  Using the infinite Projected Entangled Pair States (iPEPS) algorithm, we
study the ground-state properties of the spin-$1/2$ quantum Heisenberg
antiferromagnet on the star lattice in the thermodynamic limit. By analyzing
the ground-state energy of the two inequivalent bonds of the lattice in
different unit-cell structures, we identify two competing Valence-Bond-Solid
(VBS) phases for different antiferromagnetic Heisenberg exchange couplings.
More precisely, we observe (i) a VBS state which respects the full symmetries
of the Hamiltonian, and (ii) a resonating VBS state which, in contrast to
previous predictions, has a six-site unit-cell order and breaks $C_3$ symmetry.
We also studied the ground-state phase diagram by measuring the ground-state
fidelity and energy derivatives, and further confirmed the continuous nature of
the quantum phase transition in the system. Moreover, an analysis of the
isotropic point shows that its ground state is also a VBS as in (i), which is
as well in contrast with previous predictions.
