  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 $\alpha=J_2/J_1\approx0.241$, 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 $\alpha_c\approx0.24116$. 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.
