How fast can the spectral gap of a translation-invariant Hamiltonian close?

Consider an arbitrary local Hamiltonian defined on some lattice of size L where the local Hilbert space dimension associated with each site on the lattice is finite. If there is no constraint on the Hamiltonian besides locality, then the spectral gap between the ground state and the first excited can often be exponentially small in V (e.g. spin glasses) and the density of states exponentially large in V. The intuition is that different terms in the Hamiltonian with random signs create a complex energy landscape with many metastable states. However, such random signs are forbidden when translation invariance is imposed in addition. So one naive guess would be that the spectral gap cannot close exponentially fast and the density of states near the ground state energy can at most be a polynomial in V.

Is this guess correct? If not, would a weaker statement (Hamiltonians with exponentially large density of states near the ground state form a set of measure zero in the space of all translation-invariant local Hamiltonians) still be true?

note added: there are examples like the ferromagnetic Ising model or the toric code where the ground state and the first excited state are separated by an exponentially small gap (due to symmetry breaking order or topological order). But these models do not have a continuum density of states near the ground state energy.

• The 1D transverse field Ising model has an exponentially small gap in the ferromagnetic phase. (Admittedly, those are both ground states in the thermodynamic limit, but you will have to be more precise.) Jul 5 at 21:59
• Thanks for the comment. I have added a note to clarify this point. Jul 6 at 17:56
• If you want a continuous density of states there is natural limits on the gap, pretty much: After all, if the states are roughly equally spaced, since there are 2^N states they should have a spacing ~ 1/2^N. So for any super-small gap initially you pay with a larger gap later. Jul 6 at 19:09
• The density of states at low energy can be exponentially large in N but not 2^N (another way to say that is the entropy density at zero temperature is nonzero). This happens for example in spin glass models/SYK. But I want to ask if this is possible in local translation-invariant systems. Jul 7 at 4:09