Transfer matrix for the one-dimensional Ising model

For nearest-neighbour interactions, a two-by-two transfer matrix assigns one Boltzmann factor to every adjacent spin pair. With periodic boundary conditions, summing all spin configurations gives the partition function as the trace of a power of this matrix.

At real field and nonzero temperature, every entry of the finite-range Ising transfer matrix is positive. Its largest eigenvalue is simple by the Perron–Frobenius theorem and depends smoothly on field and temperature. The thermodynamic free energy per site is its negative logarithm times $k_{B}T$. In the nearest-neighbour chain it is analytic and the zero-field magnetization vanishes, despite exponentially long low-temperature correlations. Finite-volume spin-inversion symmetry alone would not suffice to exclude spontaneous thermodynamic order.

For $H=-2J\sum S_i^z{S}_{i+1}^z-g{\mu }_{B}B\sum S_i^z$, with dimensionless $S_i^z=\pm 1/2$, define $K=\beta J/2$ and $b=\beta g{\mu }_{B}B/2$. The transfer matrix for the one-dimensional Ising model has largest eigenvalue $e^K[\cosh b+\sqrt{{\sinh }^{2}b+e^{-4K}}]$. Differentiating its logarithm gives the displayed moment per site. In particular the zero-field susceptibility per site is $(g{\mu }_B{)}^2{e}^{\beta J}/(4k_{B}T)$. Factors of two differ from a model whose spin variables themselves take values $\pm 1$.