Physical Hamiltonians are a small family of Hamiltonians if compared to all the possible Hamiltonians. In particular physical Hamiltonians consider only local-interactions, while more general non-physical Hamiltonians consider every kind of interaction. In particular in this second case you can describe more general problems. But there is any application of this kind of objects and if yes what?
If it were possible to have these non-physical Hamiltonians, then one could have "fast scramblers" as described in http://arxiv.org/abs/1111.6580 . A fast scambler is a physical system which scrambles information about the initial state of the system (so that it is no longer recoverable) in a time that is logarithmic in the number of degrees of freedom of the system. In the above paper, examples were given to show that these non-local Hamiltonians lead to fast scrambling. The goal of the paper was to find physical Hamiltonians as you describe in your question that lead to fast scrambling, but they were unable to do so. It is believed that black holes are the fastest scramblers in nature, but this conjecture remains open.
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