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Entanglement and quantum phase transition in a mixed-spin Heisenberg chain with single-ion anisotropy
Physica A 390 (2011) 2208–2214
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Physica A journal homepage: www.elsevier.com/locate/physa
Entanglement and quantum phase transition in a mixed-spin Heisenberg chain with single-ion anisotropy E. Solano-Carrillo, R. Franco, J. Silva-Valencia ∗ Departamento de Física, Universidad Nacional de Colombia, A.A. 5997, Bogotá, Colombia
article
info
Article history: Received 10 August 2010 Received in revised form 3 December 2010 Available online 24 February 2011 Keywords: Quantum phase transitions Single-ion anisotropy Entanglement Thermal entanglement
abstract We study the ground-state and thermal entanglement in the mixed-spin (S , s) = (1, 1/2) Heisenberg chain with single-ion anisotropy D using exact diagonalization of small clusters. In this system, a quantum phase transition is revealed to occur at the value D = 0, which is the bifurcation point for the global ground state; that is, when the single-ion anisotropy energy is positive, the ground state is unique, whereas when it is negative, the ground state becomes doubly degenerate and the system has the ferrimagnetic long-range order. Using the negativity as a measure of entanglement, we find that a pronounced dip in this quantity, taking place just at the bifurcation point, serves to signal the quantum phase transition. Moreover, we show that the single-ion anisotropy helps to improve the characteristic temperatures above which the quantum behavior disappears. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, many efforts have been made to comprehend the relation between entanglement and quantum phase transitions [1–4]. An obvious advantage in determining quantum critical points using measures of entanglement is that we do not require a priori knowledge of the order parameter and the symmetries of the system. Most of the systems which have been studied previously are Heisenberg spin-1/2 systems, as there exists a good measure of entanglement for a twospin system: the concurrence [5], which is applicable to an arbitrary state of two qubits. On the other hand, entanglement in mixed-spin or higher-spin systems has not been well studied, due to the lack of good operational measures of entanglement. There have been several initial studies along this direction [6–8]; however, these works are restricted to the case of only twoparticle systems. For the case of general bipartite systems, a non-entangled state has necessarily a positive partial transpose according to the Peres–Horodecki criterion [9,10]. Fortunately, due to the SU(2) symmetry of the Heisenberg Hamiltonian, it can be shown [6] that a positive partial transpose is also sufficient for a separate state in the case of spin mixtures of the type (S , 1/2). This allows us to investigate entanglement features in this kind of mixed-spin system, such as the wellknown family of bimetallic chains [11,12] of general formula ACu(pbaOH)(H2 O)3 nH2 O with A = Ni, Co, Fe, Mn, in which the largest spin varies from S = 1 to 5/2. There are very few papers on the study of entanglement in quantum mixed-spin chain models. Li et al. [13] have considered a polymerized antiferromagnetic mixed-spin chain in which the ground-state entanglement transition found is closely related to the valence-bond solid phase transition. Sun et al. have studied the entanglement properties and their relation to quantum phase transitions in the isotropic [14] and the XXZ anisotropic [15,16] (1, 1/2) spin chain, respectively. Thermal entanglement has also been studied in more general spin mixtures such as the (S , 1/2) and the (S , 1) systems [17–19] ; here, the main interest is focused on the characteristic temperature for an entangled thermal state. The effects of external magnetic fields on the entanglement properties have also received attention recently [20–22], since they play an important role in improving the characteristic temperature and enlarging the region of entanglement in the system.
∗
Corresponding author. E-mail address: [email protected] (J. Silva-Valencia).
0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.01.013
E. Solano-Carrillo et al. / Physica A 390 (2011) 2208–2214
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Fig. 1. Schematic representation of an alternating mixed-spin (S , s) chain arranged on a ring.
To the best of our knowledge, entanglement and its relation with quantum phase transitions have never been investigated in mixed-spin chains with single-ion anisotropy. This anisotropy (also known as magnetocrystalline or crystal-field anisotropy) is very realistic from an experimental point of view, and it can be increased locally by adding non-magnetic defects in the system [23]. As an example of this anisotropy in mixed-spin chain compounds, the chain NiCu(pba)(D2 O)3 2D2 O, which has the spin mixture (1, 1/2), would possibly have single-ion anisotropy on the Ni (S = 1) ion [24]. Motivated by the above facts, in this paper we investigate the mixed-spin (S , s) = (1, 1/2) Heisenberg system with single-ion anisotropy D. We chose this particular spin mixture just for computational convenience, but we have shown, using the interacting spin-wave theory together with density matrix renormalization group calculations, that the groundstate properties for general S and s are qualitatively similar [25], so our main results here are expected not to depend on the specific spin mixture. Having said this, the Hamiltonian of our system can be written as
H =J
N − [Sj · sj + sj · Sj+1 + D(Sjz )2 ],
(1)
j =1
where Sj and sj are spin operators at the unit cell denoted by j, and we adopt periodic boundary conditions in a system with N unit cells, as sketched in Fig. 1. In what follows, the strength of the exchange interaction J is set to unity. An important result regarding this model (in any dimension) has been obtained recently by Tian and Lin [26]. They rigorously proved that, for an arbitrary bipartite lattice and spin mixture (S , s) with S > s, the global ground state is nondegenerate and has the total spin-z component Sz = 0 when D > 0. On the other hand, when D < 0, the ground state becomes doubly degenerate with Sz = ±N (S − s), and each state is antiferromagnetically ordered. When D = 0, the model reduces to the isotropic Heisenberg model, whose properties have been thoroughly studied. In particular, the Lieb–Mattis theorem [27] states that the system has total spin S = N (S − s), and therefore its ground state is highly degenerate. So the value D = 0 is a critical (bifurcation) point indicating the reconstruction of the energy spectrum of the system, and we expect that the entanglement, which is closely related to the structure of the ground state, will present a special behavior at the critical point. The aim of this paper is to investigate this characteristic behavior, which serves to indicate the quantum phase transition without reference to any specific order parameter. Moreover, we go beyond the ground state to study the effects of finite temperatures on the entanglement properties of the system. We do this by using the exact diagonalization of the Hamiltonian in Eq. (1) for small clusters. 2. Negativity as a measure of entanglement In the previous section, we commented that the Peres–Horodecki criterion is very useful in determining whether a state is entangled in high-dimensional bipartite systems. The quantitative version of the Peres–Horodecki criterion was developed by Vidal and Werner [28], who presented a measure of entanglement, called the negativity, that can be computed effectively for any mixed state ρ of an arbitrary bipartite (A ⊗ B ) system and does not increase under local manipulations of the system. It essentially measures the degree to which the partial transpose ρ TA fails to be positive definite, by summing over its negative eigenvalues µi :
− ‖ρ TA ‖ − 1 1 N (ρ) = µ= . i i 2
(2)
In the second equality, the trace norm of ρ TA is equal to the sum of the absolute values of the eigenvalues of ρ TA . With this definition, if N > 0, the system is in an entangled state. The negativity has been used to quantify the entanglement in a chain of harmonic oscillators [29,30], in distant regions of XY spin chains at criticality [31], in free one-dimensional Klein–Gordon fields [32], in the spin-chain Kondo model [33], amongst others. Another quantity which is useful in the investigation of the structure of the ground state is the purity,
P (ρ) = Tr(ρ 2 ).
(3)
This is used to measure the degree of mixedness of a state described by the density operator ρ . For a pure state, we have P = 1, whereas for a maximally mixed state, it reaches a minimum. The relation between entanglement and mixedness has attracted much attention in the last decade [34–38]. In fact, for a composite system in an global pure state, the purity is
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Fig. 2. Negativity versus the single-ion anisotropy D for different system sizes L = 2N at a low temperature (T = 0.02). The drop in the curves indicates the quantum critical point.
Fig. 3. Purity versus the single-ion anisotropy D for different system sizes L = 2N at a low temperature (T = 0.02). The minimum in the curves indicates the quantum critical point.
directly related to a measure of block entanglement: the linear entropy [39,40], which is defined as L = α(1 − P ), where α is a normalization constant depending on the partition of the system. Therefore, it is believed that the purity of states can also indicate critical behavior in some systems with quantum phase transitions. We use this concept here in order to complement and understand the results obtained with the negativity. To study the quantum critical point in our system, it suffices to consider the entanglement of two nearest-neighbor sites within a unit cell (pairwise entanglement). Since a finite system under periodic boundary conditions is translationally invariant, it does not matter which unit cell we use for the calculation. Having said this, we compute the reduced density matrix of the two-spin subsystem, which then enters into Eqs. (2) and (3). Before doing this calculation, we first have to write a convenient form for the global ground state of the system, since, as we said above, its degeneracy is not fixed, as it explores the various quantum phases. The best way to take this into account is to approximate the global ground state by considering the thermal state at low enough temperatures. This is described, at thermal equilibrium, by the Gibbs density operator ρs (T ) = exp(−H /T )/Z , where the Boltzmann constant is set to unity. Here, Z = Tr[exp(−H /T )] is the partition function, with T being the temperature. For low enough temperatures, the thermal state gives the mixture of all the degenerate ground states with equal probabilities. From this density operator, we proceed to calculate the reduced density matrix ρ by tracing out all other spin degrees of freedom in the subsystem complementary to the chosen unit cell. Although using the canonical ensemble at low temperatures is not an exact method for the determination of the ground-state properties, it can still unveil the main features of the quantum phase transition [14,15], as it does in the present case. Indeed, it has recently been shown that, even at relatively large temperatures, quantities such as the thermal quantum correlations (measured by the quantum discord) may detect the critical behavior present at zero temperature [41]. Nevertheless, one should be careful when choosing the ground state in some models with spontaneous symmetry breaking since the entanglement content of the two possible ground states, namely, the symmetric state (which is constructed obeying the symmetries of the Hamiltonian) and the broken-symmetry state (which is the realistic one in the thermodynamic limit), is different, and may lead to distinct conclusions about the quantum phase transitions [42]. In Figs. 2 and 3, we show the results for the negativity and the purity versus the single-ion anisotropy for different system sizes L = 2N = 4, 6, 8 at a relatively low temperature (T = 0.02). As expected from the discussion above, both
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Fig. 4. Effect of temperature on the dip of the negativity around the quantum critical point, for the system size L = 6.
quantities develop a special behavior at the critical point; that is, whereas the negativity (pairwise entanglement) presents a pronounced dip, the two-spin purity reaches a minimum value. It is easy to obtain analytical results for the purity of the global ground state, as the following argument shows. Since the spin degeneracy at the isotropic point (D = 0) is g = 2S + 1, each state of the degenerate set is mixed with an equal probability 1/g in the system’s density matrix ρs . Thus, in the diagonal representation, the squared ground-state density matrix is (ρs2 )ii′ = (1/g 2 )δii′ . It follows that the minimum value of the global purity at the critical point is given by Pmin = (1/g 2 )Tr(δii′ ) = 1/[L(S − s) + 1]. In the system under consideration, with S = 1 and s = 1/2, this gives Pmin = 1/(N + 1). For D < 0, the global ground state is two-fold degenerate, which gives P = 1/2, while, for D > 0, (nondegenerate case) it is a pure state (P = 1). It is remarkable that the purity of the two-spin subsystem shown in Fig. 3 exhibits the special property of the whole system at the critical point, namely, it shows a minimum which decreases when the system size is increased. Finite size effects in our calculations are clearly appreciable, as shown in Fig. 2, but what turns out to be essential is that, even in the L = 4 system, the information stored in the quantum phase transition is still captured, so we think that considering larger chains does not add any new content to the discussion, and this justifies the use of the exact diagonalization method. The effects due to finite temperatures are rather more important, as can be seen in Fig. 4 for the dip in the negativity. As the system is cooled, the dip becomes increasingly sharp, so that in the ground state (T = 0) we expect a sudden drop in which the derivative ∂D N is not analytic. This suggests that the system undergoes a second-order (or continuous) quantum phase transition at the bifurcation point D = 0, since this kind of transition is characterized by nonanalyticities in the first derivative of the entanglement measures [43]. We verified this conclusion by numerically calculating the second derivative of the ground-state energy with respect to D, which shows a divergence around the critical point, as required, by definition, for a continuous quantum phase transition. The drop in the pairwise entanglement at the quantum critical point shown in Fig. 2 can be understood on an intuitive basis. As pointed out by Fano a long time ago [44], as a system evolves from a pure state to a maximally mixed state (minimum purity), the statistical fluctuations of any physical quantity measured in the system tend to increase, which is equivalent to saying that there is a loss of information about the system. In our present case, it is easy to understand why there is a minimum of information at the quantum critical point: the transition to the isotropic point might be thought of as ‘‘adding degrees of freedom’’ to the two-spin subsystem (since a high spin degeneracy appears), similar to when a system is embedded into a larger environment. It is well known that this operation causes decoherence and therefore a loss of information, so the pairwise entanglement is expected to be reduced. A more satisfying description can be given by appealing to the entanglement-sharing scenario [2]. As we said before, when D > 0, the overall system is in a pure state, and the absence of two-spin purity (linear entropy), which is obtained from Fig. 3, measures the block entanglement between two nearest-neighbor spins and the rest of the chain. As can be seen, this quantity is maximum at the critical point (since the purity is minimum), and hence there must be a limit on the amount of entanglement that can be distributed among the sites, in the sense that an increase in the overall entanglement in the lattice appears at the expense of pairwise entanglement, as evidenced in Fig. 2. Note that for D < 0 the use of the linear entropy as a measure of block entanglement is not warranted, since the overall system is not in a pure state. However, this is not relevant to the argument; what matters is that the block entanglement increases as D → 0+ , which makes the pairwise entanglement diminish at the critical point due to entanglement sharing. In sum, by investigating the pairwise entanglement in the system, we were able to detect the quantum phase transition at D = 0 and investigate its order without using any order parameter. In the following section, we proceed to discuss in detail the effects of temperature on the pairwise entanglement as we vary the single-ion anisotropy. 3. Thermal entanglement The interest in thermal entanglement has been triggered by the possibility of using realistic systems such as solids at finite temperatures for quantum information processing [45–47]. A fundamental question in this context is: can entanglement
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Fig. 5. Negativity versus temperature for different values of the single-ion anisotropy for the system size L = 6. The inset shows the behavior of the threshold temperature (from which the thermal entanglement vanishes) as D is varied in the easy-axis regime.
Fig. 6. Negativity versus temperature for different values of the single-ion anisotropy for the system size L = 6. The inset shows the behavior of the threshold temperature (from which the thermal entanglement vanishes) as D is varied in the easy-plane regime.
and the quantum behavior in physical systems survive at arbitrary high temperatures? Positive answers to this query have been given [19], even for unusual arrangements such as an electromagnetic field mode in an optical cavity with a movable mirror in a thermal state [48]. The situation for quantum mixed-spin chains has begun to be examined in the past few years. Experimental measurements of the magnetic susceptibility (an entanglement witness) in the mixed-spin (S , 1/2) chain compounds mentioned above have determined characteristic or ‘‘threshold’’ temperatures above which the entanglement in the systems vanishes [17]. These temperatures increase with the increase of the exchange interaction J and also with the increase of S. By introducing an inhomogeneous magnetic field [22] (as required for quantum computing), the threshold temperature is also improved, so the range where the entanglement exists is broadened. Here, we consider the effect of the single-ion anisotropy on the threshold temperatures. In Figs. 5 and 6, we show the results for the pairwise entanglement (negativity) versus temperature as D is varied in the easy-axis (D < 0) and the easy-plane (D > 0) regimes, respectively. As we expected, for a fixed value of D, the negativity decreases as the temperature increases until the threshold value Tth is reached, at which the negativity vanishes. This expectation is based on the fact that the present model is qualitatively similar to the ferrimagnetic XXZ model [26], and hence the entanglement is expected to decrease monotonically with temperature [49], contrary to what could happen if an external magnetic field in the z-direction is turned on [47,50]. As can be observed in the insets, the threshold temperature Tth increases with the increase of the single-ion anisotropy, being arbitrarily high when |D| is large enough. Taking the value J = 81.4 cm−1 reported for the bimetallic NiCu (1,1/2) chain compound [12], we find, for the system size L = 6, the value Tth = 108.9 K when D = 0, which can be compared with the approximate value [17] in the thermodynamic limit Tth ≈ 122 K . On the other hand, when D = 3, for example, we obtain the value Tth = 131.1 K , which suggests that the sensitivity of the threshold temperature to variations of |D| is rather weak, as confirmed by the step-like structure of the threshold temperature versus D. We observe that, for the easy-plane case, the step-like behavior is destroyed for values around D = 2.6, and then a linear dependence arises, at least in the range of D values shown in Fig. 6. A remarkable result from our calculations is that the entanglement in the easy-plane regime is more robust against temperature than in the easy-axis regime, although, in the latter, the amount of entanglement (negativity) is larger for large enough values of |D| and low temperatures. This can be understood intuitively if we look at the qualitative effect of the
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single-ion anisotropy on the system in the ground state. When D < 0, we see from Eq. (1) that the states with high ⟨S z ⟩ are energetically favorable. Then, when we increase D in this regime, the spin-S ions tend to point along the z-axis (up or down). On the other hand, when D > 0, the states with low ⟨S z ⟩ are preferable, and thus, when D is increased in this regime, the spin-S ions turn towards a direction in the xy-plane. Since there are infinitely many directions along which the spin-S ions can point within the plane, we say that the system has more degrees of freedom in this case, and therefore if we use the decoherence argument above, we arrive at the conclusion that the entanglement in this case is less compared to the easyaxis case in which we only have two possibilities when D → −∞. Thus we conclude this section by saying that, although the entanglement is more robust against temperature for the easy-plane regime, we find that, in general, the single-ion anisotropy improves the characteristic temperature above which the quantum behavior disappears. 4. Summary We have studied the ground-state and the thermal entanglement in the mixed-spin (1, 1/2) Heisenberg chain with single-ion anisotropy using exact diagonalization of the Hamiltonian. We find that the negativity, as a measure of pairwise entanglement, exhibits a dip just at the isotropic point where the system undergoes a continuous quantum phase transition. This is explained with the help of the pairwise purity, which shows a minimum at the critical point and thus indicates a maximally mixed ground state where the quantum coherence is weakened. When finite temperatures are considered, the negativity decreases with the increase of the temperature until it vanishes at a threshold temperature. We show that in the easy-plane regime of the single-ion anisotropy the entanglement is more robust against the temperature, but, in general, the threshold temperatures increase with the increase of this anisotropy. This paper contributes to the recent efforts to find mechanisms which can improve the characteristic temperatures for an entangled state, which is of fundamental importance in the field of quantum computing. Acknowledgements This paper was supported by the Dirección de Investigaciones Sede Bogotá of the Universidad Nacional de Colombia and by the Mazda Foundation for the Arts and the Science fellowship. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
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Physica A journal homepage: www.elsevier.com/locate/physa
Entanglement and quantum phase transition in a mixed-spin Heisenberg chain with single-ion anisotropy E. Solano-Carrillo, R. Franco, J. Silva-Valencia ∗ Departamento de Física, Universidad Nacional de Colombia, A.A. 5997, Bogotá, Colombia
article
info
Article history: Received 10 August 2010 Received in revised form 3 December 2010 Available online 24 February 2011 Keywords: Quantum phase transitions Single-ion anisotropy Entanglement Thermal entanglement
abstract We study the ground-state and thermal entanglement in the mixed-spin (S , s) = (1, 1/2) Heisenberg chain with single-ion anisotropy D using exact diagonalization of small clusters. In this system, a quantum phase transition is revealed to occur at the value D = 0, which is the bifurcation point for the global ground state; that is, when the single-ion anisotropy energy is positive, the ground state is unique, whereas when it is negative, the ground state becomes doubly degenerate and the system has the ferrimagnetic long-range order. Using the negativity as a measure of entanglement, we find that a pronounced dip in this quantity, taking place just at the bifurcation point, serves to signal the quantum phase transition. Moreover, we show that the single-ion anisotropy helps to improve the characteristic temperatures above which the quantum behavior disappears. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In recent years, many efforts have been made to comprehend the relation between entanglement and quantum phase transitions [1–4]. An obvious advantage in determining quantum critical points using measures of entanglement is that we do not require a priori knowledge of the order parameter and the symmetries of the system. Most of the systems which have been studied previously are Heisenberg spin-1/2 systems, as there exists a good measure of entanglement for a twospin system: the concurrence [5], which is applicable to an arbitrary state of two qubits. On the other hand, entanglement in mixed-spin or higher-spin systems has not been well studied, due to the lack of good operational measures of entanglement. There have been several initial studies along this direction [6–8]; however, these works are restricted to the case of only twoparticle systems. For the case of general bipartite systems, a non-entangled state has necessarily a positive partial transpose according to the Peres–Horodecki criterion [9,10]. Fortunately, due to the SU(2) symmetry of the Heisenberg Hamiltonian, it can be shown [6] that a positive partial transpose is also sufficient for a separate state in the case of spin mixtures of the type (S , 1/2). This allows us to investigate entanglement features in this kind of mixed-spin system, such as the wellknown family of bimetallic chains [11,12] of general formula ACu(pbaOH)(H2 O)3 nH2 O with A = Ni, Co, Fe, Mn, in which the largest spin varies from S = 1 to 5/2. There are very few papers on the study of entanglement in quantum mixed-spin chain models. Li et al. [13] have considered a polymerized antiferromagnetic mixed-spin chain in which the ground-state entanglement transition found is closely related to the valence-bond solid phase transition. Sun et al. have studied the entanglement properties and their relation to quantum phase transitions in the isotropic [14] and the XXZ anisotropic [15,16] (1, 1/2) spin chain, respectively. Thermal entanglement has also been studied in more general spin mixtures such as the (S , 1/2) and the (S , 1) systems [17–19] ; here, the main interest is focused on the characteristic temperature for an entangled thermal state. The effects of external magnetic fields on the entanglement properties have also received attention recently [20–22], since they play an important role in improving the characteristic temperature and enlarging the region of entanglement in the system.
∗
Corresponding author. E-mail address: [email protected] (J. Silva-Valencia).
0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.01.013
E. Solano-Carrillo et al. / Physica A 390 (2011) 2208–2214
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Fig. 1. Schematic representation of an alternating mixed-spin (S , s) chain arranged on a ring.
To the best of our knowledge, entanglement and its relation with quantum phase transitions have never been investigated in mixed-spin chains with single-ion anisotropy. This anisotropy (also known as magnetocrystalline or crystal-field anisotropy) is very realistic from an experimental point of view, and it can be increased locally by adding non-magnetic defects in the system [23]. As an example of this anisotropy in mixed-spin chain compounds, the chain NiCu(pba)(D2 O)3 2D2 O, which has the spin mixture (1, 1/2), would possibly have single-ion anisotropy on the Ni (S = 1) ion [24]. Motivated by the above facts, in this paper we investigate the mixed-spin (S , s) = (1, 1/2) Heisenberg system with single-ion anisotropy D. We chose this particular spin mixture just for computational convenience, but we have shown, using the interacting spin-wave theory together with density matrix renormalization group calculations, that the groundstate properties for general S and s are qualitatively similar [25], so our main results here are expected not to depend on the specific spin mixture. Having said this, the Hamiltonian of our system can be written as
H =J
N − [Sj · sj + sj · Sj+1 + D(Sjz )2 ],
(1)
j =1
where Sj and sj are spin operators at the unit cell denoted by j, and we adopt periodic boundary conditions in a system with N unit cells, as sketched in Fig. 1. In what follows, the strength of the exchange interaction J is set to unity. An important result regarding this model (in any dimension) has been obtained recently by Tian and Lin [26]. They rigorously proved that, for an arbitrary bipartite lattice and spin mixture (S , s) with S > s, the global ground state is nondegenerate and has the total spin-z component Sz = 0 when D > 0. On the other hand, when D < 0, the ground state becomes doubly degenerate with Sz = ±N (S − s), and each state is antiferromagnetically ordered. When D = 0, the model reduces to the isotropic Heisenberg model, whose properties have been thoroughly studied. In particular, the Lieb–Mattis theorem [27] states that the system has total spin S = N (S − s), and therefore its ground state is highly degenerate. So the value D = 0 is a critical (bifurcation) point indicating the reconstruction of the energy spectrum of the system, and we expect that the entanglement, which is closely related to the structure of the ground state, will present a special behavior at the critical point. The aim of this paper is to investigate this characteristic behavior, which serves to indicate the quantum phase transition without reference to any specific order parameter. Moreover, we go beyond the ground state to study the effects of finite temperatures on the entanglement properties of the system. We do this by using the exact diagonalization of the Hamiltonian in Eq. (1) for small clusters. 2. Negativity as a measure of entanglement In the previous section, we commented that the Peres–Horodecki criterion is very useful in determining whether a state is entangled in high-dimensional bipartite systems. The quantitative version of the Peres–Horodecki criterion was developed by Vidal and Werner [28], who presented a measure of entanglement, called the negativity, that can be computed effectively for any mixed state ρ of an arbitrary bipartite (A ⊗ B ) system and does not increase under local manipulations of the system. It essentially measures the degree to which the partial transpose ρ TA fails to be positive definite, by summing over its negative eigenvalues µi :
− ‖ρ TA ‖ − 1 1 N (ρ) = µ= . i i 2
(2)
In the second equality, the trace norm of ρ TA is equal to the sum of the absolute values of the eigenvalues of ρ TA . With this definition, if N > 0, the system is in an entangled state. The negativity has been used to quantify the entanglement in a chain of harmonic oscillators [29,30], in distant regions of XY spin chains at criticality [31], in free one-dimensional Klein–Gordon fields [32], in the spin-chain Kondo model [33], amongst others. Another quantity which is useful in the investigation of the structure of the ground state is the purity,
P (ρ) = Tr(ρ 2 ).
(3)
This is used to measure the degree of mixedness of a state described by the density operator ρ . For a pure state, we have P = 1, whereas for a maximally mixed state, it reaches a minimum. The relation between entanglement and mixedness has attracted much attention in the last decade [34–38]. In fact, for a composite system in an global pure state, the purity is
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Fig. 2. Negativity versus the single-ion anisotropy D for different system sizes L = 2N at a low temperature (T = 0.02). The drop in the curves indicates the quantum critical point.
Fig. 3. Purity versus the single-ion anisotropy D for different system sizes L = 2N at a low temperature (T = 0.02). The minimum in the curves indicates the quantum critical point.
directly related to a measure of block entanglement: the linear entropy [39,40], which is defined as L = α(1 − P ), where α is a normalization constant depending on the partition of the system. Therefore, it is believed that the purity of states can also indicate critical behavior in some systems with quantum phase transitions. We use this concept here in order to complement and understand the results obtained with the negativity. To study the quantum critical point in our system, it suffices to consider the entanglement of two nearest-neighbor sites within a unit cell (pairwise entanglement). Since a finite system under periodic boundary conditions is translationally invariant, it does not matter which unit cell we use for the calculation. Having said this, we compute the reduced density matrix of the two-spin subsystem, which then enters into Eqs. (2) and (3). Before doing this calculation, we first have to write a convenient form for the global ground state of the system, since, as we said above, its degeneracy is not fixed, as it explores the various quantum phases. The best way to take this into account is to approximate the global ground state by considering the thermal state at low enough temperatures. This is described, at thermal equilibrium, by the Gibbs density operator ρs (T ) = exp(−H /T )/Z , where the Boltzmann constant is set to unity. Here, Z = Tr[exp(−H /T )] is the partition function, with T being the temperature. For low enough temperatures, the thermal state gives the mixture of all the degenerate ground states with equal probabilities. From this density operator, we proceed to calculate the reduced density matrix ρ by tracing out all other spin degrees of freedom in the subsystem complementary to the chosen unit cell. Although using the canonical ensemble at low temperatures is not an exact method for the determination of the ground-state properties, it can still unveil the main features of the quantum phase transition [14,15], as it does in the present case. Indeed, it has recently been shown that, even at relatively large temperatures, quantities such as the thermal quantum correlations (measured by the quantum discord) may detect the critical behavior present at zero temperature [41]. Nevertheless, one should be careful when choosing the ground state in some models with spontaneous symmetry breaking since the entanglement content of the two possible ground states, namely, the symmetric state (which is constructed obeying the symmetries of the Hamiltonian) and the broken-symmetry state (which is the realistic one in the thermodynamic limit), is different, and may lead to distinct conclusions about the quantum phase transitions [42]. In Figs. 2 and 3, we show the results for the negativity and the purity versus the single-ion anisotropy for different system sizes L = 2N = 4, 6, 8 at a relatively low temperature (T = 0.02). As expected from the discussion above, both
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Fig. 4. Effect of temperature on the dip of the negativity around the quantum critical point, for the system size L = 6.
quantities develop a special behavior at the critical point; that is, whereas the negativity (pairwise entanglement) presents a pronounced dip, the two-spin purity reaches a minimum value. It is easy to obtain analytical results for the purity of the global ground state, as the following argument shows. Since the spin degeneracy at the isotropic point (D = 0) is g = 2S + 1, each state of the degenerate set is mixed with an equal probability 1/g in the system’s density matrix ρs . Thus, in the diagonal representation, the squared ground-state density matrix is (ρs2 )ii′ = (1/g 2 )δii′ . It follows that the minimum value of the global purity at the critical point is given by Pmin = (1/g 2 )Tr(δii′ ) = 1/[L(S − s) + 1]. In the system under consideration, with S = 1 and s = 1/2, this gives Pmin = 1/(N + 1). For D < 0, the global ground state is two-fold degenerate, which gives P = 1/2, while, for D > 0, (nondegenerate case) it is a pure state (P = 1). It is remarkable that the purity of the two-spin subsystem shown in Fig. 3 exhibits the special property of the whole system at the critical point, namely, it shows a minimum which decreases when the system size is increased. Finite size effects in our calculations are clearly appreciable, as shown in Fig. 2, but what turns out to be essential is that, even in the L = 4 system, the information stored in the quantum phase transition is still captured, so we think that considering larger chains does not add any new content to the discussion, and this justifies the use of the exact diagonalization method. The effects due to finite temperatures are rather more important, as can be seen in Fig. 4 for the dip in the negativity. As the system is cooled, the dip becomes increasingly sharp, so that in the ground state (T = 0) we expect a sudden drop in which the derivative ∂D N is not analytic. This suggests that the system undergoes a second-order (or continuous) quantum phase transition at the bifurcation point D = 0, since this kind of transition is characterized by nonanalyticities in the first derivative of the entanglement measures [43]. We verified this conclusion by numerically calculating the second derivative of the ground-state energy with respect to D, which shows a divergence around the critical point, as required, by definition, for a continuous quantum phase transition. The drop in the pairwise entanglement at the quantum critical point shown in Fig. 2 can be understood on an intuitive basis. As pointed out by Fano a long time ago [44], as a system evolves from a pure state to a maximally mixed state (minimum purity), the statistical fluctuations of any physical quantity measured in the system tend to increase, which is equivalent to saying that there is a loss of information about the system. In our present case, it is easy to understand why there is a minimum of information at the quantum critical point: the transition to the isotropic point might be thought of as ‘‘adding degrees of freedom’’ to the two-spin subsystem (since a high spin degeneracy appears), similar to when a system is embedded into a larger environment. It is well known that this operation causes decoherence and therefore a loss of information, so the pairwise entanglement is expected to be reduced. A more satisfying description can be given by appealing to the entanglement-sharing scenario [2]. As we said before, when D > 0, the overall system is in a pure state, and the absence of two-spin purity (linear entropy), which is obtained from Fig. 3, measures the block entanglement between two nearest-neighbor spins and the rest of the chain. As can be seen, this quantity is maximum at the critical point (since the purity is minimum), and hence there must be a limit on the amount of entanglement that can be distributed among the sites, in the sense that an increase in the overall entanglement in the lattice appears at the expense of pairwise entanglement, as evidenced in Fig. 2. Note that for D < 0 the use of the linear entropy as a measure of block entanglement is not warranted, since the overall system is not in a pure state. However, this is not relevant to the argument; what matters is that the block entanglement increases as D → 0+ , which makes the pairwise entanglement diminish at the critical point due to entanglement sharing. In sum, by investigating the pairwise entanglement in the system, we were able to detect the quantum phase transition at D = 0 and investigate its order without using any order parameter. In the following section, we proceed to discuss in detail the effects of temperature on the pairwise entanglement as we vary the single-ion anisotropy. 3. Thermal entanglement The interest in thermal entanglement has been triggered by the possibility of using realistic systems such as solids at finite temperatures for quantum information processing [45–47]. A fundamental question in this context is: can entanglement
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Fig. 5. Negativity versus temperature for different values of the single-ion anisotropy for the system size L = 6. The inset shows the behavior of the threshold temperature (from which the thermal entanglement vanishes) as D is varied in the easy-axis regime.
Fig. 6. Negativity versus temperature for different values of the single-ion anisotropy for the system size L = 6. The inset shows the behavior of the threshold temperature (from which the thermal entanglement vanishes) as D is varied in the easy-plane regime.
and the quantum behavior in physical systems survive at arbitrary high temperatures? Positive answers to this query have been given [19], even for unusual arrangements such as an electromagnetic field mode in an optical cavity with a movable mirror in a thermal state [48]. The situation for quantum mixed-spin chains has begun to be examined in the past few years. Experimental measurements of the magnetic susceptibility (an entanglement witness) in the mixed-spin (S , 1/2) chain compounds mentioned above have determined characteristic or ‘‘threshold’’ temperatures above which the entanglement in the systems vanishes [17]. These temperatures increase with the increase of the exchange interaction J and also with the increase of S. By introducing an inhomogeneous magnetic field [22] (as required for quantum computing), the threshold temperature is also improved, so the range where the entanglement exists is broadened. Here, we consider the effect of the single-ion anisotropy on the threshold temperatures. In Figs. 5 and 6, we show the results for the pairwise entanglement (negativity) versus temperature as D is varied in the easy-axis (D < 0) and the easy-plane (D > 0) regimes, respectively. As we expected, for a fixed value of D, the negativity decreases as the temperature increases until the threshold value Tth is reached, at which the negativity vanishes. This expectation is based on the fact that the present model is qualitatively similar to the ferrimagnetic XXZ model [26], and hence the entanglement is expected to decrease monotonically with temperature [49], contrary to what could happen if an external magnetic field in the z-direction is turned on [47,50]. As can be observed in the insets, the threshold temperature Tth increases with the increase of the single-ion anisotropy, being arbitrarily high when |D| is large enough. Taking the value J = 81.4 cm−1 reported for the bimetallic NiCu (1,1/2) chain compound [12], we find, for the system size L = 6, the value Tth = 108.9 K when D = 0, which can be compared with the approximate value [17] in the thermodynamic limit Tth ≈ 122 K . On the other hand, when D = 3, for example, we obtain the value Tth = 131.1 K , which suggests that the sensitivity of the threshold temperature to variations of |D| is rather weak, as confirmed by the step-like structure of the threshold temperature versus D. We observe that, for the easy-plane case, the step-like behavior is destroyed for values around D = 2.6, and then a linear dependence arises, at least in the range of D values shown in Fig. 6. A remarkable result from our calculations is that the entanglement in the easy-plane regime is more robust against temperature than in the easy-axis regime, although, in the latter, the amount of entanglement (negativity) is larger for large enough values of |D| and low temperatures. This can be understood intuitively if we look at the qualitative effect of the
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single-ion anisotropy on the system in the ground state. When D < 0, we see from Eq. (1) that the states with high ⟨S z ⟩ are energetically favorable. Then, when we increase D in this regime, the spin-S ions tend to point along the z-axis (up or down). On the other hand, when D > 0, the states with low ⟨S z ⟩ are preferable, and thus, when D is increased in this regime, the spin-S ions turn towards a direction in the xy-plane. Since there are infinitely many directions along which the spin-S ions can point within the plane, we say that the system has more degrees of freedom in this case, and therefore if we use the decoherence argument above, we arrive at the conclusion that the entanglement in this case is less compared to the easyaxis case in which we only have two possibilities when D → −∞. Thus we conclude this section by saying that, although the entanglement is more robust against temperature for the easy-plane regime, we find that, in general, the single-ion anisotropy improves the characteristic temperature above which the quantum behavior disappears. 4. Summary We have studied the ground-state and the thermal entanglement in the mixed-spin (1, 1/2) Heisenberg chain with single-ion anisotropy using exact diagonalization of the Hamiltonian. We find that the negativity, as a measure of pairwise entanglement, exhibits a dip just at the isotropic point where the system undergoes a continuous quantum phase transition. This is explained with the help of the pairwise purity, which shows a minimum at the critical point and thus indicates a maximally mixed ground state where the quantum coherence is weakened. When finite temperatures are considered, the negativity decreases with the increase of the temperature until it vanishes at a threshold temperature. We show that in the easy-plane regime of the single-ion anisotropy the entanglement is more robust against the temperature, but, in general, the threshold temperatures increase with the increase of this anisotropy. This paper contributes to the recent efforts to find mechanisms which can improve the characteristic temperatures for an entangled state, which is of fundamental importance in the field of quantum computing. Acknowledgements This paper was supported by the Dirección de Investigaciones Sede Bogotá of the Universidad Nacional de Colombia and by the Mazda Foundation for the Arts and the Science fellowship. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
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