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We compute concurrence, a measure of bipartite entanglement, of the firstexcited state of the $1$-D Heisenberg frustrated $J_1$-$J_2$ spin-chain andobserve a sudden change in the entanglement of the eigen state near thecoupling strength $\alpha=J_2/J_1\approx0.241$, where a quantum phasetransition 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 isobserved asymptotically tends to a value $\alpha_c\approx0.24116$. We havecalculated the finite-size scaling exponents for spin chains with even and oddspins. It may be noted that bipartite as well as multipartite entanglementmeasures applied on the ground state of the system, fail to detect any quantumphase transition from the gapless to the gapped phase in the $1$-D Heisenbergfrustrated $J_1$-$J_2$ spin-chain. Furthermore, we measure bipartiteentanglement of first excited states for other spin models like $2$-DHeisenberg $J_1$-$J_2$ model and Shastry-Sutherland model and find similarindications of quantum phase transitions.

Entanglement in first excited states of some many-body quantum spin systems: indication of quantum phase transition in finite size systems