- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Crossover and universality breaking in the dilute Baxter-Wu model

Dimitrios Mataragkas1, Alexandros Vasilopoulos1,*, Dong-Hee Kim2,†, and Nikolaos G. Fytas1,‡

  • *Contact author: alex.vasilopoulos@essex.ac.uk
  • †Contact author: dongheekim@gist.ac.kr
  • ‡Contact author: nikolaos.fytas@essex.ac.uk
PDF

Phys. Rev. E 114, 024114 – Published 10 August, 2026

DOI: https://doi.org/10.1103/gp1c-ns8v

Abstract

The critical behavior of the Baxter-Wu model belongs to the universality class of the four-state Potts model. While the introduction of annealed vacancies does not alter the criticality of the four-state Potts model, the dilute Baxter-Wu model has remained the subject of several competing scenarios. Here we investigate the phase diagram of the spin-1 Baxter-Wu model in the presence of a crystal field using transfer-matrix calculations and large-scale Monte Carlo simulations. Our results reveal a systematic evolution of the effective critical behavior with increasing crystal field, accompanied by increasingly strong finite-size corrections near the crossover to the first-order regime. Along the line of continuous transitions, the central charge remains close to c=1, while the scaling dimensions systematically deviate from the spin-1/2 limit as the crystal field increases, consistent with either continuously varying effective critical exponents or a slow crossover between competing critical behaviors. The first-order regime is independently characterized through multicanonical simulations, which confirm the expected finite-size scaling and interfacial behavior. Taken together, our results provide a unified picture of the dilute spin-1 Baxter-Wu model, substantially narrowing the range of possible scenarios for the crossover between continuous and first-order phase transitions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (37)

  1. D. W. Wood and H. P. Griffiths, A self dual relation for an Ising model with triplet interactions, J. Phys. C: Solid State Phys. 5, L253 (1972).
  2. R. J. Baxter and F. Y. Wu, Exact solution of an Ising model with three-spin interactions on a triangular lattice, Phys. Rev. Lett. 31, 1294 (1973).
  3. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982)
  4. D. Merlini and C. Gruber, Spin-1/2 lattice system: Group structure and duality relation, J. Math. Phys. 13, 1814 (1972).
  5. E. Domany and E. K. Riedel, Phase transitions in two‐dimensional systems, J. Appl. Phys. 49, 1315 (1978).
  6. F. C. Alcaraz and J. C. Xavier, Conformal invariance studies of the Baxter-Wu model and a related site-colouring problem, J. Phys. A: Math. Gen. 30, L203 (1997).
  7. F. C. Alcaraz and J. C. Xavier, Critical and off-critical studies of the Baxter-Wu model with general toroidal boundary conditions, J. Phys. A: Math. Gen. 32, 2041 (1999).
  8. D. A. Dias, J. C. Xavier, and J. A. Plascak, Critical behavior of the spin-1 and spin-3/2 Baxter–Wu model in a crystal field, Phys. Rev. E 95, 012103 (2017).
  9. L. N. Jorge, P. H. L. Martins, C. J. DaSilva, L. S. Ferreira, and A. A. Caparica, An entropic simulational study of the spin-1 Baxter–Wu model in a crystal field, Physica A 576, 126071 (2021).
  10. B. Nienhuis, A. N. Berker, E. K. Riedel, and M. Schick, First- and second-order phase transitions in Potts models: Renormalization-group solution, Phys. Rev. Lett. 43, 737 (1979).
  11. M. L. M. Costa, J. C. Xavier, and J. A. Plascak, Phase diagram and critical behavior of the spin-1 Baxter-Wu model with a crystal field, Phys. Rev. B 69, 104103 (2004).
  12. W. Kinzel, E. Domany, and A. Aharony, Finite size scaling analysis of the dilute Baxter-Wu model, J. Phys. A: Math. Gen. 14, L417 (1981).
  13. M. L. M. Costa and J. A. Plascak, Monte Carlo study of the spin-1 Baxter-Wu model, Braz. J. Phys. 34, 419 (2004).
  14. M. L. M. Costa and J. A. Plascak, Monte Carlo study of the spin-1 Baxter-Wu model in a crystal field, J. Phys.: Conf. Ser. 686, 012011 (2016).
  15. A. Vasilopoulos, N. G. Fytas, E. Vatansever, A. Malakis, and M. Weigel, Universality in the two-dimensional dilute Baxter–Wu model, Phys. Rev. E 105, 054143 (2022).
  16. A. R. S. Macêdo, A. Vasilopoulos, M. Akritidis, J. A. Plascak, N. G. Fytas, and M. Weigel, Two-dimensional dilute Baxter-Wu model: Transition order and universality, Phys. Rev. E 108, 024140 (2023).
  17. A. R. S. Macêdo, J. A. Plascak, A. Vasilopoulos, N. G. Fytas, M. Akritidis, and M. Weigel, Universal energy and magnetisation distributions in the Blume–Capel and Baxter–Wu models, J. Stat. Mech. (2024) 103204.
  18. D. Mataragkas, A. Vasilopoulos, N. G. Fytas, and D.-H. Kim, Transfer-matrix approach to the Blume-Capel model on the triangular lattice, Phys. Rev. Res. 7, 033240 (2025).
  19. M. Jung and D.-H. Kim, First-order transitions and thermodynamic properties in the 2D Blume-Capel model: The transfer-matrix method revisited, Eur. Phys. J. B 90, 245 (2017).
  20. J. Ashkin and E. Teller, Statistics of two-dimensional lattices with four components, Phys. Rev. 64, 178 (1943).
  21. M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford, 1999).
  22. L. N. Jorge, L. S. Ferreira, and A. A. Caparica, On the order of the phase transition in the spin-1 Baxter–Wu model, Physica A 542, 123417 (2020).
  23. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).
  24. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 5th ed. (Cambridge University Press, Cambridge, 2021).
  25. J. Zierenberg, N. G. Fytas, M. Weigel, W. Janke, and A. Malakis, Scaling and universality in the phase diagram of the 2D Blume-Capel model, Eur. Phys. J. Spec. Top. 226, 789 (2017).
  26. M. Blume, Theory of the first-order magnetic phase change in UO2, Phys. Rev. 141, 517 (1966).
  27. B. A. Berg and T. Neuhaus, Multicanonical ensemble: A new approach to simulate first-order phase transitions, Phys. Rev. Lett. 68, 9 (1992).
  28. J. Zierenberg, N. G. Fytas, and W. Janke, Parallel multicanonical study of the three-dimensional Blume-Capel model, Phys. Rev. E 91, 032126 (2015).
  29. J. Zierenberg, M. Marenz, and W. Janke, Scaling properties of a parallel implementation of the multicanonical algorithm, Comput. Phys. Commun. 184, 1155 (2013).
  30. J. Gross, J. Zierenberg, M. Weigel, and W. Janke, Massively parallel multicanonical simulations, Comput. Phys. Commun. 224, 387 (2018).
  31. K. Binder and D. P. Landau, Finite-size scaling at first-order phase transitions, Phys. Rev. B 30, 1477 (1984).
  32. K. Binder, Theory of first-order phase transitions, Rep. Prog. Phys. 50, 783 (1987).
  33. J. Lee and J. M. Kosterlitz, New numerical method to study phase transitions, Phys. Rev. Lett. 65, 137 (1990).
  34. J. Lee and J. M. Kosterlitz, Finite-size scaling and Monte Carlo simulations of first-order phase transitions, Phys. Rev. B 43, 3265 (1991).
  35. D. Mataragkas, A. Vasilopoulos, N. G. Fytas, and D.-H. Kim, Tricriticality and finite-size scaling in the triangular Blume-Capel ferromagnet, Phys. Rev. Res. 7, 013214 (2025).
  36. M. S. S. Challa, D. P. Landau, and K. Binder, Finite-size effects at temperature-driven first-order transitions, Phys. Rev. B 34, 1841 (1986).
  37. W. Janke and R. Villanova, Three-dimensional 3-state Potts model revisited with new techniques, Nucl. Phys. B 489, 679 (1997).

Outline

Information

Phys. Rev. E 114, 024114– Published 10 August, 2026
  • Received 12 March 2026
  • Accepted 21 July 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation