No sliding in time

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.