Non-Symmetry-Breaking Ground State of the S=1 Heisenberg Model on the Kagome lattice

The $S=1$ antiferromagnetic Heisenberg model on a Kagome lattice is studied using the density-matrix renormalization group method. To identify the ground state, we take four kinds of clusters into account; periodic, cylindrical, and two open ones. The hexagonal singlet solid (HSS) and triangular valence bond solid (TVBS) states are artificially generated by modulating edge shapes of the open clusters. We find that the energy par sites of the HSS state is $e_0=E_0/N =-1.41095$, which is readily lower than that of the TVBS state ($e_0=-1.3912 \pm 0.0025$). This agrees well with those of the cylindrical ($e_0=-1.40988$) and periodic ($e_0=-1.409 \pm 0.005$) clusters, where no assumption as to the ordering is posed. Thus, we conclude that the HSS picture is consistent to describe the ground state of the $S=1$ Kagome Heisenberg model. This is further confirmed by finding non-symmetry-breaking state in the calculations of the dimer-dimer correlation functions as well as the entanglement entropy of cylindrical clusters. Finally, we estimate the single-triplet energy gap: The HSS ground state has $\Delta=0.1919$, while the TVBS excited state has a larger one $\Delta=0.2797$.