Corner transfer matrix

Remarks

The partition function of $\Lambda$ is $Z = \tr(ABCD)$, with the weights on the two cutting lines shared so that each is counted once. For a function $g$ of the centre spin, the expectation is $\langle g(\sigma_0) \rangle = \tr(G \, ABCD) / \tr(ABCD)$, where $G$ is the diagonal matrix with entries $g(\sigma_0)$. For an isotropic model with a reflection symmetry, all four matrices can be taken equal, and $Z = \tr(A^4)$. For the Ising model, $S = \set{-1, +1}$ and each corner transfer matrix has size $2^{m+1} \times 2^{m+1}$, with nonnegative entries. Baxter studied models whose weights satisfy the star–triangle relation. For them the corner transfer matrices become simple as $m \to \infty$. Suitably normalized, their eigenvalues are integer powers of one number that depends on the spectral parameter. Non-example: the row transfer matrix of the same model is not a corner transfer matrix. It is indexed by the spins of whole rows and builds the region row by row, not quadrant by quadrant.