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 Monte Carlo estimator for the dipole polarizability of quantum plasma

Juha Tiihonen*, David Trejo-Garcia, Tapio T. Rantala, and Marco Ornigotti

  • *Contact author: juha.tiihonen@tuni.fi

Phys. Rev. A 113, 053711 – Published 11 May, 2026

DOI: https://doi.org/10.1103/s5xt-s2fb

Abstract

We present a path-integral Monte Carlo estimator for calculating the dipole polarizability of interacting Coulomb plasma in the long-wavelength limit, i.e., the optical region. We present comprehensive details and method validation studies for our approach based on both collective and one-particle dipole autocorrelation functions in imaginary time. The simulation of thermal equilibrium in imaginary time has exact Coulomb interactions and Boltzmann quantum statistics. For reference, we use an analytically continued Drude model as the long-wavelength limit of the Lindhard response. Our collective response shows a perfect match to the analytical reference. The one-particle response is used in systematic studies of physical and numerical parameters and to discuss the phenomenological Drude scattering model.

View figure in article

Physics Subject Headings (PhySH)

Corrections

10 September, 2026

Correction: An error in Eq. (48) has been fixed.

Article Text

Supplemental Material

References (48)

  1. P. Drude, Zur elektronentheorie der metalle, Ann. Phys. (Leipzig) 306, 566 (1900).
  2. P. Drude, Zur elektronentheorie der metalle; II. Teil. Galvanometrische und thermomagnetische effekte, Ann. Phys. (Leipzig) 308, 369 (1900).
  3. I. Liberal and N. Engheta, Near-zero refractive index photonics, Nat. Photon. 11, 149 (2017).
  4. P. B. Johnson and R. W. Christy, Optical constants of the noble metals, Phys. Rev. B 6, 4370 (1972).
  5. J. R. Maack, N. A. Mortensen, and M. Wubs, Two-fluid hydrodynamic model for semiconductors, Phys. Rev. B 97, 115415 (2018).
  6. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinhart and Winston, New York, 1976).
  7. K. S. Singwi, M. P. Tosi, R. H. Land, and A. Sjölander, Electron correlations at metallic densities, Phys. Rev. 176, 589 (1968).
  8. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, Berlin, 2007).
  9. J. Wu, Z. T. Xie, Y. Sha, H. Y. Fu, and Q. Li, Epsilon-near-zero photonics: Infinite potentials, Photonics Res. 9, 1616 (2021).
  10. S. A. Schulz, R. F. Oulton, M. Kenney, A. Alu, I. Staude, A. Bashiri, Z. Fedorova, R. Kolkowski, A. F. Koenderink, X. Xiao, J. Yang, W. J. Peveler, A. W. Clark, G. Perrakis, A. C. Tasolamprou, M. Kafesaki, A. Zaleska, W. Dickson, D. Richards, A. Zayats, et al., Roadmap on photonic metasurfaces, Appl. Phys. Lett. 124, 260701 (2024).
  11. G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2008).
  12. A. Kugler, Theory of the local field correction in an electron gas, J. Stat. Phys. 12, 35 (1975).
  13. S. Moroni, D. M. Ceperley, and G. Senatore, Static response and local field factor of the electron gas, Phys. Rev. Lett. 75, 689 (1995).
  14. S. Groth, T. Dornheim, and J. Vorberger, Ab initio path integral Monte Carlo approach to the static and dynamic density response of the uniform electron gas, Phys. Rev. B 99, 235122 (2019).
  15. P. Hamann, T. Dornheim, J. Vorberger, Z. A. Moldabekov, and M. Bonitz, Dynamic properties of the warm dense electron gas based on ab initio path integral Monte Carlo simulations, Phys. Rev. B 102, 125150 (2020).
  16. T. Dornheim, Z. A. Moldabekov, K. Ramakrishna, P. Tolias, A. D. Baczewski, D. Kraus, T. R. Preston, D. A. Chapman, M. P. Böhme, T. Döppner, F. Graziani, M. Bonitz, A. Cangi, and J. Vorberger, Electronic density response of warm dense matter, Phys. Plasmas 30, 032705 (2023).
  17. T. Dornheim, S. Groth, J. Vorberger, and M. Bonitz, Ab initio path integral Monte Carlo results for the dynamic structure factor of correlated electrons: From the electron liquid to warm dense matter, Phys. Rev. Lett. 121, 255001 (2018).
  18. T. Dornheim, J. Vorberger, and M. Bonitz, Nonlinear electronic density response in warm dense matter, Phys. Rev. Lett. 125, 085001 (2020).
  19. T. Dornheim, Z. A. Moldabekov, and J. Vorberger, Nonlinear density response from imaginary-time correlation functions: Ab initio path integral Monte Carlo simulations of the warm dense electron gas, J. Chem. Phys. 155, 054110 (2021).
  20. T. Chuna, N. Barnfield, J. Vorberger, M. P. Friedlander, T. Hoheisel, and T. Dornheim, Estimates of the dynamic structure factor for the finite temperature electron liquid via analytic continuation of path integral Monte Carlo data, Phys. Rev. B 112, 125112 (2025).
  21. T. Dornheim, Z. A. Moldabekov, J. Vorberger, and B. Militzer, Path integral Monte Carlo approach to the structural properties and collective excitations of liquid He3 without fixed nodes, Sci. Rep. 12, 708 (2022).
  22. E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, Sign problem in the numerical simulation of many-electron systems, Phys. Rev. B 41, 9301 (1990).
  23. D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67, 279 (1995).
  24. M. Jarrell and J. Gubernatis, Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data, Phys. Rep. 269, 133 (1996).
  25. D. Bergeron and A.-M. S. Tremblay, Algorithms for optimized maximum entropy and diagnostic tools for analytic continuation, Phys. Rev. E 94, 023303 (2016).
  26. E. Gallicchio and B. J. Berne, The absorption spectrum of the solvated electron in fluid helium by maximum entropy inversion of imaginary time correlation functions from path integral Monte Carlo simulations, J. Chem. Phys. 101, 9909 (1994).
  27. E. Gallicchio and B. J. Berne, On the calculation of dynamical properties of solvated electrons by maximum entropy analytic continuation of path integral Monte Carlo data, J. Chem. Phys. 105, 7064 (1996).
  28. J. Tiihonen, I. Kylänpää, and T. T. Rantala, Computation of dynamic polarizabilities and van der Waals coefficients from path-integral Monte Carlo, J. Chem. Theory Comput. 14, 5750 (2018).
  29. J. Tiihonen, Thermal effects in atomic and molecular polarizabilities with path integral Monte Carlo, Ph.D. thesis, Tampere University, 2019.
  30. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  31. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  32. D. M. Ceperley and E. L. Pollock, Path-integral computation of the low-temperature properties of liquid He4, Phys. Rev. Lett. 56, 351 (1986).
  33. T. Dornheim, H. Thomsen, P. Ludwig, A. Filinov, and M. Bonitz, Analyzing quantum correlations made simple, Contrib. Plasma Phys. 56, 371 (2016).
  34. See Supplemental Material at http://link.aps.org/supplemental/10.1103/s5xt-s2fb for further benchmark figures of accuracy, which includes Refs. [11, 36, 37, 38].
  35. I. Kylänpää, First-principles finite temperature electronic structure of some small molecules, Ph.D. thesis, Tampere University of Technology, 2011.
  36. E. W. Brown, B. K. Clark, J. L. DuBois, and D. M. Ceperley, Path-integral Monte Carlo simulation of the warm dense homogeneous electron gas, Phys. Rev. Lett. 110, 146405 (2013).
  37. L. M. Fraser, W. M. C. Foulkes, G. Rajagopal, R. J. Needs, S. D. Kenny, and A. J. Williamson, Finite-size effects and Coulomb interactions in quantum Monte Carlo calculations for homogeneous systems with periodic boundary conditions, Phys. Rev. B 53, 1814 (1996).
  38. V. Natoli and D. M. Ceperley, An optimized method for treating long-range potentials, J. Comput. Phys. 117, 171 (1995).
  39. J. T. Krogel, Nexus: A modular workflow management system for quantum simulation codes, Comput. Phys. Commun. 198, 154 (2016).
  40. J. Tiihonen and D. Trejo-Garcia, Path-integral Monte Carlo estimator for the dipole polarizability of quantum plasma data repository (Version 1), Tampere University, https://doi.org/10.23729/fd-c66adb25-f567-3748-bf91-4e1f334a4278 (Ministry of Education and Culture, Helsinki, 2026).
  41. S. Chiesa, D. M. Ceperley, R. M. Martin, and M. Holzmann, Finite-size error in many-body simulations with long-range interactions, Phys. Rev. Lett. 97, 076404 (2006).
  42. N. D. Drummond, R. J. Needs, A. Sorouri, and W. M. C. Foulkes, Finite-size errors in continuum quantum Monte Carlo calculations, Phys. Rev. B 78, 125106 (2008).
  43. K. Ramakrishna, T. Dornheim, and J. Vorberger, Influence of finite temperature exchange-correlation effects in hydrogen, Phys. Rev. B 101, 195129 (2020).
  44. R. M. Martin, L. Reining, and D. M. Ceperley, Interacting Electrons: Theory and Computational Approaches (Cambridge University Press, Cambridge, 2016).
  45. D. A. R. Dalvit, in Casimir Physics, edited by P. W. Milonni, D. Roberts, and F. da Rosa, Lecture Notes in Physics, Vol. 834 (Springer, Berlin, 2011).
  46. J. Tao and A. M. Rappe, Physical adsorption: Theory of van der Waals interactions between particles and clean surfaces, Phys. Rev. Lett. 112, 106101 (2014).
  47. O. Reshef, I. De Leon, M. Z. Alam, and R. W. Boyd, Nonlinear optical effects in epsilon-near-zero media, Nat. Rev. Mater. 4, 535 (2019).
  48. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 2009).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation