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Entanglement in a two-qubit Ising model under a site-dependent external magnetic ?eld

A. F. Terzis

2

?1

and E. Paspalakis

?2

1 Physics Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece Materials Science Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece (Dated: February 1, 2008)

We investigate the ground state and the thermal entanglement in the two-qubit Ising model interacting with a site-dependent magnetic ?eld. The degree of entanglement is measured by calculating the concurrence. For zero temperature and for certain direction of the applied magnetic ?eld, the quantum phase transition observed under a uniform external magnetic ?eld disappears once a very small non-uniformity is introduced. Furthermore, we have shown analytically and con?rmed numerically that once the direction of one of the magnetic ?eld is along the Ising axis then no entangled states can be produced, independently of the degree of non-uniformity of the magnetic ?elds on each site.

PACS numbers: 03.67.-a,03.65.Ud,75.10.Jm

arXiv:quant-ph/0407230v1 28 Jul 2004

I.

INTRODUCTION

The existence of entangled states of component systems and their unique properties have attracted a lot of attention since the early days of quantum mechanics [1-5]. Entanglement has also recently been the subject of several investigations as it plays an important role in the topical areas of quantum computation and quantum information processing [6,7]. The Heisenberg magnetic spin system is one of the few physical systems that entanglement arise naturally. Nielsen [7] was the ?rst to report results on entangled states utilized in a two-spin system. He calculated the entanglement of formation of a magnetic system described by an isotropic XXX Heisenberg Hamiltonian in an external magnetic ?eld directed along the z-axis. Later, Arnesen et al. [8] systematically investigated the dependence of entanglement on temperature and on the applied external ?eld in the same 1D Heisenberg system. They showed that for ferromagnets the spins are always disentangled while entanglement is observed for antiferromagnets. In addition, they found that the degree of entanglement can be enhanced by increasing the external parameters (magnetic ?eld and/or temperature). Then, Wang [9,10] studied the e?ect of anisotropy on thermal entanglement, working on the two-qubit quantum anisotropic Heisenberg XY model [9] and on the anisotropic Heisenberg XXZ model [10]. He also investigated the isotropic Heisenberg XX model with an external magnetic ?eld applied along the z -axis [9]. Kamta and Starace [11] investigated the thermal entanglement of a two-qubit Heisenberg XY chain in the presence of an external magnetic ?eld along the z -axis. They showed that by adjusting the magnetic ?eld strength, entangled states are produced for any ?nite temperature. Sun et al. [12] extended later the work reported in Ref. [11] by introducing a non-uniform magnetic ?eld. Comparing to the uniform ?eld case, they showed that entanglement can be more e?ectively controlled via a non-uniform magnetic ?eld. The full anisotropic XYZ Heisenberg spin two-qubit system in which a magnetic ?eld is applied along the z -axis, was studied by Zhou et al. [13]. The enhancement of the entanglement for particular ?xed magnetic ?eld by increasing the z -component of the coupling coe?cient between the neighboring spins, was their main ?nding. Finally, thermal entanglement in a two-qubit Ising model assuming an applied magnetic ?eld in an arbitrary direction has been investigated by Gunlycke et al. [14]. Several, mainly numerical, works exist also on the study of pairwise thermal entanglement in the n-qubit Heisenberg spin chain. In short, the systems being studied are the XXZ three-qubit Heisenberg model with an applied magnetic ?eld in the z ? direction [15], the XYZ three-qubit Heisenberg model with an applied magnetic ?eld along the z -axis [13], the XX four-qubit Heisenberg model with an applied magnetic ?eld in the z -direction [16], the XX [17] and XXX and XXZ [18] four- and ?ve-qubit Heisenberg model and the XX m-qubit (with m up to eleven) Heisenberg model [19]. It turns to be quite interesting to study the general case of di?erent magnetic ?elds at each spin site. The control of the applied ?eld at each spin separately is very useful in order to perform quantum computations [20]. Hence in the present theoretical analysis we investigate the ground state and the thermal entanglement in an Ising model with

? ?

E-mail: terzis@physics.upatras.gr E-mail: paspalak@upatras.gr

a non-uniform and anisotropic external magnetic ?eld, i.e. the magnitude and the direction of each magnetic ?eld is di?erent in each spin. This article is organized as follows: In the following section we present the details of the theoretical analysis based on the calculation of the concurrence [21,22] of the system. Then, in section III numerical results are presented and discussed for several cases of the parameters of the system (magnetic ?elds magnitudes and directions and temperature). In addition, a general analytical result is presented in the Appendix. There, we show that once one of the applied external magnetic ?elds points along the Ising direction, no matter what is the direction and magnitude of the other ?eld the concurrence is always zero. This result is valid for zero and ?nite temperatures. Our results are summarized in section IV.

II. THEORY

The Hamiltonian studied in this work is given by

z x z x z z z z + B1 sin θ1 σ1 + B2 cos θ2 σ2 + B2 sin θ2 σ2 σ1 ) + B1 cos θ1 σ1 H = J (σ1 σ2 + σ2

,

(1)

where σ a are the Pauli operators and J is the strength of the Ising interaction. Also, B1 and B2 are the magnitudes of the external magnetic ?elds. We assume that each magnetic ?eld has an arbitrary direction, de?ned by the angles θ1 and θ2 between the ?eld and the Ising direction. It is su?cient to consider that the magnetic ?eld lies in a plane (xz ) containing the Ising z -direction, because in three spatial dimensions, the Hamiltonian possesses rotational symmetry about the z -axis. A useful and convenient quantitative tool, which has been developed to study entanglement, is the entanglement of formation [4]. The entanglement of formation of a state of a composite system is proportional to the minimum number of Bell states, which must be shared between the components of the composite system. There is no general prescription for evaluation of the entanglement of formation in arbitrary systems. In this area Wootters [21] has described an explicit method for evaluating the entanglement of formation of an arbitrary state of a two-qubit system. His explicit formula is a generalization of the expression derived by him and Hill [22] for a special class of density matrices. 2 For a pair of qubits the entanglement of formation E12 is estimated from the expression E12 = h 1 + 1 ? C12 /2, where C12 denotes the concurrence and h is the binary entropy function [21]. The concurrence is de?ned as C12 = max{λ1 ? λ2 ? λ3 ? λ4 , 0}, where the λ’s are the square roots of the eigenvalues in decreasing order of magnitude (i.e. λ1 ≥ λ2 ≥ λ3 ≥ λ4 ) of the spin-?ipped density matrix operator R12 = ρ(σ y ? σ y )ρ ? (σ y ? σ y ). Here, ρ is the density matrix operator de?ned as ρ = exp(?H/kB T )/Z , where Z is the partition function (Z = tr{exp(?H/kB T )}) and kB is the Boltzmann’s factor. As concurrence is a monotonically increasing function of E12 and both functions have values in the range 0 to 1, we practically use C12 as a measure of the entanglement. Zero concurrence corresponds to an un-entangled pair of states and unity concurrence to a maximally entangled pair of states. This type of entanglement is usually called thermal entanglement as it is described by a temperature dependent density matrix operator. In order to proceed we need to ?nd the eigenvalues (Ei ) and eigenvectors (|Ψi ) of the Hamiltonian of the Ising system in the presence of a non-uniform external magnetic ?eld, i.e. the Hamiltonian of Eq. (1). Once we have determined the eigenstates of the system, the density matrix operator can be written as ρ = Z ?1

i

exp(?Ei /kB T ) |Ψi Ψi |

.

(2)

Then, the spin-?ipped matrix operator is evaluated in a 4 × 4 matrix representation, in terms of the natural basis vectors {|00 , |01 , |10 , |11 }. In most cases, even if one obtains analytic expressions for the eigenvalues of the spin?ipped matrix operator, it is practically impossible to derive a simple analytic expression for the concurrence. The reason lies to the fact that the relative order of magnitude of the eigenvalues of R12, depends on the parameters involved. In general the concurrence can be evaluated numerically. For particular values of the parameters of the system, an analytic expression can be achieved (see in the Appendix). It is worth pointing out that the above analysis is greatly simpli?ed for the zero temperature case, and then only the ground state is populated and hence ρ = |Ψ0 Ψ0 |, where 0 is the index that denotes the ground state of the system.

III. RESULTS AND DISCUSSION

We begin our discussion with the results of the ground state of the system at zero absolute temperature (T = 0). In Fig. 1(a) we assume a uniform magnetic ?eld (i.e. same magnitude and same direction of magnetic ?eld in each

spin) and present the results of the concurrence as a function of the strength of the applied ?elds. It is clear that the entanglement is highest for nearly vanishing magnetic ?elds and decreases with increasing the ?eld’s amplitude. In this ?gure we observe that for B1 = B2 = B → 0+ , the concurrence is unity, which shows the creation of maximally entangled states. Once we set a zero value for the magnetic ?eld, the eigenstates are the same as those of the Ising model without magnetic ?eld, i.e. the standard disentangled basis {|00 , |01 , |10 , |11 }, as the Hamiltonian is diagonal in this basis. Therefore, the concurrence obtains a zero value. Then, even for an in?nitesimal increase of the magnetic ?eld, the system goes from a non-entangled state to a maximally entangled state. This is a clear evidence of a quantum phase transition (QPT) [23]. In the case that the magnetic ?eld is along the z -axis, so that θ1 = θ2 = 0, we have no entanglement at all. In this case the ground state has energy –2J and is doubly degenerate. For small, but equal, angles θ1 = θ2 , there are two energy levels, one of energy value –2J and √ the other with energy close to –2J , with corresponding states, the ?rst one equal to the Bell state ( | 01 ? | 10 ) / 2 and the other close to √ the Bell state (|01 + |10 ) / 2. Thus, we get a maximally entangled qubit pair. Hence, we conclude that even a very small component of the magnetic ?eld along the x-axis is adequate to create entangled states. In all cases we observe that the concurrence drops to zero for very strong ?elds. This is expected as for very strong ?elds the spins will be completely aligned along the ?eld direction and hence the entanglement will drop to zero. It is worth noting that the smaller the x-component of the magnetic ?eld becomes, the faster the concurrence drops to zero for B > 2J . In the case where the magnetic ?eld is along the x-axis the concurrence can be calculated analytically from the density matrix of the pure (ground) state, and we get C12 = 1 + (B /J )2

?1/2

, see short dashed line (θ1 = θ2 = π/2) in Fig.

1(a). The physical explanation of this behavior comes from the fact that the ?eld tends to align the qubit spins in a di?erent disentangled state from the spin-spin coupling. This implies that it is the trade o? between the ?eld and the Ising interaction that produces the entanglement. We now investigate the case of non-uniform magnetic ?eld, which is actually more realistic than the uniform magnetic ?eld case. In Fig. 1(b) we observe that for B2 slightly di?erent from B1 (here, B2 = 1.0005[B1 ≡ B ]) the QPT disappears in cases of small x-component of the magnetic ?elds. Now the concurrence starts from vanishing value reach a maximum value for B < 2J and then drops to zero very abruptly for B > 2J . For small, but equal, angles θ1 = θ2 , there are two energy levels one of energy value –2J and the other √ with energy value close to –2J but in this case the corresponding states are not close to the Bell states (|01 ± |10 ) / 2. As the x-component of the magnetic ?eld increases the change from the vanishing value of the concurrence at zero ?eld is very abrupt, as can be seen from the short dashed curve (θ1 = θ2 = π/2) in Fig. 1(b). For example, for θ1 = θ2 = 0.3π and for a magnetic ?eld of value B =0.01J the concurrence is practically unity. A very similar behavior is observed for slightly di?erent directions of the magnetic ?elds B1 and B2 , as can be seen in Fig.1(c). In this case, we observe that the QPT disappears for magnetic ?elds of the same magnitude but slightly di?erent direction. It is important to point out that this phenomenon is present even for directions of the magnetic ?elds very close to the x-direction. Again, we get √ that there are two lowest energy levels corresponding to states that are not close to the Bell states (|01 ± |10 ) / 2. For larger di?erences between the magnitudes of the magnetic ?elds for the two qubits, i.e. B2 = 1.05B1 , the disappearance of the QPT is even more pronounced for cases characterized by small x-component of the magnetic ?elds [see solid curve in Fig. 1(d)]. A systematic study of the ?eld e?ects is depicted in Fig.2, where we plot the concurrence for cases of small [Figs. 2(a) and (b)] and large [Figs. 2(c) and (d)] di?erence in the magnetic ?eld magnitude. We observe that the entanglement achieved for low Bx components is the one that it is more in?uenced. For example, even for a very small di?erence in the ?eld amplitudes (of the order of 1.05) the maximum entanglement is diminished. For a larger value of the amplitudes of the ?eld ratio, e.g. for 1.5, the concurrence practically disappears (not shown). The reason for this behavior is due to the fact that the eigenvectors are in this case the non-entangled pure states |01 , |10 and |00 . As we see from the same ?gure, the in?uence of the non-uniformity of the magnetic ?elds becomes smaller the more the direction of the ?elds get closer to a direction along the x-axis. In the same ?gure we investigate the in?uence of the di?erence in the ?eld directions. We note that the maximum change in the behavior is obtained for the case that Bx is small. We will now study the case of ?nite temperature, i.e. the case of thermal entanglement. We examine how the e?ects discussed in Fig. 1(a) changes for two di?erent values of temperature with B1 = B2 (= B ), θ1 = θ2 . From Figs. 3 and 4 we clearly see that the critical temperature depends on the orientation of the ?elds and that the ?eld at which the maximum entanglement occurs as the temperature increases shifts to larger values of the magnetic ?eld amplitudes. First, we study the case of the low Bx component, where we observe a sharp peak in the concurrence. We note there is a fast drop of the concurrence to zero for a low Bx component and for low values of the ?eld magnitude. The behavior found in Figs 3 and 4 has already reported and discussed in Ref. [14]. From the same plots we observe that as the direction of the ?elds gets close to an orthogonal direction with respect to the Ising axis, the maximum entanglement occurs at lower values of the magnetic ?led magnitude. In the case of θ = π/2, the concurrence does not drop to zero very fast as the eigenvectors of all eigenstates are either Bell states or a linear combination of Bell states.

Comparing Fig. 1(a) and Fig. 3(a), we clearly see that the QPT is not present in the ?nite temperature case. Moreover, we perform calculations for ?nite temperature with magnetic ?elds the same as those in Figs 1(b) and 1(c). We have found that for these values of the external parameters, the concurrence is practically the same as in Fig.3(a). This ?nding is valid once the magnitudes of the two applied ?elds are very similar. For case of larger di?erence in the ?eld magnitude, the ?nite temperature results are more in?uenced as one sees by comparing Fig.1(d) and Fig.3(b). In this case of ?nite temperature under higher anisotropy of ?elds the QPT disappears, independently of the direction of the applied ?elds. Another, general feature seen from Fig. 4 (in comparison to Fig. 2) is that in all the cases the ?nite temperature leads to a decrease in the concurrence and shifts the maximum value of the concurrence at stronger applied ?elds. The e?ect is more pronounced for directions of the ?elds along an axis perpendicular to the Ising axis. It is known [14] that there is an angle θ1 = θ2 = θ (B, T ) where the entanglement is maximum for a given temperature and amplitude (B1 = B2 ). This feature, known as the phenomenon of magnetically induced entanglement, has been explained heuristically assuming that with Bx and Bz ?xed, the entanglement should change continuously with temperature. As the increase in the temperature widens the low-entanglement zone around Bx = 0 [14] and the entanglement has to fall for large Bx , it is expected that at some intermediate value of Bx , the maximal entanglement will be reached. The preferred angle traverses from θ = 0 at zero temperature to θ = π/2 at T ? J/kB . First, we investigate the zero temperature case. For parallel directions of the magnetic ?elds, the maximum concurrence is achieved for ?elds pointing along the x-axis (not shown here). Moreover, we observe that once we ?x the direction of one of the applied ?elds the direction of the ?eld applied at the other spin depends on the relative magnitude of the ?elds. This dependence is minimal once one of the ?elds is along the x-axis (not shown here). We note that there is a cuto? value of the magnetic ?eld magnitude applied on the second spin, above which the relative angle remains practically the same. This cuto? value depends on the direction of the magnetic ?eld applied at the ?rst of the spins. We have shown that for non-uniform ?elds (B1 = J, B2 = 1.5J ) the maximum concurrence is obtained for non-parallel directions of the two applied ?elds. We observe that the smaller the Bx component of one of the ?elds becomes the more the system deviates from the parallel direction. Actually, we have found that for ?elds of equal magnitudes and smaller than 2J , no matter what is the direction of the ?eld applied at the ?rst spin, the maximum concurrence is achieved for magnetic ?eld at the second spin pointing at the same direction as in the ?rst one. This is not true for ?elds of equal magnitudes but with values larger than 2J . The deviation from parallel ?elds becomes more pronounced the more the direction of one of the ?elds gets closer to the Ising direction. It is well understood that if both ?elds get very large values the concurrence is zero. In our study we found that there is no entanglement even if either one of the ?elds is very large. How large should the ?eld be in order to destroy entanglement depends on the orientations of the ?elds. A rule of thumb is that we need larger ?elds as the directions of the ?elds gets closer to a direction perpendicular to the Ising direction. In Fig. 5(a) we con?rm that for ?elds of equal magnitude (B1 = B2 ), the maximum concurrence occurs for θ1 = θ2 . On the contrary, Fig. 5(b) shows that for non-uniform ?elds the maximum entanglement occurs for unequal angles (θ1 = θ2 ). Actually, as the di?erence in the magnitude of the two ?elds increases, the maximum entanglement occurs for direction of the larger ?eld along the x-axis independent of the direction of the other ?eld. A very similar behavior was found for a ?nite temperature as shown in Fig.6.

IV. CONCLUSIONS

In summary, in the present work, a systematic investigation was performed on the entanglement in a two-spin Ising model in a site-dependent magnetic ?eld. One of the most interesting results is the ?nding that the QPT observed at zero temperature when a uniform magnetic ?eld is applied, disappears with the introduction of a very small di?erence in the applied ?elds. This di?erence could be either a di?erence in magnitude or a di?erence in the direction of the magnetic ?elds. Moreover, we have found that for parallel ?elds with direction close to the z -axis (Ising direction), small di?erences in the magnetic ?eld magnitudes result in very weak entanglement. On the contrary, a very large asymmetry in the amplitudes of the applied ?elds has small e?ects on the well-entangled states obtained for ?elds with large x-component. We have also studied the phenomenon of the magnetically induced entanglement and observe that for equal magnitudes of the external magnetic ?elds the maximum concurrence occurs for parallel directions of the ?elds. Once a non-uniformity is introduced the maximally entangled states obtained for non-parallel orientations of the ?elds. In addition, we have shown that the concurrence drops to zero even if only one of the ?elds gets very large values. Finally, we have derived an analytic result valid for ground state entanglement and thermal entanglement. The analytic result, which has been con?rmed numerically, predicts that we get vanishing concurrence once the direction of one of the ?elds is along the Ising direction.

APPENDIX

In the Appendix we discuss the special case of one magnetic ?eld parallel to z -azis and the other in any direction. We ?nd analytic expressions that predict that the concurrence in this case is always zero. The Hamiltonian studied in this case is given by

+ ? z z z z z z /2 , H = J (σ1 σ2 + σ2 σ1 ) + B1 σ1 + B2 cos θσ2 + B2 sin θ σ1 + σ1

(3)

where σ a are the Pauli operators and J is the strength of the Ising interaction. Also, B1 and B2 are the magnitudes of the external magnetic ?elds. We assume that the ?rst spin feels a magnetic ?eld along the z -direction and the second ?eld has an arbitrary direction, de?ned by the angle θ between the ?eld and the Ising direction. The eigenenergies and the eigenstates can be calculated analytically. The eigenenergies are:

X 2 E± = ?B1 ± 4J 2 + B2 ? 4JB2 cos θ 2 Y E± = B1 ± 4J 2 + B2 + 4JB2 cos θ 1/2

,

(4) (5)

1/2

,

The corresponding normalized eigenvectors are |X± |Y± where, 1 a± = √ 2 B2 sin θ

2 ? 4JB cos θ ] [4J 2 + B2 2 1/4 2 ? 4JB cos θ ) (4J 2 + B2 2 1/2 1/2 1/2

= a± |00 + b± |01 , = c± |10 + d± |11 ,

(6) (7)

,

(8)

± (B2 cos θ ? 2J )

1 b± = ± √ 2 1 c± = √ 2

2 4J 2 + B2 ? 4JB2 cos θ

± (B2 cos θ ? 2J )

1/4

1/2

2 ? 4JB cos θ ] [4J 2 + B2 2 B2 sin θ 1/4 2 + 4JB cos θ ) (4J 2 + B2 2 1/2

,

(9) (10)

2 + 4JB cos θ ] [4J 2 + B2 2

1/2

± (B2 cos θ + 2J )

1/2

d±

1 = ±√ 2

2 4J 2 + B2 + 4JB2 cos θ

± (B2 cos θ + 2J )

1/4

1/2

2 + 4JB cos θ ] [4J 2 + B2 2

.

(11)

Now the density matrix is estimated by the following expression ρ = Z ?1 [e?βE? |X? X? | + e?βE+ |X+ X+ | + e?βE? |Y? where the partition function is de?ned as Z = e?βE? + e?βE+ + e?βE? + e?βE+ . and β ≡ 1/kB T . Then, the spin-?ipped density matrix following form ? A C ?D B R12 = ? 0 0 0 0

X X Y Y X X Y

Y? | + e?βE+ |Y+

Y

Y+ |] ,

(12)

(13)

operator R12 in the regular basis representation has the 0 0 B ?D ? 0 0 ? , ?C ? A

(14)

where, A = ρ11 ρ44 ? ρ12 ρ34 , B = ρ22 ρ33 ? ρ12 ρ34 , C = ρ33 ρ12 ? ρ11 ρ34 and D = ρ44 ρ12 ? ρ22 ρ34 . The eigenvalues, λ’s of the spin ?ip density matrix operator are easily estimated as λ1 = λ2 = [(A + B ) + (A ? B )2 + 4CD λ3 = λ4 = [(A + B ) ? (A ? B ) + 4CD

2 1/2 1/2

]/2 ]/2 .

,

(15) (16)

√ √ √ √ Hence, the concurrence de?ned as C12 = max{ λ1 ? λ2 ? λ3 ? λ4 , 0}, is always zero no matter what are the values of A, B, C and D. Therefore, we conclude that independently of the magnitudes of the magnetic ?elds and independently of the direction of one of the ?elds, if one of the ?elds points along the Ising direction there is no entanglement in the system. It is worth mentioning that the above analysis is greatly simpli?ed for the zero temperature case as only the ground state is populated and hence ρ = |Ψ0 Ψ0 |, where 0 is the index for the ground state. In this case the only nonvanishing terms in the density matrix operator are the ρ11 , ρ22 and ρ12 . Then, it is rather straightforward to show that the spin-?ip density operator matrix is the zero 4 × 4 matrix.

REFERENCES

[1] E. Schr¨ odinger, Proc. Cambridge Phil. Soc., 31, 555 (1935). [2] E. Schr¨ odinger, Naturwissenschaften, 23, 807 (1935). [3] J.S. Bell, Physics, 1, 195 (1964). [4] C.H. Benett, D.P. DiVincenzo, J.A. Smolin and W.K. Wooters, Phys. Rev. A, 54, 3824 (1996). [5] C.H. Benett and D.P. DiVincenzo, Nature (London), 404, 247 (2000). [6] M. A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000). [7] M.A. Nielsen, Ph.D. Thesis, University of New Mexico, 1998; see also LANL e-print: quant-ph/0011036. [8] M.C. Arnesen, S. Bose and V. Vedral, Phys. Rev. Lett., 87, 017901 (2001). [9] X. Wang, Phys. Rev. A, 64, 012313 (2001). [10] X. Wang, Phys. Lett. A, 281, 101 (2001). [11] G.L. Kamta and A.F. Starace, Phys. Rev. Lett., 88, 107901 (2002). [12] Y. Sun, Y. Chen and H. Chen, Phys. Rev. A, 68, 044301 (2003). [13] L. Zhou, H.S. Song, Y.Q. Guo and C. Li, Phys. Rev. A, 68, 024301 (2003). [14] D. Gunlycke, V.M. Kendon, V. Vedral and S. Bose, Phys. Rev. A, 64, 042302 (2001). [15] X. Wang, H. Fu and A.I. Solomon, J. Phys. A:Math. Gen., 34, 11307 (2001). [16] X. Wang, Phys. Rev. A, 66, 034302 (2002). [17] X.-Q. Xi, W.-X. Chen, S.-R. Hao and R.-H. Yue, Phys. Lett. A, 300, 567 (2002). [18] X. Wang and K. Mφlmer, Eur. Phys. J. D, 18, 385 (2002). [19] X. Wang, Phys. Rev. A, 66, 044305 (2002). [20] Y. Makhlin, G. Schoen and A. Shnirman, Rev. Mod. Phys., 73, 357 (2001). [21] W.K. Wootters, Phys. Rev. Lett., 80, 2245 (1998). [22] S. Hill and W.K. Wootters, Phys. Rev. Lett., 78, 5022 (1997). [23] S. Sachdev, Quantum Phase Transitions, (Cambridge University Press, Cambridge, 1999).

(a) 1 0.8

Concurrence

(b) 1 0.8

Concurrence

0.6 0.4 0.2

0.6 0.4 0.2

0

1

2 B/J (c)

3

4

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1

2 B/J (d)

3

4

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1 0.8

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0.6 0.4 0.2

0.6 0.4 0.2

0

1

2 B/J

3

4

0

1

2 B/J

3

4

FIG. 1: Plots of concurrence as a function of the magnetic ?eld magnitude B (=B1 ) in units of J at zero temperature. (a) Equal magnetic ?eld magnitudes (B1 = B2 = B ). Solid curve is for θ1 = θ2 = 0.01π , long and short dashed curves correspond to θ1 = θ2 = 0.1π and θ1 = θ2 = 0.5π , respectively. (b) The same as (a) but with unequal ?eld amplitudes (B2 =1.0005B1 ). (c) Equal magnetic ?eld magnitudes (B1 = B2 = B ). Solid curve is for θ1 = 0.01π, θ2 = 0.011π , long dashed curve is for θ1 = 0.1π, θ2 = 0.11π and short dashed curve is for θ1 = 0.5π, θ2 = 0.51π . (d) The same as (a) but with unequal ?eld amplitudes (B2 =1.05B1 )

(a) 1 0.8

Concurrence

(b) 1 0.8

Concurrence

0.6 0.4 0.2

0.6 0.4 0.2

0

1

2 B/J (c)

3

4

0

1

2 B/J (d)

3

4

1 0.8

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1 0.8

Concurrence

0.6 0.4 0.2

0.6 0.4 0.2

0

1

2 B/J

3

4

0

1

2 B/J

3

4

FIG. 2: Plots of concurrence as a function of the magnetic ?eld magnitude B (=B1 ) in units of J at zero temperature. (a) Unequal ?eld amplitudes (B2 =1.0005B1 ). Solid curve is for θ1 = 0.01π , long and short dashed curves correspond to θ1 = 0.1π and θ1 = 0.5π , respectively. Also, θ2 ? θ1 = 0.01π . (b) Same as (a) but with θ2 ? θ1 = 0.1π . (c) Same as (a) but with B2 =1.05B1 . (d) The same as (c) but with θ2 ? θ1 = 0.1π .

(a) 1 0.8

Concurrence

0.6 0.4 0.2

0

1

2 B/J (b)

3

4

1 0.8

Concurrence

0.6 0.4 0.2

0

1

2 B/J

3

4

FIG. 3: Plots of concurrence as a function of the magnetic ?eld magnitude B (=B1 ) in units of J at non-zero temperature (T = 0.01J /kB ). (a) Equal magnetic ?eld magnitudes (B1 = B2 = B ). Solid curve is for θ1 = θ2 = 0.01π , long and short dashed curves correspond to θ1 = θ2 = 0.1π and θ1 = θ2 = 0.5π , respectively. (b) Same as (a) but with unequal ?eld amplitudes (B2 =1.05B1 ).

(a) 1 0.8

Concurrence

(b) 1 0.8

Concurrence

0.6 0.4 0.2

0.6 0.4 0.2

0

1

2 B/J (c)

3

4

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2 B/J (d)

3

4

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2 B/J

3

4

0

1

2 B/J

3

4

FIG. 4: The same as in Fig. 2 but with non-zero temperature T = 0.1J /kB .

HaL

HbL

3 2.5 0 2 1.5 1 0.5 0.5 0 0 0.5 1 1.5 2 θ1Hrad L 2.5 3

3 2.5 0 2 1.5 1 0.2 0.5 0 0 0.5 1 1.5 2 θ1Hrad L 2.5 3

θ2HradL

FIG. 5: Contour plots of concurrence as a function of the directions of magnetic ?elds θ1 and θ2 (polar angles between ?eld and Ising z -axis) for zero temperature. In both plots B1 =2.1J . In (a) B2 = B1 and in (b) B2 =3B1 .

θ2HradL

HaL

HbL

3 2.5 0 2 1.5 1 0.33 0.5 0 0 0.5 1 1.5 2 θ1Hrad L 2.5 3

3 2.5 0 2 1.5 1 0.22 0.5 0 0 0.5 1 1.5 2 θ1Hrad L 2.5 3

θ2HradL

FIG. 6: Contour plots of concurrence as a function of the directions of magnetic ?elds θ1 and θ2 . In both plots T = 1J /kB and B1 =2.1J . In (a) B2 = B1 and in (b) B2 =3B1 .

θ2HradL

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