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Geometrically Constructed Markov Chain Monte Carlo Study of Quantum Spin-phonon Complex Systems pp 43–68Cite as

Monte Carlo Method for Spin-Peierls Systems

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Part of the book series: Springer Theses ((Springer Theses))

Abstract

The novel quantum Monte Carlo method for nonconserved particles will be presented in this chapter. The quantum Monte Carlo method based on the worldline representation has been applied to a wide variety of quantum spin and boson systems. Although the method has performed well for many quantum systems, the conventional methods cannot efficiently calculate particle-number nonconserving systems, such as the spin-Peierls model. Summing up the difficulties by the conventional methods, we will develop an extended worm (directed-loop) algorithm that overcomes all of the problems. The state-of-the-art quantum Monte Carlo method in the continuous time representation and the nontrivial technique for off-diagonal correlation measurement will be also presented. We will mention some programming details at the last portion.

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References

  1. Beard, B. B., & Wiese, U. J. (1996). Simulations of discrete quantum systems in continuous Euclidean time. Physical Review Letters, 77, 5130.

    Article  ADS  Google Scholar 

  2. Bray, J., Hart, H. R., Interrante, L. V., Jacobs, I. S., Kasper, J. S., Watkins, G. D., et al. (1975). Observation of a spin-Peierls transition in a Heisenberg antiferromagnetic linear-chain system. Physical Review Letters, 35, 744.

    Article  ADS  Google Scholar 

  3. Chandrasekharan, S., Cox, J., Osborn, J., & Wiese, U. J. (2003). Meron-cluster approach to systems of strongly correlated electrons. Nuclear Physics B, 673, 405.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Chandrasekharan, S., & Wiese, U. J. (1999). Meron-cluster solution of fermion sign problems. Physical Review Letters, 83, 3116.

    Article  ADS  Google Scholar 

  5. Cross, M. C., & Fisher, D. S. (1979). A new theory of the spin-Peierls transition with special relevance to the experiments on TTFCuBDT. Physical Review B, 19, 402.

    Article  ADS  Google Scholar 

  6. Evertz, H. G. (2003). The loop algorithm. Advances in Physics, 52, 1.

    Article  ADS  Google Scholar 

  7. Evertz, H. G., Lana, G., & Marcu, M. (1993). Cluster algorithm for vertex models. Physical Review Letters, 70, 875.

    Article  ADS  Google Scholar 

  8. Fukui, K., & Todo, S. (2009). Order-\(N\) cluster Monte Carlo method for spin systems with long-range interactions. Journal of Computational Physics, 228, 2629.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Fukuyama, H., Tanimoto, T., & Saito, M. (1996). Antiferromagnetic long range order in disordered spin-Peierls systems. Journal of the Physical Society of Japan, 65, 1182.

    Article  ADS  Google Scholar 

  10. Geertsma, W., & Khomskii, D. (1996). Influence of side groups on \(90^{\circ }\) superexchange: A modification of the Goodenough-Kanamori-Anderson rules. Physical Review B, 54, 3011–3014.

    Article  ADS  Google Scholar 

  11. Handscomb, D. C. (1962). The Monte Carlo method in quantum statistical mechanics. Proceedings of the Cambridge Philological Society, 58, 594.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Hase, M., Terasaki, I., & Uchinokura, K. (1993). Observation of the spin-Peierls transition in linear Cu\(^{2+}\) (spin-\(\frac{1}{2})\) chains in an inorganic compound CuGeO\(_3\). Physical Review Letters, 70, 3651.

    Article  ADS  Google Scholar 

  13. Henelius, P., Fröbrich, P., Kuntz, P. J., Timm, C., & Jensen, P. J. (2002). Quantum Monte Carlo simulation of thin magnetic films. Physical Review B, 66, 094407.

    Article  ADS  Google Scholar 

  14. Iba, Y., Chikenji, G., & Kikuchi, M. (1998). Simulation of lattice polymers with multi-self-overlap ensemble. Journal of the Physical Society of Japan, 67, 3327–3330.

    Article  ADS  Google Scholar 

  15. Kawashima, N., & Harada, K. (2004). Recent development of world-line Monte Carlo methods. Journal of the Physical Society of Japan, 73, 1379.

    Article  ADS  Google Scholar 

  16. Kühne, R. W., & Löw, U. (1999). Thermodynamical properties of a spin-\(\frac{1}{2}\) Heisenberg chain coupled to phonons. Physical Review B, 60, 12125.

    Article  ADS  Google Scholar 

  17. Matsumoto, M., Todo, S., Yasuda, C., & Takayama, H. (2002). Ground state of \(S=1\) Heisenberg ladders. Progress of Theoretical Physics, 145(Suppl.), 221.

    Google Scholar 

  18. McKenzie, R. H., Hamer, C. J., & Murray, D. W. (1996). Quantum Monte Carlo study of the one-dimensional Holstein model of spinless fermions. Physical Review B, 53, 9676–9687.

    Article  ADS  Google Scholar 

  19. Michel, F., Evertz, H.G. (2007). Lattice dynamics of the Heisenberg chain coupled to finite frequency bond phonons. cond-mat p. arXiv:0705.0799v2.

    Google Scholar 

  20. Onishi, H., & Miyashita, S. (2000). Temperature dependence of spin and bond ordering in a spin-Peierls system. Journal of the Physical Society of Japan, 69, 2634–2641.

    Article  ADS  Google Scholar 

  21. Onishi, H., & Miyashita, S. (2003). Quantum narrowing effect in a spin-Peierls system with quantum lattice fluctuation. Journal of the Physical Society of Japan, 72, 392.

    Article  ADS  Google Scholar 

  22. Prokof’ev, N. V., Svistunov, B. V., & Tupitsyn, I. S. (1998). Exact, complete, and universal continuous-time world-line Monte Carlo approach to the statistics of discrete quantum systems. Soviet Physics JETP, 87, 310.

    Article  ADS  Google Scholar 

  23. Raas, C., Löw, U., Uhrig, G. S., & Kühne, R. W. (2002). Spin-phonon chains with bond coupling. Physical Review B, 65, 144438.

    Article  ADS  Google Scholar 

  24. Rieger, H., & Kawashima, N. (1999). Application of a continuous time cluster algorithm to the two-dimensional random quantum Ising ferromagnet. European Physical Journal B: Condensed Matter Physics, 9, 233.

    Article  ADS  Google Scholar 

  25. Sandvik, A. W. (1999). Multichain mean-field theory of quasi-one-dimensional quantum spin systems. Physical Review Letters, 83, 3069.

    Article  ADS  Google Scholar 

  26. Sandvik, A. W. (1999). Stochastic series expansion method with operator-loop update. Physical Review B, 59(R14), 157.

    Google Scholar 

  27. Sandvik, A. W., & Campbell, D. K. (1999). Spin-Peierls transition in the Heisenberg chain with finite-frequency phonons. Physical Review Letters, 83, 195.

    Article  ADS  Google Scholar 

  28. Seidel, A., Marianetti, C. A., Chou, F. C., Ceder, G., & Lee, P. A. (2003). \(S=\frac{1}{2}\) chains and spin-Peierls transition in TiOCl. Physical Review B, 67, 020405R.

    Article  ADS  Google Scholar 

  29. Suzuki, M. (1976). Relationship between \(d\)-dimensional quantal spin systems and \((d+1)\)-dimensional Ising systems. Progress of Theoretical Physics, 56, 1454.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Swendsen, R. H., & Wang, J. S. (1987). Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58, 86.

    Article  ADS  Google Scholar 

  31. Syljuasen, O. F. (2003). Directed loop updates for quantum lattice models. Physical Review E, 67, 046701.

    Article  MathSciNet  ADS  Google Scholar 

  32. Syljuasen, O. F., & Sandvik, A. W. (2002). Quantum Monte Carlo with directed loops. Physical Review E, 66, 046701.

    Article  ADS  Google Scholar 

  33. Uchinokura, K. (2002). Spin-Peierls transition in \(\text{ CuGeO }_3\) and impurity-induced ordered phases in low-dimensional spin-gap systems. Journal of Physics: Condensed Matter, 14, R195–R237.

    Article  ADS  Google Scholar 

  34. Werner, R., Gros, C., & Braden, M. (1999). Microscopic spin-phonon coupling constants in \(\text{ CuGeO }_3\). Physical Review B, 59, 14356–14366.

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Boston University, Boston, MA, USA

    Hidemaro Suwa

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  1. Hidemaro Suwa
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Correspondence to Hidemaro Suwa .

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Suwa, H. (2014). Monte Carlo Method for Spin-Peierls Systems. In: Geometrically Constructed Markov Chain Monte Carlo Study of Quantum Spin-phonon Complex Systems. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54517-0_3

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