  We perform a systematic study of the antiferromagnetic Heisenberg model on
the Husimi lattice using numerical tensor-network methods based on Projected
Entangled Simplex States (PESS). The nature of the ground state varies strongly
with the spin quantum number, $S$. For $S = 1/2$, it is an algebraic (gapless)
quantum spin liquid. For $S = 1$, it is a gapped, non-magnetic state with
spontaneous breaking of triangle symmetry (a trimerized simplex-solid state).
For $S = 2$, it is a simplex-solid state with a spin gap and no
symmetry-breaking; both integer-spin simplex-solid states are characterized by
specific degeneracies in the entanglement spectrum. For $S = 3/2$, and indeed
for all spin values $S \ge 5/2$, the ground states have $120$-degree
antiferromagnetic order. In a finite magnetic field, we find that, irrespective
of the value of $S$, there is always a plateau in the magnetization at $m =
1/3$.
