Effects of anisotropy on thermal entanglement

We study the thermal entanglement in the two-qubit anisotropic XXZ model and the Heisenberg model with Dzyaloshinski– Moriya (DM) interactions. The DM interaction is another kind of anisotropic antisymmetric exchange interaction. The effects of these two kinds of anisotropies on the thermal entanglement are studied in detail for both the antiferromagnetic and ferromagnetic cases.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ud; 03.67.Lx; 75.10.Jm Recently, the concept of thermal entanglement was introduced and studied within one-dimensional isotropic Heisenberg model [1]. The state of the system described by the Hamiltonian H at thermal equilibrium is ρ(T ) = exp(−H/kT )/Z, where Z = Tr[exp(−H/kT )] is the partition function and k is the Boltzmann’s constant. As ρ(T ) represents a thermal state, the entanglement in the state is called the thermal entanglement [1]. For two-qubit isotropic Heisenberg model there exists thermal entanglement for the antiferromagnetic case and no thermal entanglement for the ferromagnetic case [1]. While for the XY model the thermal entanglement appears for both the antiferromagnetic and ferromagnetic cases [2]. It is known that the isotropic Heisenberg model and the XY model are special cases of the anisotropic Heisenberg model (see Eq. (3)). So it is worth to study the thermal entanglement in the anisotropic models and see the role of anisotropic parameters. In this Letter we consider two types of E-mail address: [email protected] (X. Wang). anisotropy and study the effects of them on the thermal entanglement. Both the antiferromagnetic and ferromagnetic cases are considered. First we briefly review a measure of entanglement, the concurrence [3]. Let ρ12 be the density matrix of a pair of qubits 1 and 2. The density matrix can be either pure or mixed. The concurrence corresponding to the density matrix is defined as (1) C12 = max{λ1 − λ2 − λ3 − λ4,0}, where the quantities λ1 λ2 λ3 λ4 are the square roots of the eigenvalues of the operator (2) 12 = ρ12(σ1y ⊗ σ2y)ρ∗ 12(σ1y ⊗ σ2y). The operators σjy (j = 1,2) are the usual Pauli operators for the qubit j . The concurrence C12 = 0 corresponds to an unentangled state and C12 = 1 corresponds to a maximally entangled state. We consider the two-qubit anisotropicXXZ Heisenberg model [4,5] H = J 2 (σ1xσ2x + σ1yσ2y +∆σ1zσ2z) 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0375-9601(01) 00 12 32 102 X. Wang / Physics Letters A 281 (2001) 101–104 (3) = J (σ1+σ2− + σ1−σ2+)+ J∆ 2 σ1zσ2z, where the coupling constants J > 0 corresponds to the antiferromagnetic case and J < 0 the ferromagnetic case. The operators σj± = (1/2)(σjx ± iσjy) (j = 1,2). The XXZ model was initiated by Bethe for the case ∆=±1 in 1931 [4] and has been studied for ∆ = ±1 since 1959 [5]. The eigenvalues and eigenvectors of H are easily obtained as H |00〉 = J∆ 2 |00〉, H |11〉 = J∆ 2 |11〉, (4) H ∣∣Ψ±〉= ( − 2 ± J ∣∣Ψ±〉, where |Ψ±〉 = (1/√2 )(|01〉 ± |10〉) are maximally entangled states and |0〉 (|1〉) denotes the ground (excited) state of a two-level particle. In the standard basis, {|00〉, |01〉, |10〉, |11〉}, the density matrix ρ(T ) is written as (k = 1) ρ(T )= 1 2(eJ∆/2T cosh(J/T )+ e−J∆/2T ) (5) ×   e−J∆/2T 0 0 0 0 eJ∆/2T cosh J T −eJ∆/2T sinh J T 0 0 −eJ∆/2T sinh J T eJ∆/2T cosh J T 0 0 0 0 e−J∆/2T   . The square roots of the four eigenvalues of the density matrix 12 are λ1 = λ2 = e −J∆/T 2(cosh(J/T )+ e−J∆/T ) , λ3 = e J/T 2(cosh(J/T )+ e−J∆/T ) , (6) λ4 = e −J/T 2(cosh(J/T )+ e−J∆/T ) . Which is the largest eigenvalue depends on the value of anisotropic parameter ∆ and sign of J. For antiferromagnetic case (J > 0) the largest eigenvalue is λ1 when ∆ −1 and λ3 when ∆>−1. Therefore the concurrences are given by CAFM(∆)= 0 for ∆ −1, CAFM(∆)= max ( sinh(J/T )− e−J∆/T cosh(J/T )+ e−J∆/T ,0 ) (7) for ∆>−1. When ∆ = 1, the anisotropic model becomes the isotropic model, and Eqs. (7) reduces to (8) CAFM(1)= max ( e2J/T−3 e2J/T+3 ,0 ) which is obtained in Ref. [1]. From the above equation we know that when the temperature is larger than the critical temperature TC = 2J/ ln 3 the thermal entanglement disappears. For the anisotropic model the critical temperature TC is determined by the nonlinear equation