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By exact diagonalization, we investigate several spin-$\frac 12$ $J_1-J_2$real clusters of molecular scale with different shapes. Our calculations showthat when the ratio, $\eta $, of next nearest neighbor to nearest neighborbonds is equal to 1, even the cluster of only 25 sites exhibits the bulkbehaviors and has the quantum phase transitions. Two effective critical pointsare around $J_2/J_1=0.3762$ and 0.612 respectively. They are very close tothose of the infinite $J_1-J_2$ square lattice. But, when $\eta \leq 0.85$, thequantum phase transition around $J_2/J_1=0.3762$ disappears. By calculating thedistributions of the average values of $S_i^z$, the bulk behaviors aredemonstrated graphically. In the intermediate phase, the sites on the cornershave distinctly different character from the other sites. The distribution isobviously centralized on the corner sites.Comment: 4 pages, 3 figures, Accepted for publication in Phys. Rev.

physical review. b, condensed matter

Quantum phase transitions in spin-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:math>frustrated molecular clusters: The numerical evidence

Quantum phase transitions in spin-12frustrated molecular clusters: The numerical evidence