- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Path-integral spin dynamics with exchange and external field

Thomas Nussle1,*, Stam Nicolis2,†, Iason Sofos1, and Joseph Barker1,‡

  • *Contact author: t.s.nussle@leeds.ac.uk
  • †Contact author: stamatios.nicolis@univ-tours.fr
  • ‡Contact author: j.barker@leeds.ac.uk
PDF

Phys. Rev. B 112, 054404 – Published 4 August, 2025

DOI: https://doi.org/10.1103/m6vx-cl3v

Abstract

In this work, we propose a path-integral-inspired formalism for computing the quantum thermal expectation values of spin systems subject to magnetic fields that can be time dependent and can accommodate the presence of Heisenberg exchange interactions between the spins. This is done by deriving an effective magnetic field from the quantum partition function of the system to use in classical atomistic spin dynamics simulations and generalizes the formalism presented in our previous work [Phys. Rev. Res. 5, 043075 (2023)]. In special cases where the effective field can be computed exactly, we compare our results with exact/numerical diagonalization methods for both ferromagnetic and antiferromagnetic coupling. We show that our method works well across a large temperature range and can reproduce quantum expectation values for antiferromagnetic coupling, which is usually not possible with classical models.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (30)

  1. R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20, 367 (1948).
  2. D. Marx, P. Nielaba, and K. Binder, Path-integral Monte Carlo study of a model adsorbate with internal quantum states, Phys. Rev. B 47, 7788 (1993).
  3. D. Marx and M. Parrinello, Ab initio path integral molecular dynamics: Basic ideas, J. Chem. Phys. 104, 4077 (1996).
  4. M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, Efficient stochastic thermostatting of path integral molecular dynamics, J. Chem. Phys. 133, 124104 (2010).
  5. L. Schulman, A path integral for spin, Phys. Rev. 176, 1558 (1968).
  6. J. R. Klauder, Path integrals and stationary-phase approximations, Phys. Rev. D 19, 2349 (1979).
  7. D. C. Cabra, A. Dobry, A. Greco, and G. L. Rossini, On the path integral representation for spin systems, J. Phys. A: Math. Gen. 30, 2699 (1997).
  8. T. Nussle, S. Nicolis, and J. Barker, Numerical simulations of a spin dynamics model based on a path integral approach, Phys. Rev. Res. 5, 043075 (2023).
  9. M. Lax, Quantum noise. XI. Multitime correspondence between quantum and classical stochastic processes, Phys. Rev. 172, 350 (1968).
  10. R. Gilmore, C. M. Bowden, and L. M. Narducci, Classical-quantum correspondence for multilevel systems, Phys. Rev. A 12, 1019 (1975).
  11. M. Gell-Mann and J. B. Hartle, Classical equations for quantum systems, Phys. Rev. D 47, 3345 (1993).
  12. E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev. 150, 1079 (1966).
  13. M. Elyasi, Y. M. Blanter, and G. E. W. Bauer, Resources of nonlinear cavity magnonics for quantum information, Phys. Rev. B 101, 054402 (2020).
  14. T. Nussle, P. Thibaudeau, and S. Nicolis, Path integral spin dynamics for quantum paramagnets, Adv. Phys. Res. 2400057 (2024).
  15. T. Nussle, S. Nicolis, J. Barker, and I. Sofos, Sources for: Path integral spin dynamics with exchange and external field, Zenodo (2025), https://doi.org/10.5281/zenodo.14936582.
  16. B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson, A method for atomistic spin dynamics simulations: Implementation and examples, J. Phys.: Condens. Matter 20, 315203 (2008).
  17. P. Thibaudeau and D. Beaujouan, Thermostatting the atomic spin dynamics from controlled demons, Physica A 391, 1963 (2012).
  18. A. Alexandru, G. Başar, P. F. Bedaque, and N. C. Warrington, Complex paths around the sign problem, Rev. Mod. Phys. 94, 015006 (2022).
  19. A. Hallam, Tensor network descriptions of quantum entanglement in path integrals, thermalisation and machine learning, Ph.D. thesis, University College London, 2019.
  20. J. Anders, C. R. J. Sait, and S. A. R. Horsley, Quantum Brownian motion for magnets, New J. Phys. 24, 033020 (2022).
  21. F. Cerisola, M. Berritta, S. Scali, S. A. R. Horsley, J. D. Cresser, and J. Anders, Quantum–classical correspondence in spin–boson equilibrium states at arbitrary coupling, New J. Phys. 26, 053032 (2024).
  22. M. Berritta, S. Scali, F. Cerisola, and J. Anders, Accounting for quantum effects in atomistic spin dynamics, Phys. Rev. B 109, 174441 (2024).
  23. R. Dirl, Clebsch–Gordan coefficients: General theory, J. Math. Phys. 20, 659 (1979).
  24. C. Wei and S. H. Curnoe, Exact diagonalization for a 16-site spin-1/2 pyrochlore cluster, J. Phys.: Condens. Matter 35, 295802 (2023).
  25. H. Q. Lin, Exact diagonalization of quantum-spin models, Phys. Rev. B 42, 6561 (1990).
  26. J. Schnack, Exact diagonalization techniques for quantum spin systems, in Computational Modelling of Molecular Nanomagnets, edited by G. Rajaraman (Springer International Publishing, Cham, 2023), pp. 155–177.
  27. F. Hiai, Trace norm convergence of exponential product formula, Lett. Math. Phys. 33, 147 (1995).
  28. M. Suzuki, Convergence of general decompositions of exponential operators, Commun. Math. Phys. 163, 491 (1994).
  29. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces: Revised and Enl. Transl. from the Romanian (Noordhoff International Publishing, Leyden, 1976).
  30. W. Rudin, Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991).

Outline

Information

Phys. Rev. B 112, 054404– Published 4 August, 2025
  • Received 27 February 2025
  • Revised 6 June 2025
  • Accepted 18 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation