Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry

We elaborate a simple classification scheme of all rank-5 SU(2)-spin rotational symmetric tensors according to i) the on-site physical spin-$S$, (ii) the local Hilbert space $V^{\otimes 4}$ of the four virtual (composite) spins attached to each site and (iii) the irreducible representations of the $C_{4v}$ point group of the square lattice. We apply our scheme to draw a complete list of all SU(2)-symmetric translationally and rotationally-invariant Projected Entangled Pair States (PEPS) with bond dimension $D⩽6$. All known SU(2)-symmetric PEPS on the square lattice are recovered and simple generalizations are provided in some cases. More generally, to each of our symmetry class can be associated a $(\mathcal{D}-1)$-dimensional manifold of spin liquids (potentially) preserving lattice symmetries and defined in terms of $\mathcal{D}$ independent tensors of a given bond dimension $D$. In addition, generic (low-dimensional) families of PEPS explicitly breaking either (i) particular point-group lattice symmetries (lattice nematics) or (ii) time reversal symmetry (chiral spin liquids) or (iii) SU(2)-spin rotation symmetry down to $U(1)$ (spin nematics or N'eel antiferromagnets) can also be constructed. We apply this framework to search for new topological chiral spin liquids characterized by well-defined chiral edge modes, as revealed by their entanglement spectrum. In particular, we show how the symmetrization of a double-layer PEPS leads to a chiral topological state with a gapless edge described by a SU(2)${}_2$ Wess-Zumino-Witten model.