  In this paper, combined with infinite time-evolving block decimation (iTEBD)
algorithm and Bell-type inequalities, we investigate multi-partite quantum
nonlocality in an infinite one-dimensional quantum spin-1/2 XXZ system. High
hierarchy of multipartite nonlocality can be observed in the gapless phase of
the model, meanwhile only the lowest hierarchy of multipartite nonlocality is
observed in most regions of the gapped anti-ferromagnetic phase. Thereby,
Bell-type inequalities disclose different correlation structures in the two
phases of the system. Furthermore, at the infinite-order QPT (or
Kosterlitz-Thouless QPT) point of the model, the correlation measures always
show a local minimum value, regardless of the length of the subchains. It
indicates that relatively low hierarchy of multi-partite nonlocality would be
observed at the infinite-order QPT point in a Bell-type experiment. The result
is in contrast to the existing results of the second-order QPT in the
one-dimensional XY model, where multi-partite nonlocality with the hierarchy
has been observed. Thus, multi-partite nonlocality provides us an alternative
perspective to distinguish between these two kinds of QPTs. Reliable clues for
the existence of tripartite quantum entanglement have also been found.
