Are there ever circumstances when exponential decay of correlations implies gappedness of the Hamiltonian?

Question

There are by now a set of well-known results showing that if a Hamiltonian has a gap, then this implies exponential decay of correlations between spatial separated observables (e.g. https://arxiv.org/abs/1206.2947).

In general, the converse cannot be true one could imagine two essentially separate, non-interacting, systems in which are considered a single system, but where the parameters are tuned so that the gap closes. Here the gap is closed but there are no correlations between the two systems.

But are there ever any assumptions we can make on a Hamiltonian where exponential decay of correlations occurs, and this implies the system is gapped? We can aprori make assumptions about the form of the terms (e.g. all spins are coupled by $O(1)$ interactions). Does it help if we assume classical/commuting/frustration free Hamiltonians?

This post imported from StackExchange Physics at 2024-09-26 17:28 (UTC), posted by SE-user Hans Schmuber

Comments

Exponential decay implies an exponential function with a dimensionless argument. Therefore, the time in the argument of the exponent must be multiplied by a dimension parameter (frequency) which is then related to a scale parameter of the theory, which it turn represents the mass gap.

This post imported from StackExchange Physics at 2024-09-26 17:29 (UTC), posted by SE-user flippiefanus