AbstractWe 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.