Some developments in the theory of modulated order. I. The role of fluctuations in the axial next-nearest-neighbor Ising model and the relevance of the Thouless-Anderson-Palmer equation

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Abstract
The general mean-field theory of systems with modulated order is extended so as to include nontrivial fluctuation effects through the use of the Kirkwood approximation. Application of the theory to the axial next-nearest-neighbor Ising (ANNNI) model reveals that fluctuations (1) decrease the size of the entire ordered region of the phase diagram, (2) decrease that fraction of the ordered region that is most highly modulated, and (3) suppress the reentrant parts of the modulated region that do survive. Physical reasons for this behavior are discussed. In addition, we note that our ANNNI mean-field equations turn out to be precisely the Thouless-Anderson-Palmer (TAP) equations of spin-glass theory despite a number of rather different assumptions made in deriving the TAP equations. Some of what this observation reveals about the TAP equations is discussed.