Spectral gaps of frustration-free spin systems with boundary

Abstract: In quantum many-body systems, the existence of a spectral gap above the ground state has far-reaching consequences. In this paper, we discuss "finite-size" criteria for having a spectral gap in frustration-free spin systems and their applications. We extend a criterion that was originally developed for periodic systems by Knabe and Gosset-Mozgunov to systems with a boundary. Our finite-size criterion says that if the spectral gaps at linear system size $n$ exceed an explicit threshold of order $n^{-3/2}$, then the whole system is gapped. The criterion takes into account both "bulk gaps" and "edge gaps" of the finite system in a precise way. The $n^{-3/2}$ scaling is robust: it holds in 1D and 2D systems, on arbitrary lattices and with arbitrary finite-range interactions. One application of our results is to give a rigorous foundation to the folklore that 2D frustration-free models cannot host chiral edge modes (whose finite-size spectral gap would scale like $n^{-1}$).