Two-dimensional Ising model with non-homogenous interactions

• Main Authors: Ganikhodjaev, Nasir, Ibrahim, Huda Husna
• Format: Conference or Workshop Item
• Language: English; English; English
• Published: American Institute of Physics 2017
• Subjects: QA Mathematics
• Online Access: http://irep.iium.edu.my/52990/54/52990%20Two-dimensional%20Ising%20Model.pdf; http://irep.iium.edu.my/52990/48/52990_Two-dimensional%20Ising%20model%20with_Scopus.pdf; http://irep.iium.edu.my/52990/60/52990_Two-dimensional%20Ising%20model%20_wos.pdf; http://irep.iium.edu.my/52990/; https://aip.scitation.org/doi/abs/10.1063/1.4980984
• Institution: Universiti Islam Antarabangsa Malaysia

In this paper we investigate the Ising model on Z2 with competing interactions. In this model we consider J1 as horizontal interactions and J2 as vertical interactions where J1,J2 > 0. We prove that this model can reach a phase transition. Onsager considered the case where horizontal interaction parameter J1 and vertical interaction parameter J2 are different. For any fixed J1 and J2, he showed that below a critical temperature Tc which depends on J1 and J2, phase transition occurs using some matrix transfer method. However in this paper we will prove the existence of phase transition using contours methods introduced by Sinai. We will show that there exists a β0 > 0 such that for β > β0 there exist at least two limit Gibbs distribution which leads to the phenomena of phase transition