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We elaborate a simple classification scheme of all rank-5 SU(2)-spinrotational 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 acomplete list of all SU(2)-symmetric translationally and rotationally-invariantProjected Entangled Pair States (PEPS) with bond dimension $D\leqslant 6$. Allknown SU(2)-symmetric PEPS on the square lattice are recovered and simplegeneralizations are provided in some cases. More generally, to each of oursymmetry class can be associated a $({\cal D}-1)$-dimensional manifold of spinliquids (potentially) preserving lattice symmetries and defined in terms of${\cal D}$ 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) timereversal symmetry (chiral spin liquids) or (iii) SU(2)-spin rotation symmetrydown to $U(1)$ (spin nematics or N\'eel antiferromagnets) can also beconstructed. We apply this framework to search for new topological chiral spinliquids characterized by well-defined chiral edge modes, as revealed by theirentanglement spectrum. In particular, we show how the symmetrization of adouble-layer PEPS leads to a chiral topological state with a gapless edgedescribed by a SU(2)$_2$ Wess-Zumino-Witten model.

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