Quantum Phase Transitions in Spin-1/2 Frustrated Molecular Cluster: the Numerical Evidence

By exact diagonalization, we investigate several spin-$\frac 12$ $J_1-J_2$ real clusters of molecular scale with different shapes. Our calculations show that when the ratio, $\eta$, of next nearest neighbor to nearest neighbor bonds is equal to 1, even the cluster of only 25 sites exhibits the bulk behaviors and has the quantum phase transitions. Two effective critical points are around $J_2/J_1=0.3762$ and 0.612 respectively. They are very close to those of the infinite $J_1-J_2$ square lattice. But, when $\eta \leq 0.85$, the quantum phase transition around $J_2/J_1=0.3762$ disappears. By calculating the distributions of the average values of $S_i^z$, the bulk behaviors are demonstrated graphically. In the intermediate phase, the sites on the corners have distinctly different character from the other sites. The distribution is obviously centralized on the corner sites.