Long-range entanglement in the XXZ Heisenberg spin chain after a local quench
aa r X i v : . [ qu a n t - ph ] J un Long-range entanglement in the XXZ Heisenberg spin chain aftera local quench
Jie Ren ∗ and Shiqun Zhu † Department of Physics and Jiangsu Laboratory of Advanced Functional Materials,Changshu Institute of Technology, Changshu, Jiangsu 215500, China School of Physical Science and Technology,Suzhou University, Suzhou, Jiangsu 215006, China (Dated: November 15, 2018)
Abstract
The long-range entanglement dynamics of an one-dimensional spin-1/2 anisotropic XXZ modelare studied using the method of the adaptive time-dependent density-matrix renormalization-group.The long-range entanglement can be generated when a local quench on one of boundary bonds isperformed in the system. The anisotropic interaction has a strong influence both on the maximalvalue of long-range entanglement and the time of reaching the maximum long-range entanglement.The local coupling has a notable impact on the long-range entanglement, but it can be neglectedin the time of reaching the maximal long-range entanglement.PACS number: 03.67.Mn, 03.65.Ud, 75.10.PqKeywords: long-range entanglement, spin chain, adaptive time-dependent density-matrixrenormalization-group ∗ E-mail: [email protected] † E-mail: [email protected] . INTRODUCTION Entanglement generation and distribution is one of the important problems in performingquantum-information tasks, such as quantum computation and quantum teleportation [1–3]. Many results showed that entanglement existed naturally in the spin chain when thetemperature is at zero [4, 5]. It is discouraging that the entanglement in many spin systemsis typically very short ranged. It exists only in the nearest neighbors and the next nearestneighbors [6]. It is interesting that some schemes for long-range entanglement, such asexploiting weak couplings of two distant spins to a spin chain were proposed [7–9]. Thesemethods have limited thermal stability or very long time scale of entanglement generation.In recent years, many researches has shifted the focus of the study in the dynamics ofentanglement [10–14]. The dynamics of long range entanglement are obtained by a timequench of magnetic field [15, 16]. The end-to-end entanglement can be shared across a chainof arbitrary range [10]. It is excited that the most common situation studies so far concernsa sudden quench of some coupling of the model Hamiltonian [17–19]. In Ref. [19], it showedthat the long-range entanglement can be engineered by a non-perturbative quenching of asingle bond in a Kondo spin chain with impurity. This is the first example that a minimallocal action on a spin chain can generate long-range entanglement dynamically. It would beinteresting to investigate the possibility of producing long-range entanglement in the XXZHeisenberg spin chain without impurity.It is well-known that the Hamiltonian of an opened chain of N spin-1 / H = N − X i =1 J [ S xi S xi +1 + S yi S yi +1 + ∆ S zi S zi +1 ] , (1)where S αi ( α = x, y, z ) are spin operators on the i -th site, N is the length of the spin chain.The parameter ∆ denotes the couplings in the z -axis.This model can be realized in theJosephson-junction[20]and optical lattices[21, 22].In the paper, the long-range entanglement in the XXZ Heisenberg spin chain after alocal quench is investigated. An opened boundary condition (OBC) is assumed becausethe antiferromagnetic Heisenberg spin chain with OBC can be achieved artificially in theexperiment [23], and the coupling J = 1 is considered for simplicity. In section II, the localquench is presented. The concurrence is used as a measurement of the entanglement. In2ection III, a more general situation and a single quench and its effect on the end-to-endqubits entanglement are analyzed. The effects of the anisotropic interaction and system sizeon the end-to-end qubits entanglement is also studied. The robustness of the entanglementagainst an increase in temperature is investigated. A discussion concludes the paper. II. LOCAL QUENCH AND ENTANGLEMENT MEASURE
In the paper, the system is assumed to be in the ground state Gs of H initially. A localquench change of the coupling between the first qubit and the second qubit with the sameanisotropy interaction ∆ in Eq. (1) is performed. The Hamiltonian of the system modifiesto H = J [ S x S x + S y S y + ∆ S z S z ] + N − X i =2 J [ S xi S xi +1 + S yi S yi +1 + ∆ S zi S zi +1 ] . (2)Since [ H , H ] = 0, the the ground state Gs of H is not one of the eigenstates of H . Thestate of the system will evolve as ψ ( t ) = exp − iH t Gs . (3)In the paper, the concurrence is chosen as a measurement of the pairwise entanglement[3]. The concurrence C is defined as C ,N = max { λ − λ − λ − λ , } , (4)where the quantities λ i ( i = 1 , , ,
4) are the square roots of the eigenvalues of the operator ̺ = ρ ( σ y ⊗ σ yN ) ρ ∗ ,N ( σ y ⊗ σ yN ). They are in descending order. The case of C ,N = 1corresponds to the maximum entanglement between the two qubits, while C ,N = 0 meansthat there is no entanglement between the two qubits. III. LONG-RANGE ENTANGLEMENT DYNAMICS
It is known that it is hard to calculate the dynamics of entanglement because of the lackof knowledge of eigenvalues and eigenvectors of the Hamiltonian. For models that are notexactly solvable, most of researchers resort to exact diagonalization to obtain the ground3tate for small system size. The method is difficult to be applied for large system size
N >
20. For a large system, the adaptive time-dependent density-matrix renormalization-group can be applied with a second order Trotter expansion of the Hamiltonian as describedin [24, 25]. In order to check the accuracy of the results of the adaptive time-dependentdensity-matrix renormalization-group, the results of exact diagonalization can be consideredas a benchmark for a small size system.The numerical error of the adaptive time-dependent density-matrix renormalization-group comes from the discarded weight and the Trotter decomposition. The error of dis-carded weight is dependent on the data precision and truncated Hilbert space. The errorof the Trotter decomposition is relied on Trotter slicing. In our numerical simulations aTrotter slicing δt = 0 .
05 and Matlab codes of the adaptive time-dependent density-matrixrenormalization-group with double precision are performed with a truncated Hilbert spaceof m = 100. In turns out that a typically discarded weight of δρ ≤ − . The error ofTrotter decomposition δ ∝ ( δt ) . These can keep the relative error δC in C below 10 − fora chain of N = 60 sites with time t ≤ /J .In the adaptive time-dependent density-matrix renormalization-group, it is hard to calcu-late the reduced density matrix ρ ,N . By making use of the relation between the correlationand the reduced density matrix, the reduced density matrix can be expressed as ρ ,N = 14 [ I ,N + X i,j = x,y,z h σ i σ jN i σ i σ jN ] , (5)where σ αk ( α = x, y, z ) are Pauli operators and I ,N is identity matrix.We obtain the reduced density matrix ρ ,N by Eq. (5), then calculate the entanglementbetween two-ends qubits. The end-to-end entanglement C ,N is plotted as a function ofthe anisotropic interaction ∆ and time t with J = − .
01 in Fig. 1(a). It is seen thatthe long-range entanglement C ,N can be generated after a short time. When the time t increases, the entanglement reaches the maximal value and then disappears quickly. Thatis, there is a peak in the entanglement. The peak of the end-to-end entanglement is plottedas a function of the anisotropic interaction ∆ in Fig. 1(b). It is seen that the peak inlong-range end-to-end entanglement increases with the anisotropic interaction ∆. It reachesthe maximal value when the anisotropic interaction ∆ = 1 [26, 27]. With the anisotropicinteraction ∆ increases further, the height of the peak decreases.4he relation between the time of reaching the maximal end-to-end entanglement labeledby T max and the anisotropic interaction ∆ can be seen in the inset of Fig. 1(b). The time ofreaching the maximal long-range entanglement decreases when the anisotropic interaction∆ increases, while they are also not a linear relationship. Similar to Refs. [10, 19], thesystem generates end-to-end entanglement periodically. In our simulations, the error of thesimulations increases when the time increases. Since the coherent time of the system is notvery long, the long-distance entanglement is not shown when the time t > /J . It is shownthat a sharp drop of long-distance entanglement occurs at ∆ = − .
5. Furthermore, thetime of the long-range entanglement reaching the maximal value goes up drastically. Thisis similar to the anomalous behavior appeared in Ref. [27].The end-to-end entanglement C ,N is plotted as a function of the interaction J and thetime t with ∆ = 1 in Fig. (2). The size of the system N = 20 is chosen. It does notinclude the case of J = 0. The long-range entanglement C ,N can be generated after ashort time. There is a peak in C ,N when C ,N is plotted as a function of time t . It is seenthat the influence of the coupling J on the maximal long-range entanglement is relativelylarge. The peak of the end-to-end entanglement increases when the coupling interaction J increases, and reaches the maximal value when J = − .
1. It seems that the changing of localinteraction from antiferromagnetic to ferromagnetic may enhance the entanglement creation[7]. When J increases further to J > .
32, the long-range entanglement disappears. It isinteresting that the coupling interaction J has a relatively small impact on the time whenthe end-to-end entanglement generates and reaches the maximal value.It is easy to obtain that [ H , P Ni =1 S zi ] = [ H , P Ni =1 S zi ] = 0. It means that the groundstates of H is a total singlet S tot = 0. The Hamiltonian H has invariant on every excitationsubspaces. During the evolution, the boundary spin at i = 2 will have a strong tendencyto form a singlet pair with its only nearest neighbor i = 3 on the right-hand side. This issimilar for spin pairs (4 ,
5) and (6 , J decides the ability of forming singlet for even bands. Thus, thelong-distance entanglement creates.The thermalization and relaxation during the period of generating long-distance entan-glement are neglected because the dynamical time scale is quite short. When initial state istaken to be the relevant thermal state, the end-to-end entanglement is plotted as a functionof temperature in Fig. 3 after bond quenching for different system sizes. It is shown that5he entanglement vanishes when kT > .
16 with the size of N = 8, and kT > .
06 with thesize of N = 10. It seems that our scheme is quite robust against temperature and is similarto that in Ref. [19]. The initial thermal state does not have long-distance entanglement inEq. (1), but the energy gap between the ground state and the first and other excited statesare larger than the system with impurity. This leads to our scheme is quite robust againsttemperature.It is interesting to investigate the long-range entanglement creation even for very longchain of large size. The maximal value of the end-to-end entanglement C ,N is plottedas a function of the size of the system for different anisotropic interaction ∆ and differentinteraction J in Fig. 4(a). It is found that the maximal value of the end-to-end entanglement C ,N decreases when the size of system increases. When the anisotropic interaction ∆ = 1and the interaction J = − .
1, the maximal value is 0.5481 for N = 40 and 0.4547 for N = 60. It is shown that the maximal value of the end-to-end entanglement label by ξ ( N )is given by [10] ξ ( N ) ≃ . N − / . (6)In Ref. [10], ξ (40) ≃ . N = 40, and ξ (40) ≃ . N = 60. While in theKondo regime, ξ (40) can be as large as 0 . N = 40. It seems that our results arealso quite good. The long distance entanglement can be obtained by performing a localquench in either one of two boundary bands. The time of reaching the maximal end-to-end entanglement labeled by T max is plotted as a function of the size N of the system fordifferent anisotropic interaction ∆ and different interaction J in Fig. 4(b). The time ofthe end-to-end entanglement reaching the maximal value T max is linear increase when thesize N increases [10, 27]. It is seen that the slope of the line is almost dependent on theanisotropic interaction ∆.In the paper, adiabatic quenches are studied and the decoherence effects in the systemare ignored. It is noted that the XXZ spin chains can be realized in the Josephson-junctionarray [20]and optical lattices [21, 22]. In Josephson-junction array, the interactions andthe anisotropy can be modulated by varying voltages [28]. The Josephson-junction arraysystem does not have significant decoherence over our time-scales ( T max ≃ N/J ) [29]. Theanisotropic interaction quenches can be achieved successfully with long decoherence time inthe optical lattices experiment [22]. 6
V. DISCUSSION
By using the method of the adaptive time-dependent density-matrix renormalization-group, the time evolution of the entanglement in a one-dimensional spin-1/2 anisotropicXXZ model is investigated when a local quench is performed in the system. The localquench is a abrupt change of interaction between the first qubit and the second qubit.The dynamics of pairwise entanglement between the two ends qubits in the spin chain isstudied. The entanglement of the two-ends spin qubits can be created after the local quenchis performed.The time when the long-range entanglement generates decreases in a nonlinearmanner with the anisotropic interaction increases. It reaches the maximal value when theanisotropic interaction ∆ = 1. The maximal value of the end-to-end entanglement increaseswith the coupling interaction J increases, and reaches the maximal when J = − .
1. Itis interesting that the coupling interaction has a relatively small impact on the time whenthe long-range entanglement reaches the maximal value. This phenomenon may be usedto control the dynamics of the entanglement by varying the anisotropic interaction and thelocal quench interaction of the Heisenberg spin chain.
Acknowledgments
It is a pleasure to thank Yinsheng Ling and Yinzhong Wu for their many helpful discus-sions. The financial support from the National Natural Science Foundation of China (GrantNo. 10774108) is gratefully acknowledged. [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cam-bridge University Press, Cambridge, England, 2000).[2] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev.Lett. , 1895 (1993).[3] W. K. Wootters, Phys. Rev. Lett. , 2245 (1998).[4] T. J. Osborne and M. A. Nielsen, Phys. Rev. A , 032110 (2002).[5] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature , 608 (2002).[6] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. , 517 (2008).[7] L. C. Venuti, C. D. E. Boschi, and M. Roncaglia, Phys. Rev. Lett. , 247206 (2006).
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1. The size of the system is N = 20. (b). The maximal value ofthe end-to-end entanglement is plotted as a function of the anisotropic interaction ∆. The insetshows the time of reaching the maximal end-to-end entanglement labeled by T max as a function ofanisotropic interaction ∆. C , N FIG. 2: The long-range entanglement C ,N is plotted as a function of interaction J and time t when there is local quench of the interaction J . The size of the system is N = 20 and theanisotropic interaction ∆ = 1. It does not include the case of J = 0. .0 0.3 0.6 0.9 1.2 1.50.00.20.40.60.81.0 N=10 C , N kT N=8
FIG. 3: The maximal value of end-to-end entanglement is plotted as a function of temperatureafter band quenching for different sizes. (b) T m a x N =1 J =0.1 =0.5 J =0.1 =1 J =-0.1 =0.5 J =-0.1 C m a x , N N =1 J =0.1 =0.5 J =0.1 =1 J =-0.1 =0.5 J =-0.1 (a) FIG. 4: (a). The maximal value of the long-range entanglement C ,N is plotted as a function ofthe size of the system for different anisotropic interaction ∆ and different interaction J . (b). Thetime of reaching the maximal long-range entanglement labeled by T max is plotted as a function ofthe size of the system for different anisotropic interaction ∆ and different interaction J . The redlines are fixed lines.. The redlines are fixed lines.