Quadratic fermionic interactions yield Hamiltonians with large ground-state energy gaps

Polynomially large ground-state energy gaps are rare in many-body quantum systems, but useful in quantum information and an interesting feature of the one-dimensional quantum Ising model. We show analytically that the gap is generically polynomially large not just for the quantum Ising model, but for one-, two-, and three-dimensional interaction lattices and Hamiltonians with certain random interactions. We extend the analysis to Hamiltonian evolutions and we use the Jordan-Wigner transformation and a related transformation for spin-3/2 particles to show that our results can be restated using spin operators in a surprisingly simple manner. These results also yield a new perspective on the one-dimensional cluster state.