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Generalized directed loop method for quantum Monte Carlo simulations

Fabien Alet1,2, Stefan Wessel1, and Matthias Troyer1,2

  • 1Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland
  • 2Computation Laboratory, ETH Zürich, CH-8092 Zürich, Switzerland

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Phys. Rev. E 71, 036706 – Published 22 March, 2005

DOI: https://doi.org/10.1103/PhysRevE.71.036706

Abstract

Efficient quantum Monte Carlo update schemes called directed loops have recently been proposed, which improve the efficiency of simulations of quantum lattice models. We propose to generalize the detailed balance equations at the local level during the loop construction by accounting for the matrix elements of the operators associated with open world-line segments. Using linear programming techniques to solve the generalized equations, we look for optimal construction schemes for directed loops. This also allows for an extension of the directed loop scheme to general lattice models, such as high-spin or bosonic models. The resulting algorithms are bounce free in larger regions of parameter space than the original directed loop algorithm. The generalized directed loop method is applied to the magnetization process of spin chains in order to compare its efficiency to that of previous directed loop schemes. In contrast to general expectations, we find that minimizing bounces alone does not always lead to more efficient algorithms in terms of autocorrelations of physical observables, because of the nonuniqueness of the bounce-free solutions. We therefore propose different general strategies to further minimize autocorrelations, which can be used as supplementary requirements in any directed loop scheme. We show by calculating autocorrelation times for different observables that such strategies indeed lead to improved efficiency; however, we find that the optimal strategy depends not only on the model parameters but also on the observable of interest.

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References (29)

  1. R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987); U. Wolff, ibid. 62, 361 (1989).
  2. H. G. Evertz, G. Lana, and M. Marcu, Phys. Rev. Lett. 70, 875 (1993).
  3. H. G. Evertz, Adv. Phys. 52, 1 (2003).
  4. B. B. Beard and U. J. Wiese, Phys. Rev. Lett. 77, 5130 (1996).
  5. V. A. Kashurnikov, N. V. Prokof’ev, B. V. Svistunov, and M. Troyer, Phys. Rev. B 59, 1162 (1999).
  6. N. V. Prokof’ev, B. V. Svistunov, and I. S. Tupitsyn, Phys. Lett. A 238, 253 (1998); JETP 87, 310 (1998).
  7. A. W. Sandvik and J. Kurkijärvi, Phys. Rev. B 43, 5950 (1991); A. W. Sandvik, J. Phys. A 25, 3667 (1992).
  8. D. C. Handscomb, Proc. Cambridge Philos. Soc. 58, 594 (1962); 60, 115 (1964).
  9. A. W. Sandvik, Phys. Rev. B 59, R14157 (1999).
  10. M. Troyer, F. Alet, S. Trebst, and S. Wessel, in The Monte Carlo Method in the Physical Sciences, edited by James E. Gubernatis, AIP Conf. Proc. No. 690 (AIP, Melville, NY, 2003), p. 156.
  11. A. W. Sandvik, R. R. P. Sing, and D. K. Campbell, Phys. Rev. B 56, 14510 (1997).
  12. O. F. Syljuåsen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002).
  13. O. F. Syljuåsen, Phys. Rev. E 67, 046701 (2003).
  14. K. Harada and N. Kawashima, Phys. Rev. E 66, 056705 (2002); 67, 039903(E) (2003).
  15. J. Smakov, K. Harada, and N. Kawashima, Phys. Rev. E 68, 046708 (2003).
  16. S. Bergkvist, P. Henelius, and A. Rosengren, Phys. Rev. B 66, 134407 (2002).
  17. M. V. Zyubin and V. A. Kashurnikov, Phys. Rev. E 69, 036701 (2004).
  18. F. Alet and E. Sørensen, Phys. Rev. E 68, 026702 (2003).
  19. O. Nohadani, S. Wessel, B. Normand, and S. Haas, Phys. Rev. B 69, 220402 (2004).
  20. N. Kawashima and K. Harada, J. Phys. Soc. Jpn. 73, 1379 (2004).

  21. In principle, both the head and tail of the worm can move freely, but in this work, we restrict ourselves for simplicity to the case where the tail remains still.

  22. Formally, we should write the initial state as ∣s1⟩ and the new state as T̃0∣s1⟩, but for reasons of clarity, we use the simpler notation s1 and T̃0(s1).

  23. F. Alet and E. Sørensen, Phys. Rev. E 67, 015701(R) (2003).
  24. A. Dorneich and M. Troyer, Phys. Rev. E 64, 066701 (2001).
  25. G. B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, 1963).
  26. N. Kawashima, in The Monte Carlo Method in the Physical Sciences (Ref. [10]), p. 216.
  27. S. Todo and K. Kato, Phys. Rev. Lett. 87, 047203 (2001).
  28. L. Pollet et al., Phys. Rev. E 70, 056705 (2004).
  29. F. Alet et al., e-print cond-mat/0410407; M. Troyer, B. Ammon, and E. Heeb, Lect. Notes Comput. Sci. 1505, 191 (1998). Source codes can be obtained from http://alps.comp-phys.org/

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Phys. Rev. E 71, 036706– Published 22 March, 2005
  • Received 22 June 2004

© 2005 American Physical Society

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