Ground-state energy and spin gap of spin-$\frac{1}{2}$ Kagomé-Heisenberg antiferromagnetic clusters: Large-scale exact diagonalization results

We present a comprehensive list of the ground-state energies and the spin gaps of finite Kagomé clusters with up to 42 spins obtained using large-scale exact diagonalization techniques. This represents the current limit of this exact approach. For a fixed number of spins $N$, we study several cluster shapes under periodic boundary conditions in both directions resulting in a toroidal geometry. The clusters are characterized by their side length and diagonal as well as the shortest “Manhattan” diameter of the torii. A finite-size scaling analysis of the ground-state energy as well as the spin gap is then performed in terms of the shortest toroidal diameter as well as the shortest “Manhattan” diameter. The structure of the spin-spin correlations further supports the importance of short loops wrapping around the torii.

Figure 1

The two 42-site Kagomé clusters used in this study. The shortest Manhattan diameters, $d_M$, are shown as thick dashed lines.

Figure 2

Ground-state energy per site plotted as a function of the inverse shortest geometrical diameter. Remarkably good agreement with recent DMRG results[26] on long cylinders with the same diameter is observed. For comparison, we also display the MERA upper bound,[24] series expansion results for the Marston–Zeng valence bond crystal,[16, 17] as well as the DMRG upper bound.[26]

Figure 3

Spin gap of even and odd Kagomé samples obtained by ED and plotted as a function of the inverse shortest geometrical diameter. For comparison, recent DMRG data[26] obtained on long cylinders with the same diameter are shown.

Figure 4

Magnified spin-spin correlations in the ground state of the 30 and 36d Kagomé samples. The 30-site cluster with its low symmetry requires three distinct reference sites to be considered. The diameter of the circle is proportional to the magnitude of the spin-spin correlation with the reference site (black filled square). Dark blue (light red) color denotes positive (negative) correlations. The nearest-neighbor correlations (all antiferromagnetic) have been dropped for clarity. One observes pronounced staggered spin-spin correlations along selected loops wrapping around the torus in the left two panels and the lower right panel.