  The ground state phase diagrams and thermodynamic properties of a spin-1
bond-alternating Heisenberg antiferromagnetic chain with a single-ion
anisotropy in longitudinal and transverse magnetic fields are investigated
jointly by means of the infinite time evolving block decimation, the linearized
tensor renormalization group and the density matrix renormalization group
methods. It is found that in the magnetic field-bond alternating ratio plane
six phases such as singlet dimer, Haldane, two Tomonaga-Luttinger liquid, 1/2
magnetization plateau and spin polarized phases are identified in a
longitudinal field, while in a transverse field there are fives phases
including singlet dimer, Haldane, Z$_2$ symmetry breaking, quasi-1/2
magnetization plateau and quasi spin polarized phases. A reentrant behavior of
the staggered magnetization in a transverse field is observed. The quantum
critical behaviors in longitudinal and transverse magnetic fields are disclosed
to fall into different universality classes with corresponding conformal field
central charge $c=1$ and 1/2, respectively. The experimental data of the
compound NTENP under both longitudinal and transverse magnetic fields are
nicely fitted, and the Luttinger liquid behavior of low temperature specific
heat experimentally observed is also confirmed.
