  String correlations are investigated in an infinite-size XXZ spin-1 chain. By
using the infinite matrix product state representation, we calculate a
long-range string order. In the XY phase, the string correlations decay within
a relatively very large lattice distance, which makes a finite-size study
difficult to verify the non-existence of the string order. Thus, in the Haldane
phase, the non-vanishing string correlations in the limit of a very large
distance allow to characterize the phase boundaries to the XY phase and the
Neel phase, which implies that the transverse long-range string order is the
order parameter for the Haldane phase. In addition, the singular behaviors of
the von Neumann entropy and the fidelity per lattice site are shown to capture
clearly the phase transition points that are consistent with the results from
the string order. The estimated critical points including a BKT transition from
the XY phase to the Haldane phase agree well with the previous results:
$\Delta_{c2} = 0$ for the XY-Haldane phase transition and $\Delta_{c3} = 1.185$
for the Haldane-N'eel phase transition from the density renormalization group.
From a finite-entanglement scaling of the von Neumann entropy with respect to
the truncation dimension, the central charges are found to be $c \simeq 1.0$ at
$\Delta_{c2} = 0$ and $c \simeq 0.5$ at $\Delta_{c3} = 1.185$, respectively,
which shows that the XY-Haldane phase transition at $\Delta_{c2} = 0$ belongs
to the Heisenberg universality class, while the Haldane-Neel phase transition
at $\Delta_{c2} = 1.185$ belongs to the two-dimensional classical Ising
universality class. It is also shown that, the long-range order parameters and
the von Neumann entropy, as well as the fidelity per site approach, can be
applied to characterize quantum phase transitions as a universal phase
transition indicator for one-dimensional lattice many-body systems.
