Connection of edge states to bulk topological invariance in a quantum spin Hall state

We propose a topological understanding of general characteristics of edge states in a quantum spin Hall phase without considering any symmetries. It follows from the requirement of gauge invariance that either the energy gap or the gap in the spectrum of the projected spin operator needs to close on the sample edges. Based upon the Kane-Mele model with a uniform Zeeman field and a smooth confining potential near the sample boundaries, we demonstrate the existence of gapless edge states in the absence of time-reversal symmetry and their robust properties against impurities. These gapless edge states are protected by the band topology alone, rather than any symmetries.