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arXiv:quant-ph/0502121 · Physical Review A

Characterizing entanglement by momentum jump in the frustrated Heisenberg ring at a quantum phase transition

Xiao‐Feng Qian, Tao Shi, Ying Li, Z. Song, C. P. Sun

Published Jul 26, 2005• 34 citations
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Abstract

We study the pairwise concurrences, a measure of entanglement, of the ground states for the frustrated Heisenberg ring to explore the relation between entanglement and quantum phase transition associated with the momentum jump. The ground-state concurrences between any two sites are obtained analytically and numerically. It shows that the summation of all possible pairwise concurrences is an appropriate candidate to depict the phase transition. We also investigate the role that the momentum takes in the jump of concurrence at the critical points. We find that an abrupt momentum change results in the maximal concurrence difference of two degenerate ground states.

DOI
10.1103/physreva.72.012333

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We study the pairwise concurrences, a measure of entanglement, of the ground states for the frustrated Heisenberg ring to explore the relation between entanglement and quantum phase transition associated with the momentum jump. The ground-state concurrences between any two sites are obtained analytically and numerically. It shows that the summation of all possible pairwise concurrences is an appropriate candidate to depict the phase transition. We also investigate the role that the momentum takes in the jump of concurrence at the critical points. We find that an abrupt momentum change results in the maximal concurrence difference of two degenerate ground states.
In this paper, we consider some frustrated spin models for which the ground states are known exactly. The concurrence, a measure of the amount of entanglement can be calculated exactly for entangled spin pairs. Quantum phase transitions involving macroscopic magnetization changes at critical values of the magnetic field are accompanied by macroscopic jumps in the T=0T=0T=0 entanglement. A specific example is given in which magnetization plateaus give rise to a plateau structure in the amount of entanglement associated with nearest-neighbor bonds. We further show that macroscopic entanglement changes can occur in quantum phase transitions brought about by the tuning of exchange interaction strengths.
Abstract 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 α = J 2 / J 1 ≈ 0.241, 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 α at which the change in entanglement is observed, asymptotically tends to a value α c ≈ 0.24116. 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.
We compute concurrence, a measure of bipartite entanglement, of the first excited state of the 111-D Heisenberg frustrated J1J_1J1​-J2J_2J2​ spin-chain and observe a sudden change in the entanglement of the eigen state near the coupling strength α=J2/J1≈0.241\alpha=J_2/J_1\approx0.241α=J2​/J1​≈0.241, 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 888 sites to 161616 sites, and the value of α\alphaα at which the change in entanglement is observed asymptotically tends to a value αc≈0.24116\alpha_c\approx0.24116αc​≈0.24116. 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 111-D Heisenberg frustrated J1J_1J1​-J2J_2J2​ spin-chain. Furthermore, we measure bipartite entanglement of first excited states for other spin models like 222-D Heisenberg J1J_1J1​-J2J_2J2​ model and Shastry-Sutherland model and detect quantum phase transitions.
We show that, for an exactly solvable quantum spin model, a discontinuity in the first derivative of the ground-state concurrence appears in the absence of a quantum phase transition. It is opposed to the popular belief that the nonanalyticity property of ground-state concurrence can be used to determine quantum phase transitions. We further point out that the analyticity property of the ground-state concurrence in general can be more intricate than that of the ground-state energy. Thus there is no one-to-one correspondence between quantum phase transitions and the nonanalyticity property of the concurrence. Moreover, we show that the von Neumann entropy, as another measure of entanglement, cannot reveal quantum phase transitions in the present model. Therefore, in order to link with quantum phase transitions, some other measures of entanglement are needed.