Analyticity of the one-dimensional Ising free energy

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.