Introduced by Rodney Baxter in 1968 as an extension of the Kramers-Wannier row-to-row transfer matrix, it provides a powerful method of studying lattice models.
The Kronecker delta in the expression ensures that σ1 = σ'1, so by ordering the configurations appropriately we may cast A as a block diagonal matrix: Corner transfer matrices are related to the partition function in a simple way.
This recursion relation allows, in principle, the iterative calculation of the corner transfer matrix for any lattice quadrant of finite size.
When using corner transfer matrices to perform calculations, it is both analytically and numerically convenient to work with their diagonal forms instead.
For the full lattice given earlier, the spin expectation value at the central site is given by With the configurations ordered such that A is block diagonal as before, we may define a 2m×2m diagonal matrix such that Another important quantity for lattice models is the partition function per site, evaluated in the thermodynamic limit and written as In our example, this reduces to since tr Ad4 is a convergent sum as m → ∞ and Ad becomes infinite-dimensional.
In other words, the partition function per site is given exactly by the diagonalised recursion relation for corner transfer matrices in the thermodynamic limit; this allows κ to be approximated via the iterative process of calculating Ad for a large lattice.