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,
Aij. The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as
χ∼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 calculate exactly the thermodynamics of ensembles of magnetic impurities with randomly distributed host-impurity couplings in the critical region of the uniaxial spin-1/2 quantum chain. We derive a finite set of integral equations describing the complete temperature and magnetic field dependence of the system. Exact numerical results for arbitrary values of temperature and external magnetic field are obtained. We show that for strong disorder the quenching of the impurity moments is absent. For weak disorder the screening persists, but with a critical non-Fermi-liquid behaviour of the magnetic susceptibility and specific heat. We point out that the `easy-plane' magnetic anisotropy does not qualitatively change the behaviour of the thermodynamic characteristics of the disordered impurities in the antiferromagnetic chain. On the other hand, anisotropy of ferromagnetic type can enhance the tendency to quenching of the impurities for strong disorder.