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. The nature of the ground state varies strongly with the spin quantum number $S$. For $S=\frac{1}{2}$, it is an algebraic (gapless) quantum spin liquid. For $S=1$, it is a gapped, nonmagnetic 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 $S=\frac{3}{2}$, and indeed for all spin values $S\ensuremath{\ge}\frac{5}{2}$, the ground states have ${120}^{\ensuremath{\circ}}$ 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 $m=\frac{1}{3}$.