An exact alternative solution method of 1D Ising model with Block-spin transformation at $H=0$

TL;DR

This paper presents an exact solution for the 1D Ising model using block-spin transformations, deriving correlation functions without boundary conditions, and confirming results consistent with the transfer matrix method.

Contribution

It introduces a novel exact explicit solution for the 1D Ising chain via block-spin transformations, avoiding boundary conditions and providing new correlation relations.

Findings

Correlation functions match transfer matrix results

Magnetization relation derived as <σ>^2 = <σ₀σ_N>

Second order phase transition only at infinite coupling K

Abstract

An alternative exact explicit solution of 1D Ising chain is presented without using any boundary conditions (or free boundary condition) by the help of applying successively block-spin transformation. Exact relation are obtained between spin-spin correlation functions in the absence of external field. To evaluate average magnetization (or the order parameter), it is assumed that the average magnetization can be related to infinitely apart two spin correlation function as $<\sigma>^{2}=<\sigma_{0}\sigma_{N}>$. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that $<\sigma_{0}\sigma_{1}>=\tanh{K}$, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of $\{\sigma_{i}\sigma_{i+1}\}=\{s_{i}\}$ and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that $\{\sigma_{0}\sigma_{j}\}=(\tanh{K})^{j}$, here $j$ assumes the values of $j=2^n$, here $n$ is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of $N\rightarrow\infty$ to $<\sigma_{i}>=(\tanh{K})^{N/2}$, indicating that the second order phase transition is only possible in the limit of $K\rightarrow\infty$. The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.