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

Abstract

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 $\chi \sim \beta / k$ 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.