= 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 $k_BT$. 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.
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