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Simplex Z_2 spin liquids on the Kagome lattice with Entangled Pair States: spinon and vison coherence lengths, topological entropy and gapless edge modes

Gapped Z_2 spin liquids have been proposed as candidates for the ground-state of the S=1/2 quantum antiferromagnet on the Kagome lattice. We extend the use of Projected Entangled Pair States to construct (on the cylinder)Resonating Valence Bond (RVB) states including both nearest-neighbor and next-nearest neighbor singlet bonds. Our ansatz -- dubbed "simplex spin liquid" -- allows for an asymmetry between the two types of triangles (of order 2-3% in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z_2 structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB Minimally Entangled States. The spinon and vison coherence lengths are extracted from a finite size scaling w.r.t the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless suggesting the occurrence, on a cylinder with {\it real} open boundaries, of gapless edge modes formally similar to Luttinger liquid (non-chiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalisation studies.