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Suppose one engineers an artificial antiferromagnet on an array of {\emisolated} wells, and then increases the tunneling between wells slowly, willthe system finally become an {\em equilibrium} antiferromagnet? Here, we showthat due to the intrinsic non-adiabaticity at the start of this process, andthat the atoms in the initial state in different wells are completelyuncorrelated, the final equilibrium state will have a temperature $T_{f}$ farabove the Neel temperature. Constructing other strongly correlated states (withcharacteristic energy per particle $E^{\ast}$ comparable to the hopping or thevirtual hopping scale) with the same method will suffer the same problem, i.e.$T_{f}>>E^{\ast}$.

The Intrinsic Difficulties of Constructing Strongly Correlated States of  Lattice Quantum Gases by Connecting Up Pre-engineered Isolated Atomic  Clusters