Topological edge states in the spin 1 bilinear-biquadratic model

The spin 1 bilinear-biquadratic model H = ∑(ij)[cosφS(i)·S(j) + sinφ(S(i)·S(j))(2)] on a square lattice in the region 0 < φ < π/4 is studied in a fermion representation with a p-wave pairing Bardeen-Cooper-Schrieffer type of mean-field theory. Our results show there may exist a non-trivial gapped spin liquid with time-reversal symmetry spontaneously breaking. This exotic state manifests its topological nature by forming chiral states at the edges. To show this more clearly, we set up and solved a ribbon system. We got a gapless dispersion representing the edge modes beneath the bulk modes. The edge modes with nonzero longitudinal momentum (k(x) ≠ 0) convect in opposite directions at the two edges, which leads to a twofold degeneracy, while the modes with zero longitudinal momentum (k(x) = 0) turn out to be Majorana fermion states. The edge spin correlation functions are found to decay following a power law with increasing distance. We also calculated the contribution of the edge modes to the specific heat and obtained a linear law at low temperatures.