Magnetic susceptibility of the square lattice Ising model

In this work, we obtained an analytical relation for the susceptibility of the square lattice Ising model. Our investigation is based on an average magnetization interrelation which was recently obtained by us. To proceed further, we have to make a mathematical conjecture about the three-site correlation function appearing in the average magnetization interrelation. We presented the conjectured mathematical form of the three spin correlation function with the relation, ⟨σ1σ2σ3⟩=a(K,H)⟨σ⟩+[1-a(K,H)]⟨σ⟩(1+β-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \sigma _{1}\sigma _{2}\sigma _{3}\rangle =a(K,H)\langle \sigma \rangle +[1-a(K,H)]\langle \sigma \rangle ^{(1+\beta ^{-1})}$$\end{document}. Here, β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} denotes the critical exponent for the average magnetization and a(K, H) is a function whose behavior will be described around the critical point with an arbitrary constant. To elucidate the relevance of the method, we have first calculated the susceptibility of the 1D chain as an example, and the obtained susceptibility expression for the 1D chain is equivalent to the result of the susceptibility obtained by the conventional method. Applying the same method, we obtained the values of the magnetic critical exponent γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} of the square lattice Ising model. The values of γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} are obtained as γ=1.72\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1.72$$\end{document} for T>Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\!>\!T_{c}$$\end{document}, and γ=0.91\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.91$$\end{document} for T<Tc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\!<\!T_{c}$$\end{document}.