=1/2 as a function of temperature.

python/figure3.py Calculates and plots the quantum analytic results and approximate results for the classical limit, first and second quantum corrections using the logarithmic expression of the field, after taking the difference from the expected classical limit for said field using the atomistic approximation method. Expectation value of Sz for s=2 as a function of temperature.

python/figure4{a,b}.py Calculates and plots the quantum analytic results and exact quantum field of the atomistic approximation method using results from exact diagonalisation. Expectation value of Sz for s={1/2,2} as a function of temperature.

python/figure5{a,b}.py Calculates and plots the quantum analytic results and exact quantum field of the atomistic approximation method using results from exact diagonalisation, this time for antiferromagnetically coupled spins. Expectation value of Sz for s={1/2,2} as a function of temperature.

python/figure6.py Calculates and plots the quantum analytic results and approximate results for the classical limit, first and second quantum corrections using the high temperature approximation of the atomistic approximation method. Expectation value of Sz for s=2 as a function of temperature.

python/generating_moments.py Defines python function to compute the moments required to compute the effective field up to a given order from using sympy and non-commutative algebra. Moments up to order 5 coded hard for computational efficiency, but given enough time and computational resources, all moments up to this order and a few higher orders can be computed.

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

./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/figureX_data 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 15.2 (Sequoia). Python and associated packages were installed using conda. The installed package versions were:

• python=3.10.16
• 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 brotli 1.1.0 hd74edd7_2 conda-forge brotli-bin 1.1.0 hd74edd7_2 conda-forge bzip2 1.0.8 h99b78c6_7 conda-forge ca-certificates 2024.12.14 hf0a4a13_0 conda-forge certifi 2024.12.14 pyhd8ed1ab_0 conda-forge contourpy 1.3.1 py310h7f4e7e6_0 conda-forge cycler 0.12.1 pyhd8ed1ab_1 conda-forge fonttools 4.55.3 py310hc74094e_1 conda-forge freetype 2.12.1 hadb7bae_2 conda-forge gmp 6.3.0 h7bae524_2 conda-forge gmpy2 2.1.5 py310h805dbd7_3 conda-forge kiwisolver 1.4.7 py310h7306fd8_0 conda-forge lcms2 2.16 ha0e7c42_0 conda-forge lerc 4.0.0 h9a09cb3_0 conda-forge libblas 3.9.0 20_osxarm64_openblas conda-forge libbrotlicommon 1.1.0 hd74edd7_2 conda-forge libbrotlidec 1.1.0 hd74edd7_2 conda-forge libbrotlienc 1.1.0 hd74edd7_2 conda-forge libcblas 3.9.0 20_osxarm64_openblas conda-forge libcxx 19.1.7 ha82da77_0 conda-forge libdeflate 1.23 hec38601_0 conda-forge libffi 3.4.2 h3422bc3_5 conda-forge libgfortran 5.0.0 13_2_0_hd922786_3 conda-forge libgfortran5 13.2.0 hf226fd6_3 conda-forge libjpeg-turbo 3.0.0 hb547adb_1 conda-forge liblapack 3.9.0 20_osxarm64_openblas conda-forge libllvm11 11.1.0 hfa12f05_5 conda-forge liblzma 5.6.3 h39f12f2_1 conda-forge libopenblas 0.3.25 openmp_h6c19121_0 conda-forge libpng 1.6.45 h3783ad8_0 conda-forge libsqlite 3.48.0 h3f77e49_0 conda-forge libtiff 4.7.0 h551f018_3 conda-forge libwebp-base 1.5.0 h2471fea_0 conda-forge libxcb 1.17.0 hdb1d25a_0 conda-forge libzlib 1.3.1 h8359307_2 conda-forge llvm-openmp 19.1.7 hdb05f8b_0 conda-forge llvmlite 0.39.1 py310h1e34944_1 conda-forge matplotlib 3.6.2 py310hb6292c7_0 conda-forge matplotlib-base 3.6.2 py310h78c5c2f_0 conda-forge mpc 1.3.1 h8f1351a_1 conda-forge mpfr 4.2.1 hb693164_3 conda-forge mpmath 1.3.0 pyhd8ed1ab_1 conda-forge munkres 1.1.4 pyh9f0ad1d_0 conda-forge ncurses 6.5 h5e97a16_2 conda-forge numba 0.56.4 py310h3124f1e_1 conda-forge numpy 1.23.5 py310h5d7c261_0 conda-forge openjpeg 2.5.3 h8a3d83b_0 conda-forge openssl 3.4.0 h81ee809_1 conda-forge packaging 24.2 pyhd8ed1ab_2 conda-forge pillow 11.1.0 py310h61efb56_0 conda-forge pip 24.3.1 pyh8b19718_2 conda-forge pthread-stubs 0.4 hd74edd7_1002 conda-forge pyparsing 3.2.1 pyhd8ed1ab_0 conda-forge python 3.10.16 h870587a_1_cpython conda-forge python-dateutil 2.9.0.post0 pyhff2d567_1 conda-forge python_abi 3.10 5_cp310 conda-forge readline 8.2 h92ec313_1 conda-forge scipy 1.9.3 py310ha0d8a01_2 conda-forge setuptools 75.8.0 pyhff2d567_0 conda-forge six 1.17.0 pyhd8ed1ab_0 conda-forge sympy 1.11.1 pypyh9d50eac_103 conda-forge tk 8.6.13 h5083fa2_1 conda-forge tornado 6.4.2 py310h078409c_0 conda-forge tzdata 2025a h78e105d_0 conda-forge unicodedata2 16.0.0 py310h078409c_0 conda-forge wheel 0.45.1 pyhd8ed1ab_1 conda-forge xorg-libxau 1.0.12 h5505292_0 conda-forge xorg-libxdmcp 1.1.5 hd74edd7_0 conda-forge zstd 1.5.6 hb46c0d2_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/figure1.py: 0.453 (s)
• python/figure3.py: 11219.177 (s)
• python/figure4a.py: 6848.936 (s)
• python/figure4b.py: 76669.533 (s)
• python/figure5a.py: 66635.302 (s)
• python/figure5b.py: 84818.633 (s)
• python/figure6.py: 4259.530 (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 {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 {classical-limit, quantum-approximation-sympy, quantum-exact, exact-bz-two-spins}
                        Approximation scheme to use
  --order ORDER         Order of the approximation scheme (up to order 4 for, "quantum-exact" and "quantum-approximation-sympy" methods)
  
  --spin SPIN           Quantum spin value (should normally be an integer multiple of 1/2)
  
  --from_difference True/False  Computing effective Hamiltonian from difference to its classical limit
  
  --exchange EXCHANGE   Value for the exchange parameter J (in units of g mu_B)

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. Only use field along z-direction in case of "exact-bz-two-spins"
• the range and increments in temperature for atomistic simulations by changing np.linspace(<starting-temperature>, <final-temperature>, <number-of-temperatures-in-range>)
• the initial orientation for both spins orientation s0 (must be of unit norm and different from np.array([[0, 0, 1], [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.

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).

This work is an extension of the method for a single spin in a constant magnetic field with a quadratic term for a uniaxial anistropy from: Thomas Nussle, Pascal Thibaudeau and Stam Nicolis, "Path Integral Spin Dynamics for Quantum Paramagnets", Adv. Phys. Research, 202400057 (2024).

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.