Transfer matrix method

Remarks

If $\lambda_1 \ge \abs{\lambda_2} \ge \dots$ are the eigenvalues of $T$, then $Z_N = \sum_k \lambda_k^N$. Since $T$ has positive entries, the Perron–Frobenius theorem makes $\lambda_1$ simple and strictly larger than the other moduli, so $-\beta f = \lim_N \frac{1}{N} \log Z_N = \log \lambda_1$. Connected correlations typically decay like $(\abs{\lambda_2}/\lambda_1)^r$ at distance $r$, so the correlation length is $1/\log(\lambda_1/\abs{\lambda_2})$. Example: the Ising chain with $E(a, b) = -J a b - h (a + b)/2$ on $S = \set{-1, +1}$ has $$T = \begin{pmatrix} e^{\beta (J + h)} & e^{-\beta J} \\ e^{-\beta J} & e^{\beta (J - h)} \end{pmatrix},$$ and for $h = 0$ its eigenvalues are $2 \cosh \beta J$ and $2 \sinh \beta J$. Non-example: a chain in which each site interacts with all other sites is not covered. Its energy is not a sum of terms in neighbouring pairs, and one would need states of unbounded size. In two dimensions one takes $S$ to be the set of configurations of a whole row, which turns the method into a large eigenvalue problem.