Corner Transfer Matrix - Definition

Definition

Consider an IRF (interaction-round-a-face) model, i.e. a square lattice model with a spin σi assigned to each site i and interactions limited to spins around a common face. Let the total energy be given by

where for each face the surrounding sites i, j, k and l are arranged as follows:



For a lattice with N sites, the partition function is

where the sum is over all possible spin configurations and w is the Boltzmann weight


To simplify the notation, we use a ferromagnetic Ising-type lattice where each spin has the value +1 or −1, and the ground state is given by all spins up (i.e. the total energy is minimised when all spins on the lattice have the value +1). We also assume the lattice has 4-fold rotational symmetry (up to boundary conditions) and is reflection-invariant. These simplifying assumptions are not crucial, and extending the definition to the general case is relatively straightforward.

Now consider the lattice quadrant shown below:



The outer boundary sites, marked by triangles, are assigned their ground state spins (+1 in this case). The sites marked by open circles form the inner boundaries of the quadrant; their associated spin sets are labelled {σ1,…,σm} and {σ'1,…,σ'm}, where σ1 = σ'1. There are 2m possible configurations for each inner boundary, so we define a 2m×2m matrix entry-wise by


The matrix A, then, is the corner transfer matrix for the given lattice quadrant. Since the outer boundary spins are fixed and the sum is over all interior spins, each entry of A is a function of the inner boundary spins. 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:

\begin{array}{cccc} & & \begin{array}{ccccc} \sigma_{1}'=+1 & & & & \sigma_{1}'=-1\end{array}\\ A & = & \left[\begin{array}{ccccccc} & & & |\\ & A_{+} & & | & & 0\\ & & & |\\ - & - & - & | & - & - & -\\ & & & |\\ & 0 & & | & & A_{-}\\ & & & |\end{array}\right] & \begin{array}{c} \sigma_{1}=+1\\ \\\\\\\sigma_{1}=-1\end{array}\end{array}


Corner transfer matrices are related to the partition function in a simple way. In our simplified example, we construct the full lattice from four rotated copies of the lattice quadrant, where the inner boundary spin sets σ, σ', σ" and σ'" are allowed to differ:


The partition function is then written in terms of the corner transfer matrix A as

Read more about this topic:  Corner Transfer Matrix

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