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Entanglement and quantum phase transition in alternating XY spin chain with next-nearest neighbour interactions

By using the method of density-matrix renormalization-group to solve the different spin-spin correlation functions, the nearest-neighbouring entanglement(NNE) and next-nearest-neighbouring entanglement(NNNE) of one-dimensional alternating Heisenberg XY spin chain is investigated in the presence of alternating nearest neighbour interactions of exchange couplings, external magnetic fields and next-nearest neighbouring interactions. For dimerized ferromagnetic spin chain, NNNE appears only above the critical dimerized interaction, meanwhile, the dimerized interaction effects quantum phase transition point and improves NNNE to a large value. We also study the effect of ferromagnetic or antiferromagnetic next-nearest neighboring (NNN) interactions on the dynamics of NNE and NNNE. The ferromagnetic NNN interaction increases and shrinks NNE below and above critical frustrated interaction respectively, while the antiferromagnetic NNN interaction always decreases NNE. The antiferromagnetic NNN interaction results to a larger value of NNNE in comparison to the case when the NNN interaction is ferromagnetic.


I. INTRODUCTION
Entanglement is one of the most profound features of quantum mechanics and has been considered as an important resource in quantum information processing (QIP) including teleportation, [1] cryptography and secure direct communication, [2,3] and quantum communication and computation. [4] In order to realize quantum information process, great effort has been devoted to the generation of entanglement in linear optics, [5] cavity QED [6−8] and ion trap [9,10] schemes. In recent years much attention has been focused on the spin system with Heisenberg exchange interaction, which is a typical quantum system, and the entanglement properties have been extensively investigated in spin systems, such as Ising model [11] and isotropic and anisotropic Heisenberg models. [12−15] However; as far as we know, most discussions mentioned above merely focused on the models with nearest neighbour interaction of exchange couplings, while next-nearest neighbour interaction has not been taken into account. Dimerized systems and frustrated systems with next-nearest-neighbour (NNN) interactions [16−18] play an important role in condensed matter theory, and entanglement in dimerized and frustrated systems has been considered. [19−23] Sun et al [19] investigate entanglement properties in dimerized and frustrated spin-one models by applying the concept of negativity. In Ref. [20], entanglement is studied in an open alternating chain of nuclear spins s = 1/2 with spin-spin couplings in an external magnetic field under the thermodynamic equilibrium conditions. Gu et al [21] have investigated entanglement in frustrated spin-half Heisenberg chains. Chen Yan et al [22] analyzed sublattice entanglement and quantum phase transitions in antiferromagnetic spin chains, Chen Shu et al [23] study the fidelity and quantum phase transition for the Heisenberg chain with next-nearest-neighbouring interaction. In fact, some one-dimensional and two-dimensional compounds (CuGeO 3 , NaV 2 O 5 ) have manifested such interactions. Therefore, it is worthwhile to include next-nearest neighbouring interaction in the studies of spin chain entanglement. The 1D Heisenberg model is a simple but realistic and extensively studied in solid state system.
Recently, Osterloh et al [24] examined the entanglement between two spins of position i and j in the spin chains for the pure case, Huang et al, [25] Osenda etal [26] and we [27] have demonstrated that entanglement can be controlled by introducing impurities into the systems. Meanwhile, the entanglement shares many features with quantum phase transition (QPT) for a many-body system. [28,29] QPT, which occurs at absolute zero temperature and is purely driven by quantum fluctuation, is the structural change in the properties of the ground state. The associated level crossings lead to the presence of non-analyticities in the energy spectrum. Therefore, the knowledge about the ground-state entanglement, the nonlocal correlation in quantum systems, is considered as a key to further understand QPT. In this paper, we study the pairwise ground state entanglement between the nearest-neighbor sites and that of the next-nearest neighbouring sites in one-dimensional s = 1 2 Heisenberg XY spin chain with dimerised exchange couplings and next-nearest neighbour coupling, to our knowledge, which has not been reported before. The present study will help us to further understand the behavior of the entanglement and QPT in one-dimensional alternating Heisenberg XY model with next-nearest neighbour interactions, and this model can display a variety of interesting physical phenomena. More interestingly, we can control or manipulate the NNE and NNNE in QPT point with the help of dimerised exchange couplings, dimerised external magnetic fields and frustrated exchange interaction.

II. SOLUTION OF THE XY MODEL AND SPIN-SPIN CORRELATION FUNCTIONS
We consider one-dimensional alternating Heisenberg XY model of N spin-1 2 particles with nearest-neighbour interactions and next-nearest neighbour interactions. One-dimensional Hamiltonian can be written as where J i,i+1 and J i,i+2 are alternating exchange interaction and nearest-neighbour interactions respectively, h i is the strength of external magnetic field on site i, σ x,y,z are the Pauli matrices, γ is a dimensionless parameter characterizing the anisotropy of the model and N is the total number of sites. Furthermore, the periodic boundary conditions satisfy σ The spectrum of this Hamiltonian can be determined exactly by a straightforward application of the standard methods. The first step in the procedure is to perform a Jordan-Wigner transformation [30] by introducing Fermi operators c + j , c j , as a result, the Hamiltonian (1) is mapped in the free fermion Hamiltonian In this paper, the exchange interaction and external magnetic field have the form J 2i−1,2i = J 1 = J, 2 ) respectively, where α = J 2 /J 1 introduces the dimerised parameter, the ratio of next-nearest (J i,i+2 = J 3 ) to nearest-neighbor interaction coefficients κ = J 3 /J 1 is called the frustration parameter. J < 0 and J > 0 correspond to the antiferromagnetic and the ferromagnetic cases, respectively. For J 3 = 0, Eq. (2) reduces to one-dimensional alternating(dimerised) Heisenberg model, for J 1 = J 2 , it reduces to onedimensional frustrated Heisenberg model.
The density-matrix renormalization-group (DMRG)method [31,32] is applied to obtain the spin correlation function. The two-point reduced density matrix obtained by tracing the full density matrix of the system over all sites except the pair {i, j} has the form In this part, we choose the concurrence defined by Wootters [33] as a measurement of the pairwise entanglement. For a pure or mixed state of two qubits described by the density matrix ρ, the concurrence C may be calculated explicitly as where λ 1 , λ 2 , λ 3 , λ 4 are the eigenvalues in descending order of the spin-flipped density operator R, which is defined by R = √ ρρ √ ρ , whereρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ),ρ denotes the complex conjugate of ρ, σ y is the usual Pauli matrix. All the matrix elements in the density matrix can be calculated from the different spin-spin correlation functions.

III. RESULTS AND DISCUSSIONS
In this paper, we focus our discussion on the transverse Ising model with γ = 1 and introduce the dimensionless parameter λ = J/2h. The goal of our present study is to find dynamic characteristics of the entanglement in varying the dimerised parameter of exchange couplings, external magnetic fields and NNN interaction. The NNN interaction has a different frustration effect on C(1,2). For frustrated ferromagneticferromagnetic(F-F) spin chain κ > 0, by the comparison among the different curves in Fig.4(a), it is interesting to find that the entanglement peak between the nearest neighbours in the dotted line increase to a value larger than those in the solid line. With the increasing of κ, in the dash-dotted line, the concurrence decreases. We can imagine there must be a critical frustration strength (κ c ), below κ c , NNN interaction enhances entanglement, while above κ c , NNN interaction shrinks entanglement. This behaviour is due to the energy level cross at the point κ = 0.5, as seen clearly from Fig.4(a). Before and after the critical point, the entanglements display distinct behaviours.
However, for frustrated ferromagnetic-antiferromagnetic(F-AF) spin chain κ < 0, the situation becomes different due to the frustration, with the increase in absolute value of κ and weak parameter λ, antiferromagnetic NNN interaction always shrinks entanglement, which is different from the results obtained from Fig.4(a). Another important character revealed by Fig.4(b) is that, as parameter λ increases, the concurrence increases and tends to move to infinity(> 0) by increasing the value of the parameter κ, the strong κ is helpful to keeping the better entanglement.
It is the frustrated interaction that leads to considerable different evolutions of the entanglement, For a ferromagnetic-antiferromagnetic frustrated spin chain, the NNN interaction is predominant in the competing role and can enhance the entanglement to a steady value. The ferromagnetic NNN interaction increases and shrinks NNE below and above critical frustrated interaction respectively, while the antiferromagnetic NNN interaction always decreases NNE. Meanwhile, the antiferromagnetic NNN interaction can generate a larger value of NNNE in comparison to the case when the NNN interaction is ferromagnetic. So we can employ NNN interaction strength as well as dimerized interaction to realize quantum entanglement control. As for the case of γ = 1(XY model), we will present further reports in the future.