  In this letter, we analyse the following apparent paradox: As has been
recently proved by Hastings (cond-mat/0305505), under a general set of
conditions, if a local Hamiltonian has a spectral gap above its (unique) ground
state (GS), all connected equal-time correlation functions of local operators
decay exponentially with distance. On the other hand, statistical mechanics
provides us with examples of 3D models displaying so-called sliding phases
(O'Hern et al., cond-mat/9904415) which are characterised by the algebraic
decay of correlations within 2D layers and exponential decay in the third
direction. Interpreting this third direction as time would imply a gap in the
corresponding (2+1)D quantum Hamiltonian which would seemingly contradict
Hastings' theorem. The resolution of this paradox lies in the non-locality of
such a quantum Hamiltonian.
