No sliding in time

In this letter, we analyse the following apparent paradox: as has been recently proved by Hastings (2004 Phys. Rev. 69 104431), 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 1999 Phys. Rev. Lett. 83 2745) which are characterized 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. (letter to the editor)