Checkerboard decimation of the square-lattice Ising model ($x+y\equiv 0(\mathrm{mod}2)\text{retained}$)

Checkerboard spin decimation retains one parity sublattice of a square-lattice Ising model and sums over the other. The retained primitive vectors are $a(1,1)$ and $a(1,-1)$, so the coarse lattice is again square with spacing $\sqrt{2}a$. A rotation and length rescaling restore the original geometry. Every original nearest-neighbour bond joins the two parity classes, whereas a diagonal next-nearest-neighbour bond stays within one class.

For dimensionless nearest-neighbour coupling $K$ and diagonal coupling $L$, the second cumulant in checkerboard decimation of the square-lattice Ising model generates $K^2$ for each pair of retained neighbours of an eliminated centre. A retained diagonal pair shares two centres, and an axial pair at distance $2a$ shares one. The displayed two-coupling recursion neglects $K^4$, $K^{2}L$, and $L^2$, treating $L$ as order $K^2$. Its finite nontrivial renormalization-group fixed point is $(K,L)=(1/3,1/9)$. Higher orders generate additional operators, so the closure is a property of the truncation.