Magnetic susceptibility of the square lattice Ising model

In this work, the susceptibility of the square lattice Ising model is investigated using the recently obtained average magnetization interrelation, which is given by $\langle\sigma_{0, i}\rangle= \langle\tanh[K(\sigma_{1,i}+\sigma_{2,i}+\dots +\sigma_{z,i})+H]\rangle$. Here, $z$ is the number of nearest neighbors, $\sigma_{0,i}$ denotes the central spin at the $i^{th}$ site while $\sigma_{l,i}$, $l=1,2,\dots,z$, are the nearest neighbor spins around the central spin, $K=J/(k_{B}T)$, where $J$ is the nearest neighbor coupling constant, $k_{B}$ is the Boltzmann's constant and $T$ is the temperature of the system. In our investigation, inevitably we have to make a conjecture about the three-site correlation function appearing in the obtained relation of this paper. The conjectured form of the the three spin correlation function is given by the relation, $\langle\sigma_{1}\sigma_{2}\sigma_{3}\rangle=a(K,H)\langle\sigma\rangle+[1-a(K,H)]\langle\sigma\rangle^{(1+\beta^{-1})}$. Here $\beta$ 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 used in this paper, we have first calculated the susceptibility of the 1D chain as an example, and the obtained susceptibility expression is seen as equivalent to the result of the susceptibility of the conventional method. The magnetic critical exponent $\gamma$ of the square lattice Ising model is obtained as $\gamma=1.72$ for $T\!>\!T_{c}$, and $\gamma=0.91$ for $T\!<\!T_{c}$.