The Monte Carlo computer experiment to study the order and phase transitions in the mixed phase region based on the example of the 3D Ashkin-Teller model
Wojtkowiak Zbigniew, Musiał Grzegorz *
Adam Mickiewicz University
Faculty of Physics
ul. Uniwersytetu Poznańskiego 2
61-614 Poznań, Poland
*E-mail: gmusial@amu.edu.pl
Received:
Received: 21 June 2021; accepted: 19 September 2021; published online: 28 September 2021
DOI: 10.12921/cmst.2021.0000019
Abstract:
In this paper, we demonstrate a new way of performing Monte Carlo (MC) simulations in a mixed phase region that is difficult to study, where with certain probabilities there are different ordering ways. That results in a large oscillation of the values of the computed thermodynamic quantities, which makes their interpretation very problematic. Our results are presented on the example of the 3D Askin-Teller (AT) model, where within a certain range of parameters with equal probabilities there are two different, but equivalent, ways of ordering two of the three order parameters showing independent behavior. The use of our new approach in an MC computer experiment allowed us to use Binder cumulant as well as Challa- and the Lee-Kosterlitz-like cumulants. This made it possible to locate phase transitions precisely enough to be able to use the energy distribution histogram method. According to the most effective strategy in the critical region we use our recently proposed cluster MC algorithm and the Metropolis algorithm beyond it, which are suitable for both the first-order and the continuous phase transitions in the 3D AT model. The new approach was demonstrated by determining smooth curves of magnetization and internal energy, and as a consequence by determining the location and character of the phase transition on the line between the mixed phase region and the paramagnetic phase.
Key words:
3D Ashkin-Teller model, lattice spin systems, mixed phase region, Monte Carlo computer experiment, phase transitions
References:
[1] J. Ashkin, E. Teller, Statistics of Two-Dimensional Lattices with Four Components, Phys. Rev. 64, 178 (1943).
[2] C. Fan, On critical properties of the Ashkin-Teller model, Phys. Lett. 39A, 136 (1972).
[3] S.-P. Li, Z.-H. Sun, Local and intrinsic quantum coherence in critical systems, Phys. Rev. A 98, 022317 (2018).
[4] J.P. Santos, J.A.J. Avila, D.S. Rosa, R.M. Francisco, Multicritical phase diagram of the three-dimensional Ashkin-Teller model including metastable and unstable phases, J. Magn. Magn. Mater. 469, 35–39 (2019).
[5] G. Delfino, N. Lamsen, Critical points of coupled vector-Ising systems. Exact results, J. Phys. A: Math. Theor. 52, 35LT02 (2019).
[6] P. Patil, A.W. Sandvik, Hilbert space fragmentation and Ashkin-Teller criticality in fluctuation coupled Ising models, Phys. Rev. B 101, 014453 (2020).
[7] A. Benmansour, S. Bekhechi, B.N. Brahmi, N. Moussa, H. Ez-Zahraouy, Monte Carlo study of thin magnetic Ashkin Teller films at the special point, J. Magn. Magn. Mater. 511, 166944 (2020).
[8] T. Lovorn, S.K. Sarker, Complex Quasi-Two-Dimensional Crystalline Order Embedded in VO2 and Other Crystals, Phys. Rev. Lett. 119, 045501 (2017).
[9] S.S. Lee, B.J. Kim, Confusion scheme in machine learning detects double phase transitions and quasi-long-range order, Phys. Rev. E 99, 043308 (2019).
[10] G.-Y. Zhu, G.-M. Zhang, Gapless Coulomb State Emerging from a Self-Dual Topological Tensor-Network State, Phys. Rev. Lett. 122, 176401 (2019).
[11] A. Kumar, J.G.M. Guy, L. Zhang, J. Chen, J.M. Gregg, J.F. Scott, Nanodomain patterns in ultra-tetragonal lead titanate (PbTiO3), Appl. Phys. Lett. 116, 182903 (2020).
[12] R.M. Francisco, J.P. Santos, Magnetic properties of the Ashkin-Teller model on a hexagonal nanotube, Phys. Lett. A 383, 1092–1098 (2019).
[13] C. Zhe, W. Ping, Z. Ying-Hong, Ashkin-Teller Formalism for Elastic Response of DNA Molecule to External Force and Torque, Commun. Theor. Phys. 49, 525 (2008).
[14] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press, Oxford (1999).
[15] G. Musiał, D. Jeziorek-Knioła, Z. Wojtkowiak, Monte Carlo examination of first-order phase transitions in a system with many independent order parameters: Three-dimensional Ashkin-Teller model, Phys. Rev. E 103, 062124 (2021).
[16] J.P. Santos, F.C. Sá Barreto, Upper Bounds on the Critical Temperature of the Ashkin-Teller Model, Braz. J. Phys. 46, 70–77 (2016).
[17] Ü. Akıncı, Nonequilibrium phase transitions in isotropic Ashkin-Teller model, Physica A 469, 740–749 (2017).
[18] Z. Wojtkowiak, G. Musiał, Wide crossover in the 3D Ashkin-Teller model, Physica A 513, 104–111 (2019).
[19] D. Jeziorek-Knioła, G. Musiał, Z. Wojtkowiak, Arbitrarily Weak First Order Phase Transitions in the 3D Standard Ashkin-Teller Model by MC Computer Experiments, Acta Phys. Polon. A 127, 327 (2015).
[20] D. Jeziorek-Knioła, G. Musiał, L. Dębski, J. Rogiers, S. Dylak, On Non-Ising Phase Transitions in the 3D Standard Ashkin-Teller Model, Acta Phys. Polon. A 121, 1105 (2012).
[21] G. Szukowski, G. Kamieniarz, G. Musiał, Verification of Ising phase transitions in the three-dimensional Ashkin-Teller model using Monte Carlo simulations, Phys. Rev. E 77, 031124 (2008).
[22] G. Musiał, Monte Carlo analysis of the tricritical behavior in a three-dimensional system with a multicomponent order parameter: The Ashkin-Teller model, Phys. Rev. B 69, 024407 (2004).
[23] P. Arnold, Y. Zhang, Monte Carlo study of very weakly first-order transitions in the three-dimensional Ashkin-Teller model, Nuclear Phys. B 501, 803–837 (1997).
[24] R.V. Ditzian, J.R. Banavar, G.S. Grest, L.P. Kadanoff, Phase diagram for the Ashkin-Teller model in three dimensions, Phys. Rev. B 22, 2542 (1980).
[25] G. Musiał, A Monte Carlo study of the mixed phase region in the 3D Ashkin-Teller model, Phys. Status Solidi B 236, 486– 489 (2003).
[26] G. Musiał, J. Rogiers, On the possibility of nonuniversal behavior in the 3D Ashkin-Teller model, Phys. Status Solidi B 243, 335–338 (2006).
[27] G. Musiał, L. Dębski, G. Kamieniarz, Monte Carlo simulations of Ising-like phase transitions in the three-dimensional Ashkin-Teller model, Phys. Rev. B 66, 012407 (2002).
[28] C. Fan, F.Y. Wu, General Lattice Model of Phase Transitions, Phys. Rev. B 2, 723 (1970).
[29] Z. Wojtkowiak, G. Musiał, Cluster Monte Carlo method for the 3D Ashkin-Teller model, J. Magn. Magn. Mater. 500, 166365 (2020).
[30] U. Wolff, Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62, 361 (1989).
[31] G. Musiał, L. Dębski, Monte Carlo Method with Parallel Computation of Phase Transitions in the Three-Dimensional Ashkin-Teller Model, Lect. Notes in Comp. Sci. 2328, 535–543 (2002).
[32] T. Preis, P. Virnau, W.Paul, J.J. Schneider, GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model, J. Comput. Phys. 228, 4468–4477 (2009).
[33] Y. Komura, Y. Okabe, GPU-based single-cluster algorithm for the simulation of the Ising model, J. Comput. Phys. 231, 1209–1215 (2012).
[34] T.A. Kampmann, H.-H. Boltz, J. Kierfeld, Parallelized event chain algorithm for dense hard sphere and polymer systems, J. Comput. Phys. 281, 864–875 (2015).
[35] G. Kamieniarz, R. Dekeyser, G. Musiał, L. Dębski, M. Bieliński, Modified effective-field approach to low-dimensional spin-1/2 systems, Phys. Rev. E 56, 144 (1997).
[36] K. Binder, D.P. Landau, Finite-size scaling at first-order phase transitions, Phys. Rev. B 30, 1477 (1984).
[37] M.S.S. Challa, D.P. Landau, K. Binder, Finite-size effects at temperature-driven first-order transitions, Phys. Rev. B 34, 1841 (1986).
[38] J. Lee, J.M. Kosterlitz, Finite-size scaling and Monte Carlo simulations of first-order phase transitions, Phys. Rev. B 43, 3265 (1991).
[39] K. Binder, Applications of Monte Carlo methods to statistical physics, Rep. Prog. Phys. 60, 487 (1997).
[40] C. Borgs, R. Kotecky, S. Miracle-Sole, Finite-size scaling for Potts models, J. Stat. Phys. 62, 529–551 (1991).
[41] D. Jeziorek-Knioła, Z. Wojtkowiak, G. Musiał, Computation of Latent Heat based on the Energy Distribution Histogram in the 3D Ashkin-Teller Model, Acta Phys. Polon. A 133, 435–437 (2018).
[42] A. Billoire, T. Neuhaus, B.A. Berg, A determination of interface free energies, Nuclear Phys. B 413, 795-812 (1994).
In this paper, we demonstrate a new way of performing Monte Carlo (MC) simulations in a mixed phase region that is difficult to study, where with certain probabilities there are different ordering ways. That results in a large oscillation of the values of the computed thermodynamic quantities, which makes their interpretation very problematic. Our results are presented on the example of the 3D Askin-Teller (AT) model, where within a certain range of parameters with equal probabilities there are two different, but equivalent, ways of ordering two of the three order parameters showing independent behavior. The use of our new approach in an MC computer experiment allowed us to use Binder cumulant as well as Challa- and the Lee-Kosterlitz-like cumulants. This made it possible to locate phase transitions precisely enough to be able to use the energy distribution histogram method. According to the most effective strategy in the critical region we use our recently proposed cluster MC algorithm and the Metropolis algorithm beyond it, which are suitable for both the first-order and the continuous phase transitions in the 3D AT model. The new approach was demonstrated by determining smooth curves of magnetization and internal energy, and as a consequence by determining the location and character of the phase transition on the line between the mixed phase region and the paramagnetic phase.
Key words:
3D Ashkin-Teller model, lattice spin systems, mixed phase region, Monte Carlo computer experiment, phase transitions
References:
[1] J. Ashkin, E. Teller, Statistics of Two-Dimensional Lattices with Four Components, Phys. Rev. 64, 178 (1943).
[2] C. Fan, On critical properties of the Ashkin-Teller model, Phys. Lett. 39A, 136 (1972).
[3] S.-P. Li, Z.-H. Sun, Local and intrinsic quantum coherence in critical systems, Phys. Rev. A 98, 022317 (2018).
[4] J.P. Santos, J.A.J. Avila, D.S. Rosa, R.M. Francisco, Multicritical phase diagram of the three-dimensional Ashkin-Teller model including metastable and unstable phases, J. Magn. Magn. Mater. 469, 35–39 (2019).
[5] G. Delfino, N. Lamsen, Critical points of coupled vector-Ising systems. Exact results, J. Phys. A: Math. Theor. 52, 35LT02 (2019).
[6] P. Patil, A.W. Sandvik, Hilbert space fragmentation and Ashkin-Teller criticality in fluctuation coupled Ising models, Phys. Rev. B 101, 014453 (2020).
[7] A. Benmansour, S. Bekhechi, B.N. Brahmi, N. Moussa, H. Ez-Zahraouy, Monte Carlo study of thin magnetic Ashkin Teller films at the special point, J. Magn. Magn. Mater. 511, 166944 (2020).
[8] T. Lovorn, S.K. Sarker, Complex Quasi-Two-Dimensional Crystalline Order Embedded in VO2 and Other Crystals, Phys. Rev. Lett. 119, 045501 (2017).
[9] S.S. Lee, B.J. Kim, Confusion scheme in machine learning detects double phase transitions and quasi-long-range order, Phys. Rev. E 99, 043308 (2019).
[10] G.-Y. Zhu, G.-M. Zhang, Gapless Coulomb State Emerging from a Self-Dual Topological Tensor-Network State, Phys. Rev. Lett. 122, 176401 (2019).
[11] A. Kumar, J.G.M. Guy, L. Zhang, J. Chen, J.M. Gregg, J.F. Scott, Nanodomain patterns in ultra-tetragonal lead titanate (PbTiO3), Appl. Phys. Lett. 116, 182903 (2020).
[12] R.M. Francisco, J.P. Santos, Magnetic properties of the Ashkin-Teller model on a hexagonal nanotube, Phys. Lett. A 383, 1092–1098 (2019).
[13] C. Zhe, W. Ping, Z. Ying-Hong, Ashkin-Teller Formalism for Elastic Response of DNA Molecule to External Force and Torque, Commun. Theor. Phys. 49, 525 (2008).
[14] M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press, Oxford (1999).
[15] G. Musiał, D. Jeziorek-Knioła, Z. Wojtkowiak, Monte Carlo examination of first-order phase transitions in a system with many independent order parameters: Three-dimensional Ashkin-Teller model, Phys. Rev. E 103, 062124 (2021).
[16] J.P. Santos, F.C. Sá Barreto, Upper Bounds on the Critical Temperature of the Ashkin-Teller Model, Braz. J. Phys. 46, 70–77 (2016).
[17] Ü. Akıncı, Nonequilibrium phase transitions in isotropic Ashkin-Teller model, Physica A 469, 740–749 (2017).
[18] Z. Wojtkowiak, G. Musiał, Wide crossover in the 3D Ashkin-Teller model, Physica A 513, 104–111 (2019).
[19] D. Jeziorek-Knioła, G. Musiał, Z. Wojtkowiak, Arbitrarily Weak First Order Phase Transitions in the 3D Standard Ashkin-Teller Model by MC Computer Experiments, Acta Phys. Polon. A 127, 327 (2015).
[20] D. Jeziorek-Knioła, G. Musiał, L. Dębski, J. Rogiers, S. Dylak, On Non-Ising Phase Transitions in the 3D Standard Ashkin-Teller Model, Acta Phys. Polon. A 121, 1105 (2012).
[21] G. Szukowski, G. Kamieniarz, G. Musiał, Verification of Ising phase transitions in the three-dimensional Ashkin-Teller model using Monte Carlo simulations, Phys. Rev. E 77, 031124 (2008).
[22] G. Musiał, Monte Carlo analysis of the tricritical behavior in a three-dimensional system with a multicomponent order parameter: The Ashkin-Teller model, Phys. Rev. B 69, 024407 (2004).
[23] P. Arnold, Y. Zhang, Monte Carlo study of very weakly first-order transitions in the three-dimensional Ashkin-Teller model, Nuclear Phys. B 501, 803–837 (1997).
[24] R.V. Ditzian, J.R. Banavar, G.S. Grest, L.P. Kadanoff, Phase diagram for the Ashkin-Teller model in three dimensions, Phys. Rev. B 22, 2542 (1980).
[25] G. Musiał, A Monte Carlo study of the mixed phase region in the 3D Ashkin-Teller model, Phys. Status Solidi B 236, 486– 489 (2003).
[26] G. Musiał, J. Rogiers, On the possibility of nonuniversal behavior in the 3D Ashkin-Teller model, Phys. Status Solidi B 243, 335–338 (2006).
[27] G. Musiał, L. Dębski, G. Kamieniarz, Monte Carlo simulations of Ising-like phase transitions in the three-dimensional Ashkin-Teller model, Phys. Rev. B 66, 012407 (2002).
[28] C. Fan, F.Y. Wu, General Lattice Model of Phase Transitions, Phys. Rev. B 2, 723 (1970).
[29] Z. Wojtkowiak, G. Musiał, Cluster Monte Carlo method for the 3D Ashkin-Teller model, J. Magn. Magn. Mater. 500, 166365 (2020).
[30] U. Wolff, Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62, 361 (1989).
[31] G. Musiał, L. Dębski, Monte Carlo Method with Parallel Computation of Phase Transitions in the Three-Dimensional Ashkin-Teller Model, Lect. Notes in Comp. Sci. 2328, 535–543 (2002).
[32] T. Preis, P. Virnau, W.Paul, J.J. Schneider, GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model, J. Comput. Phys. 228, 4468–4477 (2009).
[33] Y. Komura, Y. Okabe, GPU-based single-cluster algorithm for the simulation of the Ising model, J. Comput. Phys. 231, 1209–1215 (2012).
[34] T.A. Kampmann, H.-H. Boltz, J. Kierfeld, Parallelized event chain algorithm for dense hard sphere and polymer systems, J. Comput. Phys. 281, 864–875 (2015).
[35] G. Kamieniarz, R. Dekeyser, G. Musiał, L. Dębski, M. Bieliński, Modified effective-field approach to low-dimensional spin-1/2 systems, Phys. Rev. E 56, 144 (1997).
[36] K. Binder, D.P. Landau, Finite-size scaling at first-order phase transitions, Phys. Rev. B 30, 1477 (1984).
[37] M.S.S. Challa, D.P. Landau, K. Binder, Finite-size effects at temperature-driven first-order transitions, Phys. Rev. B 34, 1841 (1986).
[38] J. Lee, J.M. Kosterlitz, Finite-size scaling and Monte Carlo simulations of first-order phase transitions, Phys. Rev. B 43, 3265 (1991).
[39] K. Binder, Applications of Monte Carlo methods to statistical physics, Rep. Prog. Phys. 60, 487 (1997).
[40] C. Borgs, R. Kotecky, S. Miracle-Sole, Finite-size scaling for Potts models, J. Stat. Phys. 62, 529–551 (1991).
[41] D. Jeziorek-Knioła, Z. Wojtkowiak, G. Musiał, Computation of Latent Heat based on the Energy Distribution Histogram in the 3D Ashkin-Teller Model, Acta Phys. Polon. A 133, 435–437 (2018).
[42] A. Billoire, T. Neuhaus, B.A. Berg, A determination of interface free energies, Nuclear Phys. B 413, 795-812 (1994).
