I having difficulties in understanding "transfer matrix" in the paper Metastability in the two-dimensional Ising model.
They consider a periodic $N \times \infty$ lattice with the energy
$$ E = -J \sum_{nn} \sigma \sigma – H \sum \sigma $$
for spins $\sigma = \pm 1$, where the first summation occurs over nearest neighbours.
Now they say, "<…>
The associated $2^N \times 2^N$ symmetric transfer matrix $L$ is defined as follows.
for two column configurations $\vert \mu \rangle = (\sigma_1, \cdots, \sigma_n)$ and $\vert \mu' \rangle = (\sigma_1', \cdots, \sigma_n')$
$$ \langle \mu \vert L \vert \mu' \rangle = \exp \bigg\{{ \frac{\nu}{2} \sum_{i=1}^{N} \sigma_i \sigma_{i+1} + \sigma_i' \sigma_{i+1}' + \frac{1}{2} h \sum{i=1}^{N} (\sigma_i + \sigma_i') + \nu \sum_{i=1}^N \sigma_i \sigma_i' \bigg\}} $$
where $\nu = J/T$ and $h = H/T$ <…>".
I cannot understand this definition, let alone reconcile it with the one I am used to.
First of all, the column configuration $\mu$ has $N$ components, while $L$ is a $2^N \times 2^N$ matrix, so I cannot make even basic sense of the left-hand side, what operation it represents.
For a say one-dimensional lattice model with nearest-neighbours interaction $U_{ij} = U(\sigma_i, \sigma_j)$ the transfer matrix $V$ is a $2^N \times 2^N$ matrix and has components
$$ V_{ij} = -\exp \big\{ -\frac{U_{ij}}{kT} \big\} $$
What does the first notation mean? Hopefully I will then see how it relates to the latter definition, thanks
Best Answer
There are $2^N$ possible spin configurations $|\mu\rangle= (\sigma_1,\sigma_2,\ldots, \sigma_N)$, when $\sigma_i=\pm1$. So although the sum on the RHS is only over $i=1,\ldots N$ there are $2^N$ by $2^N$ possible expressions that can be evaluated to give a matrix entry on the LHS. The matrix $L$ defined by the array of numerical entries $\langle \mu|L|\mu'\rangle$ is therefore $2^N$-by-$2^N$. Indeed, you will need to sum over all $2^N$ possible spin configurations at each intermediate level, and that is exactly what the trace of powers of the $2^N$-by-$2^N$ matrix $L$ is doing..