  We investigate the quantum phase transition of the Su-Schrieffer-Heeger (SSH)
model by inspecting the two-site entanglements in the ground state. It is shown
that the topological phase transition of the SSH model is signified by a
nonanalyticity of local entanglement, which becomes discontinuous for finite
even system sizes, and that this nonanalyticity has a topological origin. Such
a peculiar singularity has a universal nature in one-dimensional topological
phase transitions of noninteracting fermions. We make this clearer by pointing
out that an analogous quantity in the Kitaev chain exhibiting the identical
nonanalyticity is the local electron density. As a byproduct, we show that
there exists a different type of phase transition, whereby the pattern of the
two-site entanglements undergoes a sudden change. This transition is
characterised solely by quantum information theory and does not accompany the
closure of the spectral gap. We analyse the scaling behaviours of the
entanglement in the vicinities of the transition points.
