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

TL;DR

This paper provides an exact analytical solution for the 1D Ising model with periodic impurity fields, deriving explicit formulas for thermodynamic quantities and correlation functions, and analyzing the effects of impurity spacing.

Contribution

It introduces a novel exact solution for the 1D Ising model with periodic impurities, including explicit eigenvalues, magnetization, susceptibility, and correlation functions.

Findings

Susceptibility scales as χ ∼ β / k for large k

Correlation strengths show strong anisotropy

Explicit solutions for k=1,2,3 are derived

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.