Physica A Statistical Mechanics and its Applications

Exact solutions of the 1D Ising chain under periodic modulations: Fixed spins and periodic fields

Malek Telfah, Abdalla A. Obeidat

Published Apr 30, 2026• 2 citations

Publisher

DOI: 10.1016/j.physa.2026.131611

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Physica A Statistical Mechanics and its Applications

Exact solutions of the 1D Ising chain under periodic modulations: Fixed spins and periodic fields

Malek Telfah, Abdalla A. Obeidat

Published Apr 30, 2026• 2 citations

Publisher

DOI: 10.1016/j.physa.2026.131611

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Jan 1, 20250 citations

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Exact Analytical Results for the 1D Ising Chain with Periodic Impurity Fields

Telfah, Malek, Obeidat, Abdalla

We present an exact analytical solution for the one-dimensional Ising model in the presence of an external magnetic field applied periodically to every $k$ -th site. The problem is handled using the symmetrized transfer matrix approach, we derive a compact closed-form expression for the system's eigenvalues for arbitrary period $k$ . From the resulting free energy, we obtain exact expressions for the magnetization and zero-field susceptibility. Explicit results are presented for $k = 1$ , $k = 2$ , and $k = 3$ which is considered a novel result. We further analyze the spin-spin correlation functions, deriving the correlation length and the set of position-dependent correlation strength prefactors, $A_{ij}$ . The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as $χ\sim β/ k$ for large $k$ , offering insights into diluted magnetic systems and serving as a benchmark for quasiperiodic modulations. The correlation strengths exhibit a strong anisotropy, revealing a complex, non-local structure of spin fluctuations. These results provide a complete and rigorous benchmark for understanding the effects of periodic modulation in 1D systems.We present an exact analytical solution for the one-dimensional Ising model in the presence of an external magnetic field applied periodically to every $k$-th site. The problem is handled using the symmetrized transfer matrix approach, we derive a compact closed-form expression for the system's eigenvalues for arbitrary period $k$. From the resulting free energy, we obtain exact expressions for the magnetization and zero-field susceptibility. Explicit results are presented for $k = 1$, $k = 2$, and $k = 3$ which is considered a novel result. We further analyze the spin-spin correlation functions, deriving the correlation length and the set of position-dependent correlation strength prefactors, $A_{ij}$. The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as $χ\sim β/ k$ for large $k$, offering insights into diluted magnetic systems and serving as a benchmark for quasiperiodic modulations. The correlation strengths exhibit a strong anisotropy, revealing a complex, non-local structure of spin fluctuations. These results provide a complete and rigorous benchmark for understanding the effects of periodic modulation in 1D systems.

arXiv:2511.15399Nov 19, 20250 citations

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

Tuncer Kaya

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 $^{2}=$ . A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that $=\tanh{K}$ , which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of $\{σ_{i}σ_{i+1}\}=\{s_{i}\}$ and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that $\{σ_{0}σ_{j}\}=(\tanh{K})^{j}$ , 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 $=(\tanh{K})^{N/2}$ , 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.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 $^{2}=$. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that $=\tanh{K}$, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of $\{σ_{i}σ_{i+1}\}=\{s_{i}\}$ and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that $\{σ_{0}σ_{j}\}=(\tanh{K})^{j}$, 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 $=(\tanh{K})^{N/2}$, 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.

arXiv:1801.04452Jan 13, 20180 citations

The exact solution of the Ising quantum chain with alternating single and sector defects

Degang Zhang, Bo‐Zang Li, Yun Li

Physics Letters AMar 1, 19943 citations

We present an exact analytical solution for the one-dimensional Ising model in the presence of an external magnetic field applied periodically to every

k

-th site. The problem is handled using the symmetrized transfer matrix approach, we derive a compact closed-form expression for the system's eigenvalues for arbitrary period

k

. From the resulting free energy, we obtain exact expressions for the magnetization and zero-field susceptibility. Explicit results are presented for

k = 1

k = 2

, and

k = 3

which is considered a novel result. We further analyze the spin-spin correlation functions, deriving the correlation length and the set of position-dependent correlation strength prefactors,

A_{ij}

. The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as

χ\sim β/ k

for large

k

, offering insights into diluted magnetic systems and serving as a benchmark for quasiperiodic modulations. The correlation strengths exhibit a strong anisotropy, revealing a complex, non-local structure of spin fluctuations. These results provide a complete and rigorous benchmark for understanding the effects of periodic modulation in 1D systems.

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

H=0

^{2}=

. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that

=\tanh{K}

, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of

\{σ_{i}σ_{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

\{σ_{0}σ_{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

=(\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.