Abstract
By exact diagonalization, we investigate several spin-$\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ real clusters of molecular scale with different shapes. Our calculations show that when the ratio $\ensuremath{\eta}$ of next-nearest-neighbor to nearest-neighbor bonds is equal to 1, even a cluster of only 25 sites exhibits bulk behaviors and has 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 ${J}_{1}\ensuremath{-}{J}_{2}$ square lattice. But when $\ensuremath{\eta}<~0.85,$ 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.
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