Non-analyticity of the Correlation Length in Systems with Exponentially Decaying Interactions

We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $$\mathbb {Z}^d$$ Z d with long-range interactions of the form $$J_x = \psi (x) e^{-|x|}$$ J x = ψ ( x ) e - | x | , where $$\psi (x) = e^{{\mathsf o}(|x|)}$$ ψ ( x ) = e o ( | x | ) and $$|\cdot |$$ | · | is an arbitrary norm. We characterize explicitly the prefactors $$\psi $$ ψ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $$\beta $$ β , magnetic field $$h$$ h , etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $$\psi $$ ψ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein–Zernike theory of correlations.