# Transfer matrix

1. Jul 31, 2014

### LagrangeEuler

You can use transfer matrix method for 2d Ising model. Is there a list of models for which this method could be of use. Such as classical Heisenberg perhaps? Or Potts?

2. Aug 1, 2014

### t!m

For a 1D Ising model, the transfer matrix is $2\times 2$. For a 2D Ising model on an $N\times N$ lattice, the transfer matrix is $2^N\times 2^N$, and the thermodynamic limit requires infinitely large matrices, $N \rightarrow \infty$! Nonetheless, this can be done and used to derive the exact solution of the 2D Ising model. Most books on statistical physics and critical phenomena should cover this. For a generalized (Potts-like) model with $m$ possible states for each lattice site, the transfer matrix will be $m^N \times m^N$.

EDIT: I mis-read your first sentence as a question. All of the above above applies to a Potts-like model with discrete degrees of freedom. For something like the Heisenberg model (or 2D Potts/XY model), where each site can take an infinite number of values $m\rightarrow \infty$, I imagine a transfer matrix solution would become intractable.

Last edited: Aug 1, 2014