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Finite-size scaling in ferromagnetic spin systems on the pyrochlore lattice


Soldatov K.S., Padalko M.A., Strongin V.S., Kapitan D.Yu., Vasiliev E.V., Rybin A.E., Kapitan V.Yu., Nefedev K.V.

2020, issue 2, Ñ. 255–266
DOI: https://doi.org/10.47910/FEMJ202026


Abstract
In this paper we present the results of the high-performance computations for the Ising model, the XY-model and the classical Heisenberg model for the pyrochlore lattice. We used Wolff and Swendsen-Wang cluster algorithms with GPU parallelization for the calculations. We obtained critical exponents and critical temperatures using finite-size scaling approach.

Keywords:
phase transitions, critical temperature, finite-size scaling, pyrochlore lattice, cluster algorithms

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References

[1] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford Univ. Press, New York, 1971.
[2] C.-K. Hu, “Historical Review on Analytic, Monte Carlo, and Renormalization Group Ap- proaches to Critical Phenomena of Some Lattice Models”, Chinese J. Phys., 52:1, (2014), 1–76.
[3] M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. Zeiske, K. W. Godfrey, “Geometrical Frustration in the Ferromagnetic Pyrochlore Ho2T i2O7”, Phys. Rev. Lett., 79:13, (1997).
[4] A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan, B. S. Shastry, “Zero-point entropy inspinice”, Nature (London), 399, (1999), 333–335.
[5] S. T. Bramwell, J. P. Gingras, “Spin Ice State in Frustrated Magnetic Pyrochlore Materials”, Science, 294:5546, (2001), 1495–1501.
[6] X. Ke, R. S. Freitas, B. G. Ueland, G. C. Lau, M. L. Dahlberg, R. J. Cava, R. Moessner, P. Schiffer, “Nonmonotonic Zero-Point Entropy in Diluted Spin Ice”, Phys. Rev. Lett., 99, (2007), 137203.
[7] Y. Shevchenko, K. Nefedev, Y. Okabe, “Entropy of diluted antiferromagnetic Ising models on frustrated lattices using the Wang-Landau method”, Physical Review E, 95:5, (2017), 052132.
[8] D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 3rd edition, Cambridge University Press, Cambridge, 2009.
[9] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of State Calculations by Fast Computing Machines”, J. Chem. Phys., 21:6, (1953), 1087.
[10] R. H. Swendsen, J. S. Wang, “Nonuniversal critical dynamics in Monte Carlo simulations”, Phys. Rev. Lett., 58, (1987), 86.
[11] U. Wolff, “Collective Monte Carlo Updating for Spin Systems”, Phys. Rev. Lett., 62, (1989), 361.
[12] Y. Komura, Y. Okabe, “GPU-based Swendsen-Wang multi-cluster algorithm for the simulation of two-dimensional classical spin systems”, Comput. Phys. Comm., 183, (2012), 1155.
[13] K. A. Hawick, A. Leist, D. P. Playne, “Parallel Graph Component Labelling with GPUs and CUDA”, Parallel Computing, 36, (2010), 655.
[14] O. Kalentev, A. Rai, S. Kemnitzb, R. Schneider, “Connected component labeling on a 2D grid using CUDA”, J. Parallel Distrib. Comput., 71, (2011), 615.
[15] Y. Komura, “A generalized GPU-based connected component labeling algorithm”, Comput. Phys. Comm., 194, (2015), 54, arXiv: 1603.08357.
[16] Y. Komura, Y. Okabe, “CUDA programs for the GPU computing of the Swendsen–Wang multi-cluster spin flip algorithm: 2D and 3D Ising, Potts, and XY models”, Comput. Phys. Comm., 185, (2014), 1038–1043.
[17] Y. Komura, Y. Okabe, “Improved CUDA programs for GPU computing of Swendsen–Wang multi-cluster spin flip algorithm: 2D and 3D Ising, Potts, and XY models”, Comput. Phys. Comm., 200, (2016), 400–401.
[18] Y. Komura, Y. Okabe, “Large-Scale Monte Carlo Simulation of Two-Dimensional Classical XY Model Using Multiple GPUs”, J. Phys. Soc. Jpn., 81, (2012), 113001.
[19] Y. Komura, Y. Okabe, “High-Precision Monte Carlo Simulation of the Ising Models on the Penrose Lattice and the Dual Penrose Lattice”, J. Phys. Soc. Jpn., 85, (2016), 044004.
[20] H. Shinaoka, Y. Tomita, Y. Motome, “Effect of magnetoelastic coupling on spin-glass be- havior in Heisenberg pyrochlore antiferromagnets with bond disorder”, Phys. Rev. B, 90, (2014), 165119.
[21] J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, Oxford University Press, O, 1932.
[22] M. E. Fisher, “The Theory of critical point singularities”, Proc. 1970 E. Fermi Int. School of Physics, v. 51, ed. M.S. Green, Academic, New York, 1971, 1–99.
[23] K. Binder, “Finite size scaling analysis of ising model block distribution functions”, Z. Phys. B: Condens. Matter, 43, (1981), 119.
[24] L. D. Landau, E. M. Lifshits, Teoreticheskaia fizika, V 10 t. T. 5 (V 2 ch. Ch.1) Stati- sticheskaia fizika, Fizmatlit, M., 2013.
[25] C.-K. Hu, C.-Y. Lin, J.-A. Chen, “Universal Scaling Functions in Critical Phenomena”, Phys. Rev. Lett., 75, (1995), 193.
[26] Y. Okabe, M. Kikuchi, “Universal finite-size-scaling functions”, International Journal of Modern Physics C, 7, (1996), 287–294.
[27] Y. Okabe, K. Kaneda, M. Kikuchi, C.-K. Hu, “Universal finite-size scaling functions for critical systems with tilted boundary conditions”, Phys. Rev. E, 59, (1999), 1585.
[28] Y. Tomita, Y. Okabe, C.-K. Hu, “Cluster analysis and finite-size scaling for Ising spin systems”, Phys. Rev. E, 60, (1999), 2716.
[29] M.-C. Wu, C.-K. Hu, N. Sh. Izmailian, “Universal finite-size scaling functions with exact nonuniversal metric factors”, Phys. Rev. E, 67, (2003), 065103(R).
[30] H. W. J. Bl?ote, E. Luijten, J. R. Heringa, “Ising universality in three dimensions: a Monte Carlo study”, J. Phys. A: Math. Gen., 28, (1995), 6289.
[31] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, E. Vicari, “Critical behavior of the three-dimensional XY universality class”, Phys. Rev. B, 63, (2001), 214503.
[32] A. P. Gottlob, M. Hasenbusch, S. Meyer, “Critical behaviourof the 3D XY-model: A Monte Carlo study”, Nucl. Phys. B (Proc. Suppl.), 30, (1993), 838.
[33] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, E. Vicari, “Critical exponents and equation of state of the three-dimensional Heisenberg universality class”, Phys. Rev. B, 65, (2002), 144520.
[34] C. Holm, W. Janke, “Critical exponents of the classical three-dimensional Heisenberg model: A single-cluster Monte Carlo study”, Phys. Rev. B, 48, (1993), 936.

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