In this section, we assume that the D vector of DM interaction in Hamiltonian  (1) is along the z axis, i.e., D = D_ze_z. As mentioned above, the states of neighboring Heisenberg spin dimers are disentangled due to a classical character of the coupling between them via the Ising spin.^[39] Thus, we can calculate the concurrence for each of the dimers individually. We consider the n-th cluster consisting of Heisenberg spin dimer (S_n1 and S_n2) and two Ising spins (σ _n and σ _{n+ 1}), and would like to emphasize that each Ising spin belongs simultaneously to two clusters. Then, the sixteen eigenvalues of the n-th cluster Hamiltonian H_n(1) can be given as

where, and the corresponding eigenvectors are expressed as

where θ = arctan (J₀/D_Z). It is noted that here the sequence of the standard basis vectors of the Hilbert space associated with the n-th cluster is defined as {S_n1, σ _{n+ 1}, S_n2, σ _n}.

The density matrix ρ _n describing the n-th cluster state at temperature T is written as

with the partition function Z_n

We construct the reduced density matrix ρ _S12 describing the Heisenberg spin dimer {S_n1, S_n2} by tracing out two Ising spins {σ _{n+ 1}, σ _n} in the n-th cluster. In the basis vectors {| 00〉 , | 01〉 , | 10〉 , | 11〉 }, the reduced density matrix ρ _S12 can be given by

where

The square roots of the four eigenvalues of the operator ρ defined in Eq.  (3) are calculated

so the concurrence as a measure of Heisenberg spin dimer entanglement is given by

We can discuss the effects of DM interaction on the thermal entanglement of the Heisenberg spin dimer by calculating the concurrence  (8). Firstly, we consider the case without the magnetic field (i.e., h = 0). In Fig.  2, the variations of concurrence C with the temperature T for different values of D_z are plotted in three different cases (i.e., J − J₀ < 0, J − J₀ = 0, and J − J₀ > 0). Disregarding the DM interaction (i.e., letting D_z = 0), one can obtain results presented by the black solid lines, which are consistent with the results of the previous literature.^[39] For the case with J₀ = 1 and J = 0.5 (see Fig.  2 (a)), the concurrence C is equal to 1 at zero temperature, monotonically decreases with temperature increasing and reaches zero at the critical temperature T_c corresponding to the vanishing of entanglement. Considering the effect of DM interaction, it is found that the critical temperature T_c increases with | D_z| . For instance, T_c = 0.867, 0.887, 0.941, 1.022, and 1.123, for D_z = 0, 0.25, 0.5, 0.75 and 1.0, respectively. It is also seen that at a fixed temperature (lower than the critical temperature) the concurrence value increases with the DM interaction. The variation of T_c with D_z is shown in Fig.  3, where the black dotted line corresponds to the case with J₀ = 1 and J = 0.5. It is concluded that the DM interaction favors the formation of thermal entanglement. This consequence is due to the fact that DM interaction induces the quantum correlation between two Heisenberg spins in the dimer, when the DM interaction is turned on, the low-lying excited states tend to the more correlated.^[48]

Fig.  2.

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	Fig.  2. The concurrence C versus the temperature T for different values of Dz with h = 0, J0 = 1, and (a) J = 0.5; (b) J = 1; (c) J = 1.1.

Figure  2 (b) shows the case with J₀ = 1 and J = 1. When D_z = 0, the concurrence C starts from 1/3 other than 1 at T = 0, and then with temperature increasing it gradually deceases to zero at the critical temperature T_c = 0.693.^[39] When the DM interaction is considered, the concurrence C near zero temperature(e.g., T = 0.005) increases with | D_z| , and gradually reaches its maximum values of 1. The variation of the concurrence C with D_z at T = 0.005 is shown in Fig.  4, where the red dotted line corresponds to the case with J₀ = 1 and J = 1. In this case, the maximum entangled state with C = 1 appears when the DM interaction parameter | D_z| is larger than 0.4. In Fig.  3, it is shown that the critical temperature T_c (red dashed-line) also increases with | D_z| , but T_c of the case with J = 1 is smaller than that of the case with J = 0.5 for the same value of the DM interaction parameter D_z.

Fig.  3.

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	Fig.  3. The dependence of the critical temperature Tc on Dz for different values of J with h = 0 and J0 = 1.

Fig.  4.

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	Fig.  4. The dependence of the concurrence C at T = 0.005 on Dz for different values of J with h = 0 and J0 = 1.

The third case (J − J₀ > 0, e.g., J₀ = 1 and J = 1.1) is plotted in Fig.  2 (c). In the case of D_z = 0, the Heisenberg spin dimer is not entangled (i.e., the concurrence C = 0) at T = 0. By increasing the temperature from zero, the Heisenberg spin dimer remains unentangled up to the first critical temperature T_c1 = 0.145. As soon as the temperature increases from T_c1, the concurrence C > 0 (i.e., the thermal entanglement regains, that is so-called entanglement revival) and takes a maximum value and then decreases and reaches zero at the second critical temperature T_c2 = 0.617. The existence of the first critical temperature arises from mixing the entangled excited states with the unentangled ground state due to thermal fluctuations, while sufficiently large thermal fluctuations will destroy all quantum correlations so that the second critical temperature occurs. It is observed that with | D_z| increasing, T_c1 gradually decreases to zero, while T_c2 increases. Thus, the width of the temperature interval within which the Heisenberg spin dimer becomes entangled (i.e., the concurrence C > 0) gets larger by increasing | D_z| . As a result, different from the first two cases (J = 0.5 and J = 1), there are two critical temperature lines for the case with J₀ = 1 and J = 1.1 in Fig.  3 (shown by two blue solid lines). Further, from blue solid lines in Fig.  3 it is found that if | D_z| is larger than 0.663, the DM interaction can induce the entanglement at T = 0. In fact, at zero temperature, when the ground state contains twofold degenerate states | ψ ₈〉 and | ψ ₉〉 , which are unentangled states with C = 0 since these states are factorable. However, when , the ground state becomes fourfold degenerate, containing | ψ ₄〉 , | ψ ₅〉 , | ψ ₁₂〉 , and | ψ ₁₃〉 , which are the maximum entangled states in the subspace of the Heisenberg spin dimer, corresponding to the concurrence C = 1. Thus, there exists the quantum phase transition from unentangled state to maximum entangled state at zero temperature by changing the DM interaction parameter D_z. In Fig.  4, it is shown by the blue dashed-line that the concurrence C sharply changes from 0 to 1 at the critical DM interaction parameter value .

Now, we proceed to study the role of the magnetic field h on the thermal entanglement with the DM interaction D_z. The variations of concurrence C with the magnetic field h for different values of D_z at temperature T = 0.1 are plotted in Fig.  5. Figure  5 (a) shows the case with J − J₀ < 0 (here, J₀ = 1 and J = 0.5). With | h| increasing, the concurrence C gradually decreases to zero, and moreover the critical magnetic field | h_c| corresponding to the vanishing of entanglement increases with | D_z| . For instance, | h_c| = 2.25, 2.40, and 2.81, for D_z = 0.5, 1.0, and 2.0, respectively. Here, it is assumed that C with the value less than 10^{− 3} corresponds to the entanglement disappearing. The dependence of the critical magnetic field h_c on D_z is shown in Fig.  6, in which the black dashed-line corresponds to the case with J₀ = 1 and J = 0.5.

The case with J₀ = 1 and J = 1 is shown in Fig.  5 (b). When D_z = 0, with increasing the magnetic field h the concurrence C firstly increases from 0.333 to a maximum 0.493, and then gradually vanishes. It is found that this maximum of the concurrence C increases with | D_z| up to 1. For instance, the maximums of C are 0.639, 0.887, and 0.998 for D_z = 0.5, 1.0, and 2.0, respectively. Likewise, the critical magnetic field | h_c| also increases with | D_z| , but | h_c| of the case with J = 1 (also see red dotted line in Fig.  6) is higher than that of the case with J = 0.5 for the same value of the DM interaction parameter D_z.

Fig.  5.

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	Fig.  5. The concurrence C versus the magnetic field h for different values of Dz with T = 0.1, J0 = 1, and (a) J = 0.5; (b) J = 1; (c) J = 1.1.

Fig.  6.

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	Fig.  6. The dependence of the critical magnetic field hc on Dz for different values of J with J0 = 1 and T = 0.1.(Here, it is assumed that C with the value less than 10− 3 corresponds to the entanglement disappearing.)

Figure  5 (c) shows the case with J₀ = 1 and J = 1.1. In the absence of the DM interaction D_z and the magnetic field h, the concurrence C is zero and Heisenberg spin dimer is unentangled. By increasing | h| from zero, Heisenberg spin dimer still keeps unentangled up to the first critical magnetic field | h_c1| = 0.23. When the magnetic field increases from | h_c1| , the thermal entanglement(at T = 0.1) revives, and the concurrence C takes a maximum value 0.264 at | h| = 1.20, and then gradually deceases to zero at the second critical magnetic field | h_c2| = 2.78. As mentioned above, when D_z = 0 and h = 0, the ground state contains twofold degenerate states | ψ ₈〉 and | ψ ₉〉 , which are unentangled states. At T = 0.1, with increasing the magnetic field due to admixture of entangled excited states into the mix state, the thermal entanglement regains at the first critical magnetic field | h_c1| . The existence of the second critical magnetic field is completely natural, since sufficiently large magnetic field will induce the ground state to be factorable state | ψ ₁₀〉 = | 1111〉 or | ψ ₁₁〉 = | 0000〉 . It is seen that with | D_z| increasing, | h_c1| gradually decreases to zero, while | h_c2| increases. Thus, for the case with J₀ = 1 and J = 1.1, distinct from the first two cases with J = 0.5 and J = 1, the dependence of the critical magnetic field h_c on the DM interaction parameter D_z is presented by three blue solid lines in Fig.  6.

In this section we shall study the case that the D vector of DM interaction in the Hamiltonian  (1) is along the x axis, i.e., D = D_xe_x. In principle, the problem of 4-qubit spin models can be solved analytically. However, for the case with the D vector along the x axis the analytical expressions of eigenfunctions and the corresponding eigenvalues of the Hamiltonian  (1) are too complex to be compactly given. Thus, in order to obtain explicit physical results, we resort to a numerical calculation method based on the standard process given above.

First, we shall start with discussion of the thermal entanglement in the absence of a magnetic field(i.e., h = 0). Figure  7 shows the variations of concurrence C with the temperature T for different values of D_x according to three different cases, i.e., J − J₀ < 0, J − J₀ = 0, and J − J₀ > 0. For the case with J₀ = 1 and J = 0.5 (see Fig.  7 (a)), due to the effect of the DM interaction D_x, the maximum of the concurrence C is not located at zero temperature any more, but shifts to a finite T value, and moreover is smaller than 1. For instance, the maximum of the concurrence C is 0.964 at T = 0.115 for D_x = 1.0. This non-monotonic variation of C with T in the case with the D vector along the x axis is obviously different from the case with the D vector along the z axis. For the cases with J − J₀ = 0 and J − J₀ > 0, the concurrence C increases with D_x, and the non-monotonic variation of C with T is also observed (see Figs.  7 (b) and 7 (c)).

Fig.  7.

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	Fig.  7. The concurrence C versus the temperature T for different values of Dx with h = 0, J0 = 1, and (a) J = 0.5; (b) J = 1; (c) J = 1.1.

After the concurrence reaches the maximum, it decreases gradually with temperature increasing and vanishes at the critical temperature T_c. The dependence of the critical temperature T_c on D_x for different values of J is plotted in Fig.  8. Compared with the case D_z (see Fig.  3), the effect of D_x on the thermal entanglement has a distinct property in the case with J − J₀ > 0 (see blue solid line in Fig.  8), in which the DM interaction D_x can induce the entanglement characterized by a nonzero concurrence near zero temperature. Specifically, for very small | D_x| , there exist three critical temperature values in the case with J₀ = 1 and J = 1.1.

Fig.  8.

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	Fig.  8. The dependence of the critical temperature Tc on Dx for different values of J with h = 0 and J0 = 1.

Additionally, the dependence of the concurrence C on D_x for different values of J near zero temperature (e.g., T = 0.01) is shown in Fig.  9. For the case with J₀ = 1 and J = 0.5, with | D_x| increasing, the concurrence C firstly decreases from 1 to a minimum of 0.940 at | D_x| = 1.29, and then gradually increases, while for the case with J − J₀ ≥ 0, it is observed that the concurrence C increases with | D_x| . The results given by Fig.  9 (for the DM interaction D_x) are obviously different from those of the case D_z (see Fig.  4).

Fig.  9.

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	Fig.  9. The dependence of the concurrence C at T = 0.01 on Dx for different values of J with h = 0 and J0 = 1.

Now, we concentrate on the thermal entanglement in the presence of a magnetic field. The variations of concurrence C (at T = 0.1) with the magnetic field h for different values of D_x are plotted in Fig.  10. Figure  10 (a) shows the case with J − J₀ < 0 (here, J₀ = 1 and J = 0.5). Due to the effect of D_x, with | h| increasing, the concurrence C firstly increases to a maximum, and then gradually decreases to zero. Likewise, the critical magnetic field | h_c| increases with | D_x| , and the variation of h_c with D_x is also shown in Fig.  11 (here, the black dashed line corresponds to the case with J₀ = 1 and J = 0.5). It is found that for the same value of the DM interaction the critical magnetic field | h_c| of the case D_x is higher than that of the case D_z.

Figures  10 (b) and 10 (c) show the cases with J − J₀ = 0 and J − J₀ > 0, respectively. Compared with the case D_z (see Figs.  5 (b) and 5 (c)), the concurrence C shows a different variation tendency in Figs.  10 (b) and 10 (c). At T = 0.1, for the case with J − J₀ > 0, there exist two critical magnetic fields | h_c1| and | h_c2| (| h_c1| < | h_c2| ) without the DM interaction (see black solid line in Fig.  10 (c)). With | D_x| increasing, the first critical magnetic field | h_c1| gradually decreases, while the second critical magnetic field | h_c2| increases (see blue solid lines in Fig.  11). Noted that for the same value of the DM interaction, | h_c1| of the case D_x is smaller than that of the case D_z, but | h_c2| of the case D_x is higher than that of the case D_z.

Fig.  10.

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	Fig.  10. The concurrence C versus the magnetic field h for different values of Dx with T = 0.1, J0 = 1, and (a) J = 0.5; (b) J = 1; (c)  J = 1.1.

Fig.  11.

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	Fig.  11. The dependence of the critical magnetic field hc on Dx for different values of J with J0 = 1 and T = 0.1. (Here, it is assumed that C with the value less than 10− 3 corresponds to the entanglement disappearing).