Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction

The features of the concurrences between the nearest-neighbors and that of the next-nearest-neighbors for the one-dimensional Heisenberg model with next-nearest-neighbor coupling J are studied as functions of temperature and J. The two concurrences exhibit a different dependence on J at the ground state, which could be interpreted from the point of view of the correlation functions. The threshold temperature at which the concurrence is zero and the temperature effect on the two concurrences for systems up to 12 sites are studied numerically.