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<p>In this study, we undertake a rigorous examination of the three-state Chui–Weeks model on a third-order Cayley tree, marking its first application in this context. The Chui–Weeks model, initially introduced by S.&#xa0;T.&#xa0;Chui and J.&#xa0;D.&#xa0;Weeks in Phys. Rev. B 14, 4978–4982 (1976), is characterized by an infinitely dimensional transfer matrix, posing significant analytical challenges. Prior investigations, such as those presented in Cuesta and Sanchez, J. Stat. Phys. (2004), have predominantly focused on the study of phase transition phenomena within one-dimensional systems. However, to date, the structural and statistical mechanical properties of the Chui–Weeks model on a Cayley tree remain unexplored. In this work, we systematically characterize and classify all translation-invariant and two-periodic ground states associated with this model on a third-order Cayley tree. Furthermore, we establish the existence of Gibbs measures corresponding to the constructed ground states by applying the contour method and Peierls-type arguments, and then, we develop the boundary law approach, deriving recursive relations that characterize Gibbs measures, and compare these solutions with those obtained by the contour method. By extending the theoretical framework of the Chui–Weeks model to hierarchical lattice structures, we aim to contribute novel insights into its equilibrium properties and phase behavior. The findings of this study provide a foundational basis for further investigations into critical phenomena and phase transitions in complex hierarchical systems.</p>

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Periodic ground states for the Chui-Weeks model on the Cayley tree of order three

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Muhayyo A. Rasulova,
  • Muslima A. Hakimova

摘要

In this study, we undertake a rigorous examination of the three-state Chui–Weeks model on a third-order Cayley tree, marking its first application in this context. The Chui–Weeks model, initially introduced by S. T. Chui and J. D. Weeks in Phys. Rev. B 14, 4978–4982 (1976), is characterized by an infinitely dimensional transfer matrix, posing significant analytical challenges. Prior investigations, such as those presented in Cuesta and Sanchez, J. Stat. Phys. (2004), have predominantly focused on the study of phase transition phenomena within one-dimensional systems. However, to date, the structural and statistical mechanical properties of the Chui–Weeks model on a Cayley tree remain unexplored. In this work, we systematically characterize and classify all translation-invariant and two-periodic ground states associated with this model on a third-order Cayley tree. Furthermore, we establish the existence of Gibbs measures corresponding to the constructed ground states by applying the contour method and Peierls-type arguments, and then, we develop the boundary law approach, deriving recursive relations that characterize Gibbs measures, and compare these solutions with those obtained by the contour method. By extending the theoretical framework of the Chui–Weeks model to hierarchical lattice structures, we aim to contribute novel insights into its equilibrium properties and phase behavior. The findings of this study provide a foundational basis for further investigations into critical phenomena and phase transitions in complex hierarchical systems.