  We investigate the quantum phase transitions of spin systems in one and two
dimensions by employing trace distance and multipartite entanglement along with
real-space quantum renormalization group method. As illustration examples, a
one-dimensional and a two-dimensional $XY$ models are considered. It is shown
that the quantum phase transitions of these spin-chain systems can be revealed
by the singular behaviors of the first derivatives of renormalized trace
distance and multipartite entanglement in the thermodynamics limit. Moreover,
we find the renormalized trace distance and multipartite entanglement obey
certain universal exponential-type scaling laws in the vicinity of the quantum
critical points.
