Local magnetic field distributions: Two-dimensional Ising models

We show that the statistical mechanics of Ising models can be conveniently reformulated in terms of the local magnetic field probability distribution function $P\left(h\right)\mathrm{}$. It is shown that for arbitrary exchange interactions $J_{\mathrm{ij}}$, which may or may not be random, both thermodynamic quantities such as magnetization, specific heat, etc., and the neutron scattering law $S(\overrightarrow{k},\omega )$ can be obtained from $P\left(h\right)\mathrm{}$. Indeed $S(\overrightarrow{k},\omega )$ provides a direct measurement of the symmetric part of $P\left(h\right)\mathrm{}$ which also determines the energy, specific heat, etc., while the magnetization can be obtained from the antisymmetric part of $P\left(h\right)\mathrm{}$. As an example, specific results for $P\left(h\right)\mathrm{}$ are presented for the honeycomb, square, and triangular lattices with constant nearest-neighbor interactions. All three lattices exhibit a pronounced dip in the center of $P\left(h\right)\mathrm{}$ at the transition temperature.