  We study correlations in fermionic lattice systems with long-range
interactions in thermal equilibrium. We prove a bound on the correlation decay
between anti-commuting operators and generalize a long-range Lieb-Robinson type
bound. Our results show that in these systems of spatial dimension $D$ with,
not necessarily translation invariant, two-site interactions decaying
algebraically with the distance with an exponent $\alpha \geq 2\,D$,
correlations between such operators decay at least algebraically with an
exponent arbitrarily close to $\alpha$ at any non-zero temperature. Our bound
is asymptotically tight, which we demonstrate by a high temperature expansion
and by numerically analyzing density-density correlations in the 1D quadratic
(free, exactly solvable) Kitaev chain with long-range pairing.
