We compute concurrence, a measure of bipartite entanglement, of the first excited state of the $1$-D Heisenberg frustrated $J_1$-$J_2$ spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength $alpha=J_2/J_1approx0.241$, where a quantum phase transition from spin-fluid phase to dimer phase has been previously reported. We numerically observe this phenomena for spin-chain with $8$ sites to $16$ sites, and the value of $alpha$ at which the change in entanglement is observed asymptotically tends to a value $alpha_capprox0.24116$. We have calculated the finite-size scaling exponents for spin chains with even and odd spins. It may be noted that bipartite as well as multipartite entanglement measures applied on the ground state of the system, fail to detect any quantum phase transition from the gapless to the gapped phase in the $1$-D Heisenberg frustrated $J_1$-$J_2$ spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like $2$-D Heisenberg $J_1$-$J_2$ model and Shastry-Sutherland model and find similar indications of quantum phase transitions.
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