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Fully ergodic simulations using radial updates

Finn L. Temmen1,*, Evan Berkowitz1,2,3,4,5,†, Anthony Kennedy6,‡, Thomas Luu1,7,§, Johann Ostmeyer7,∥, and Xinhao Yu6,¶

  • *Contact author: f.temmen@fz-juelich.de
  • †Contact author: evan.berkowitz@uvi.edu
  • ‡Contact author: Tony.kennedy@ed.ac.uk
  • §Contact author: t.luu@fz-juelich.de
  • ∥Contact author: ostmeyer@hiskp.uni-bonn.de
  • Contact author: X.Yu-34@sms.ed.ac.uk
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Phys. Rev. B 112, 045134 – Published 21 July, 2025

DOI: https://doi.org/10.1103/9zst-bvmk

Abstract

A sensible application of the hybrid Monte Carlo (HMC) method is often hindered by the presence of large—or even infinite—potential barriers. These potential barriers separate the configuration space into distinct sectors and can lead to ergodicity violations that bias measurements. In this work, we address this problem by augmenting HMC with a multiplicative Metropolis-Hastings update in a “radial direction” of the fields which enables crossing the potential barriers and ensures ergodicity of the sampling algorithm at comparably low computational cost. We demonstrate the algorithm on a simple toy model and show how it can be applied to the fermionic Hubbard model describing physics ranging from an exactly solvable two-site system to the C20H12 perylene molecule. Our numerical results show that the radial updates successfully remove ergodicity violations, while simultaneously reducing autocorrelation times.

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References (31)

  1. B. Alles, G. Boyd, M. D'Elia, A. Di Giacomo, and E. Vicari, Hybrid Monte Carlo and topological modes of full QCD, Phys. Lett. B 389, 107 (1996).
  2. J.-L. Wynen, E. Berkowitz, C. Körber, T. A. Laehde, and T. Luu, Avoiding ergodicity problems in lattice discretizations of the Hubbard model, Phys. Rev. B 100, 075141 (2019).
  3. S. R. White, R. L. Sugar, and R. T. Scalettar, Algorithm for the simulation of many-electron systems at low temperatures, Phys. Rev. B 38, 11665 (1988).
  4. S. Beyl, F. Goth, and F. F. Assaad, Revisiting the hybrid quantum Monte Carlo method for Hubbard and electron-phonon models, Phys. Rev. B 97, 085144 (2018).
  5. P. Buividovich, D. Smith, M. Ulybyshev, and L. von Smekal, Hybrid Monte Carlo study of competing order in the extended fermionic Hubbard model on the hexagonal lattice, Phys. Rev. B 98, 235129 (2018).
  6. P. Buividovich, D. Smith, M. Ulybyshev, and L. von Smekal, Numerical evidence of conformal phase transition in graphene with long-range interactions, Phys. Rev. B 99, 205434 (2019).
  7. D. Schuh, J. Kreit, E. Berkowitz, L. Funcke, T. Luu, K. A. Nicoli, and M. Rodekamp, Simulating the Hubbard model with equivariant normalizing flows, PoS LATTICE2024, 069 (2025).
  8. A. D. Kennedy and X. Yu, On the geometric convergence of HMC on Riemannian manifolds, PoS LATTICE2024, 064 (2025).
  9. S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195, 216 (1987).
  10. S. Dutta, Multiplicative random walk Metropolis-Hastings on the real line, Sankhya B 74, 315 (2012).
  11. S. Dutta and S. Bhattacharya, Markov-chain Monte Carlo based on deterministic transformations, Stat. Methodol. 16, 100 (2014).
  12. J. Ostmeyer, Exponential speed up in Monte Carlo sampling through radial updates, J. Phys. A: Math. Theor. 58, 185201 (2025).
  13. M. Rodekamp et al., Nanosystem simulation library (NSL), GitHub repository (2024), https://github.com/Marcel-Rodekamp/NSL.
  14. J. Ostmeyer, comp-avg: Compare averages of time series and more, Zenodo, https://github.com/j-ostmeyer/comp-avg.
  15. F. Temmen, Radial updates, GitHub repository (2025), https://github.com/ftemmen/Radial-Updates.
  16. J. W. Negele and H. Orland, Quantum Many-particle Systems, Frontiers in Physics (CRC Press, Boca Raton, FL, 1988).
  17. R. Brower, C. Rebbi, and D. Schaich, Hybrid Monte Carlo simulation on the graphene hexagonal lattice, PoS LATTICE2011, 056 (2011).
  18. D. Smith and L. von Smekal, Monte-Carlo simulation of the tight-binding model of graphene with partially screened Coulomb interactions, Phys. Rev. B 89, 195429 (2014).
  19. T. Luu and T. A. Lähde, Quantum Monte Carlo calculations for carbon nanotubes, Phys. Rev. B 93, 155106 (2016).
  20. M. Rodekamp, Machine learning for path deformation and Bayesian data analysis in selected lattice field theories, Ph.D. thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 2024.
  21. J. Ostmeyer and P. Buividovich, Minimal autocorrelation in hybrid Monte Carlo simulations using exact Fourier acceleration, Comput. Phys. Commun. 313, 109624 (2025).
  22. P. Sinilkov and E. Berkowitz, Correlators exact diagonalization (beehive), GitHub repository (2025), https://github.com/petio9671/Correlators-Exact-Diagonalization, DOI: https://doi.org/10.5281/zenodo.15045126.
  23. U. Wolff, Monte Carlo errors with less errors, Comput. Phys. Commun. 156, 143 (2004).
  24. A. Sokal, Monte Carlo methods in statistical mechanics: Foundations and new algorithms, in Functional Integration: Basics and Applications, edited by C. DeWitt-Morette, P. Cartier, and A. Folacci (Springer US, Boston, MA, 1997), pp. 131–192.
  25. C. W. Tang, Two–layer organic photovoltaic cell, Appl. Phys. Lett. 48, 183 (1986).
  26. W. Ni, L. Sun, and G. G. Gurzadyan, Ultrafast spectroscopy reveals singlet fission, ionization and excimer formation in perylene film, Sci. Rep. 11, 5220 (2021).
  27. J. Cao and S. Yang, Progress in perylene diimides for organic solar cell applications, RSC Adv. 12, 6966 (2022).
  28. Q. Guo, Y. Xu, B. Xiao, B. Zhang, E. Zhou, F. Wang, Y. Bai, T. Hayat, A. Alsaedi, and Z. Tan, Effect of energy alignment, electron mobility, and film morphology of perylene diimide based polymers as electron transport layer on the performance of perovskite solar cells, ACS Appl. Mater. Interfaces 9, 10983 (2017).
  29. A. Dodabalapur, H. E. Katz, and L. Torsi, Molecular orbital energy level engineering in organic transistors, Adv. Mater. 8, 853 (1996).
  30. M. Rodekamp, E. Berkowitz, C. Gäntgen, S. Krieg, T. Luu, J. Ostmeyer, and G. Pederiva, Single-particle spectrum of doped C20H12-perylene, Eur. Phys. J. B 98, 36 (2025).
  31. P. Thörnig, JURECA: Data centric and booster modules implementing the modular supercomputing architecture at Jülich supercomputing centre, J. Large-scale Res. Facil. 7, (2021).

Outline

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Phys. Rev. B 112, 045134– Published 21 July, 2025
  • Received 15 April 2025
  • Revised 23 June 2025
  • Accepted 24 June 2025
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