  We use a tensor network strong-disorder renormalization group (tSDRG) method
to study spin-1 random Heisenberg antiferromagnetic chains. The ground state of
the clean spin-1 Heisenberg chain with uniform nearest-neighbor couplings is a
gapped phase known as the Haldane phase. Here we consider disordered chains
with random couplings, in which the Haldane gap closes in the strong disorder
regime. As the randomness strength is increased further and exceeds a certain
threshold, the random chain undergoes a phase transition to a critical
random-singlet phase. The strong-disorder renormalization group method
formulated in terms of a tree tensor network provides an efficient tool for
exploring ground-state properties of disordered quantum many-body systems.
Using this method we detect the quantum critical point between the gapless
Haldane phase and the random-singlet phase via the disorder-averaged string
order parameter. We determine the critical exponents related to the average
string order parameter, the average end-to-end correlation function and the
average bulk spin-spin correlation function, both at the critical point and in
the random-singlet phase. Furthermore, we study energy-length scaling
properties through the distribution of energy gaps for a finite chain. Our
results are in closer agreement with the theoretical predictions than what was
found in previous numerical studies. As a benchmark, a comparison between tSDRG
results for the average spin correlations of the spin-1/2 random Heisenberg
chain with those obtained by using unbiased zero-temperature QMC method is also
provided.
