Regularly spaced blocks of impurities in the Ising model: Critical temperature and specific heat

We consider the thermodynamics of a square Ising lattice which is constructed from unit cells of height $n$ lattice spacings and width $m$ lattice spacings. In a particular unit cell the values of the $2mn$ exchange energies can be chosen at will. Once specified this basic unit cell is repeated to make up the entire planar lattice. For arbitrary $n$ and for $m=2\mathrm{}$ and $m=3\mathrm{}$ we obtain the exact equations for the critical point. Furthermore, we show that when $n$ is finite and $m=1\mathrm{}$ or $m=2\mathrm{}$ the specific heat has a singularity of the form $A\ln |1-\frac{T}{T_c}|$ and we derive an explicit formula for $A$. Our method can be extended to values of $m>\mathrm{}2\mathrm{}$. A numerical evaluation of our results shows that the average of $A$ over impurity configurations is smaller for $m=1\mathrm{}$ than for $m=2\mathrm{}$.