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

TL;DR

We solve the 1D Ising chain with a periodic external field of period $k$ using a symmetrized transfer-matrix method, obtaining exact eigenvalues $λ_k^{±}$ and closed-form thermodynamics. The magnetization and zero-field susceptibility are derived from these eigenvalues, with explicit results for $k=1,2$, and novel expressions for $k=3$. The correlation function exhibits an exponential decay with a correlation length $ξ_k = \frac{k}{\ln(λ_k^+/λ_k^-)}$ and a rich, anisotropic set of correlation-strength prefactors $A_{ij}$, showing non-local spin-fluctuation pathways. The periodic modulation systematically suppresses magnetic response, and in the large-$k$ limit the susceptibility scales as $χ_k(0) \sim \frac{β}{k}$, providing a rigorous benchmark for periodic and quasi-periodic modulation in 1D spin systems and potential relevance to engineered quantum materials.

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 $χ\sim β/ k$ 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.

Figures (5)

• Figure 1: Magnetization $m$ as a function of field $h$ at $\beta J = 1.5$, for $k=1,2,3$.
• Figure 2: Zero-field susceptibility $\chi(0)$ as a function of temperature $T$, for $k=1,2,3$.
• Figure 3: Correlation length $\xi_k$ vs. Temperature ($T$ in $J/k_B$) for $h=0.5J$. This shows the characteristic peak for $h>0$.
• Figure 4: Correlation length $\xi_k$ vs. Field Strength ($h/J$) at $T=1.5 J/k_B$. For $h>0$, correlations persist over longer distances for larger $k$.
• Figure 5: The full matrix of correlation strengths $|A_{ij}|$ vs. Temperature for $k=3$, at fixed field strengths. All prefactors exhibit a peak, but their magnitudes vary significantly, demonstrating strong anisotropy.