  It is a generalized belief that there are no thermal phase transitions in
short range 1D quantum systems. However, the only known case for which this is
rigorously proven is for the particular case of finite range translational
invariant interactions. The proof was obtained by Araki in his seminal paper of
1969 as a consequence of pioneering locality estimates for the time-evolution
operator that allowed him to prove its analiticity on the whole complex plane,
when applied to a local observable. However, as for now there is no
mathematical proof of the abscence of 1D thermal phase transitions if one
allows exponential tails in the interactions. In this work we extend Araki's
result to include exponential (or faster) tails. Our main result is the
analyticity of the time-evolution operator applied on a local observable on a
suitable strip around the real line. As a consequence we obtain that thermal
states in 1D exhibit exponential decay of correlations above a threshold
temperature that decays to zero with the exponent of the interaction decay,
recovering Araki's result as a particular case. Our result however still leaves
open the possibility of 1D thermal short range phase transitions. We conclude
with an application of our result to the spectral gap problem for Projected
Entangled Pair States (PEPS) on 2D lattices, via the holographic duality due to
Cirac et al.
