By using the method of density-matrix renormalization-group to solve the different spin–spin correlation functions, the nearest-neighbouring entanglement (NNE) and the next-nearest-neighbouring entanglement (NNNE) of one-dimensional alternating Heisenberg XY spin chain are investigated in the presence of alternating the-nearest-neighbouring interaction of exchange couplings, external magnetic fields and the next-nearest neighbouring interaction. For a dimerised ferromagnetic spin chain, the NNNE appears only above a critical dimerized interaction, meanwhile, the dimerized interaction a effects a quantum phase transition point and improves the NNNE to a large extent. We also study the effect of ferromagnetic or antiferromagnetic next-nearest neighbouring (NNN) interaction on the dynamics of NNE and NNNE. The ferromagnetic NNN interaction increases and shrinks the NNE below and above a critical frustrated interaction respectively, while the antiferromagnetic NNN interaction always reduces the NNE. The antiferromagnetic NNN interaction results in a large value of NNNE compared with the case where the NNN interaction is ferromagnetic.
A series of symmetry-protected topological(SPT) and trivial spin-gap phases in the spin-1/2 ferromagnetic--antiferromagnetic alternating Heisenberg chain with alternating next-nearest-neighbour interaction are investigated using two kinds of entanglement spectra defined by different divisions of the whole chain. In the case one of the next-nearest-neighbor interaction vanishes, the model reduces to the
Δ-chain in which a series of spin-gap phases are found in J. Phys. Soc. Jpn. 77, 044707 (2008). From the degeneracy of the entanglement spectra, these phases are identified as the SPT and trivial phases. It is found that the ground state phases boundaries are insensitive to the strength of the alternation in next-nearest neighbor interaction. These results are consistent with the analysis based on the nonlinear
σ model and exact solution on the ferromagnetic-nonmagnetic phase boundary.