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Journal of Physics A Mathematical and General

Statistical mechanics of the 1D ferromagnetic ANNNI chain under an external field: revisited

N Bhattacharyya, Subinay Dasgupta

Published Aug 21, 1991• 3 citations
Publisher

Abstract

Presents an analysis of the exact solution for the statistical mechanics of the one-dimensional ferromagnetic ANNNI chain under an external magnetic field by the transfer matrix method. Expressions in closed form are derived for the free energy, magnetic moment, disorder line, susceptibility and the temperature and field derivatives of susceptibility. The variation of the wavevector and correlation length with the external field shows interesting features.

DOI
10.1088/0305-4470/24/16/027

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Presents an analysis of the exact solution for the statistical mechanics of the one-dimensional ferromagnetic ANNNI chain under an external magnetic field by the transfer matrix method. Expressions in closed form are derived for the free energy, magnetic moment, disorder line, susceptibility and the temperature and field derivatives of susceptibility. The variation of the wavevector and correlation length with the external field shows interesting features.
The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.
We present an exact analytical solution for the one-dimensional Ising model in the presence of an external magnetic field applied periodically to every kkk-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 kkk. From the resulting free energy, we obtain exact expressions for the magnetization and zero-field susceptibility. Explicit results are presented for k=1k = 1k=1, k=2k = 2k=2, and k=3k = 3k=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, AijA_{ij}Aij​. The framework highlights how impurity spacing suppresses thermodynamic responses, with susceptibility scaling as χ∼β/kχ\sim β/ kχ∼β/k for large kkk, 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.