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Breaking the curse of dimensionality: Solving configurational integrals for crystalline solids by tensor networks

Duc P. Truong1, Benjamin Nebgen1, Derek DeSantis1, Dimiter N. Petsev2, Kim Ø. Rasmussen1, and Boian S. Alexandrov1

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Phys. Rev. Materials 9, 083802 – Published 27 August, 2025

DOI: https://doi.org/10.1103/xrbw-xr49

Abstract

Accurately evaluating configurational integrals for dense solids remains a central and difficult challenge in the statistical mechanics of condensed systems. Here, we present a tensor network approach that reformulates the high-dimensional configurational integral for identical-particle crystals into a sequence of computationally efficient summations. We represent the integrand as a high-dimensional tensor and apply tensor-train (TT) decomposition together with a custom TT-cross interpolation. This approach circumvents the need to explicitly construct the full tensor. We introduce tailored rank-1 and rank-2 schemes optimized for sharply peaked Boltzmann probability densities, typical for identical-particle crystals. When applied to the calculation of internal energy and pressure-temperature curves for crystalline Cu and Ar at high (GPa) pressures, as well as the alpha-to-beta phase transition diagram of Sn, our method accurately reproduces molecular dynamics simulation results using tight-binding, machine learning, hierarchical interacting particle–neural network, and modified embedded atom method potentials,all within seconds of computation time.

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References (55)

  1. J. C. Maxwell, Philos. Trans. R. Soc. London 157, 49 (1867).
  2. L. Boltzmann, Lectures on Gas Theory (Dover, New York, 2011).
  3. J. W. Gibbs, Elementary Principles in Statistical Mechanics (Cambridge University Press, New York, 2010).
  4. R. Bellman, Science 153, 34 (1966).
  5. B. Alexandrov, G. Manzini, E. W. Skau, P. M. D. Truong, and R. G. Vuchkov, Mathematics 11, 534 (2023).
  6. J. G. Kirkwood, J. Chem. Phys. 3, 300 (1935).
  7. C. H. Bennett, J. Comput. Phys. 22, 245 (1976).
  8. D. Frenkel and A. J. Ladd, J. Chem. Phys. 81, 3188 (1984).
  9. Y. Sugita and Y. Okamoto, Chem. Phys. Lett. 314, 141 (1999).
  10. A. Laio and M. Parrinello, Proc. Natl. Acad. Sci. USA 99, 12562 (2002).
  11. G. M. Torrie and J. P. Valleau, J. Comput. Phys. 23, 187 (1977).
  12. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).
  13. W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).
  14. J. Behler and M. Parrinello, Phys. Rev. Lett. 98, 146401 (2007).
  15. C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
  16. D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Elsevier, Amsterdam, 2002).
  17. J. Ihm, Rep. Prog. Phys. 51, 105 (1988).
  18. H. J. Berendsen, in Computational Molecular Dynamics: Challenges, Methods, Ideas: Proceedings of the 2nd International Symposium on Algorithms for Macromolecular Modelling, Berlin, May 21–24, 1997 (Springer, New York, 1999), pp. 3–36.
  19. M. K. Cowles and B. P. Carlin, J. Am. Stat. Assoc. 91, 883 (1996).
  20. A. Cichocki, N. Lee, I. Oseledets, A.-H. Phan, Q. Zhao, D. P. Mandic et al., Found. Trends Mach. Learn. 9, 249 (2016).
  21. I. Oseledets and E. Tyrtyshnikov, Linear Algebra Appl. 432, 70 (2010).
  22. V. Khoromskaia, B. N. Khoromskij, and R. Schneider, SIAM J. Sci. Comput. 35, A987 (2013).
  23. S. Dolgov and D. Savostyanov, Comput. Phys. Commun. 246, 106869 (2020).
  24. G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015).
  25. M. S. J. Tepaske and D. J. Luitz, Phys. Rev. Res. 3, 023236 (2021).
  26. J. Colbois, Tensor network investigation of frustrated ising models, Master's thesis, École Polytechnique Fédérale de Lausanne, 2022.
  27. F. F. Song, T. Y. Lin, and G. M. Zhang, Phys. Rev. B 108, 224404 (2023).
  28. B. Lanthier, J. Côté, and S. Kourtis, Front. Phys. 12, 1431810 (2024).
  29. A. A. Klishin and G. van Anders, Soft Matter 16, 6523 (2020).
  30. M. C. Bañuls, Annu. Rev. Condens. Matter Phys. 14, 173 (2023).
  31. L. Ying, Multiscale Model. Simul. 15, 1423 (2017).
  32. C.-Y. Huang, S.-H. Chan, Y.-J. Kao, and P. Chen, Phys. Rev. B 107, 205123 (2023).
  33. B. Vanhecke, J. Colbois, L. Vanderstraeten, F. Verstraete, and F. Mila, Phys. Rev. Res. 3, 013041 (2021).
  34. D. P. Truong, M. I. Ortega, I. Boureima, G. Manzini, K. Ø. Rasmussen, and B. S. Alexandrov, J. Comput. Phys. 507, 112943 (2024).
  35. M. E. Danis, D. Truong, I. Boureima, O. Korobkin, K. Ø. Rasmussen, and B. S. Alexandrov, J. Comput. Phys. 529, 113891 (2025).
  36. I. V. Oseledets, SIAM J. Sci. Comput. 33, 2295 (2011).
  37. S. A. Goreinov, E. E. Tyrtyshnikov, and N. L. Zamarashkin, Linear Algebra Appl. 261, 1 (1997).
  38. M. W. Mahoney and P. Drineas, Proc. Natl. Acad. Sci. USA 106, 697 (2009).
  39. D. V. Savostyanov, Linear Algebra Appl. 458, 217 (2014).
  40. S. A. Goreinov and E. E. Tyrtyshnikov, Contemp. Math. 280, 47 (2001).
  41. https://github.com/oseledets/TT-Toolbox.
  42. I. Oseledets, Comput. Methods Appl. Math. 11, 382 (2011).
  43. S. V. Dolgov and D. V. Savostyanov, SIAM J. Sci. Comput. 36, A2248 (2014).
  44. T. Cui, S. Dolgov, and O. Zahm, J. Comput. Phys. 485, 112103 (2023).
  45. F. Cleri and V. Rosato, Phys. Rev. B 48, 22 (1993).
  46. N. Lubbers, J. S. Smith, and K. Barros, J. Chem. Phys. 148, 241715 (2018).
  47. A. H. Larsen, J. J. Mortensen, J. Blomqvist, I. E. Castelli, R. Christensen, M. Dułak, J. Friis, M. N. Groves, B. Hammer, C. Hargus et al., J. Phys.: Condens. Matter 29, 273002 (2017).
  48. A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in 't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, Comput. Phys. Commun. 271, 108171 (2022).
  49. B.-J. Lee, M. I. Baskes, H. Kim, and Y. K. Cho, Phys. Rev. B 64, 184102 (2001).
  50. W.-S. Ko, D.-H. Kim, Y.-J. Kwon, and M. H. Lee, Metals 8, 900 (2018).
  51. S. Menon, Y. Lysogorskiy, J. Rogal, and R. Drautz, Phys. Rev. Mater. 5, 103801 (2021).
  52. V. Rosato, M. Guillope, and B. Legrand, Philos. Mag. A 59, 321 (1989).
  53. M. Chigaev, J. S. Smith, S. Anaya, B. Nebgen, M. Bettencourt, K. Barros, and N. Lubbers, J. Chem. Phys. 158, 184108 (2023).
  54. Y. Shao, Z. Gan, E. Epifanovsky, A. T. Gilbert, M. Wormit, J. Kussmann, A. W. Lange, A. Behn, J. Deng, X. Feng et al., Mol. Phys. 113, 184 (2015).
  55. D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 (1993).

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Phys. Rev. Materials 9, 083802– Published 27 August, 2025
  • Received 28 May 2025
  • Accepted 13 August 2025
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