Heisenberg antiferromagnet on the Husimi lattice

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 $S=1\mathrm{/}2$, 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 $S=3\mathrm{/}2$, and indeed for all spin values $S\ge 5\mathrm{/}2$, 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 $m = 1/3$.