The entanglement of the XY spin chain in a random magnetic field
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The entanglement of the XY spin chain in a random magnetic field Masashi
Fujinaga and Naomichi Hatano ∗ Department of Physics, University of Tokyo, Komaba, Meguro, Tokyo 153-8505 Institute of Industrial Science, University of Tokyo, Komaba, Meguro, Tokyo 153-8505
We investigate the entanglement of the ferromagnetic XY model in a random magneticfield at zero temperature and in the uniform magnetic field at finite temperatures. We use theconcurrence to quantify the entanglement. We find that, in the ferromagnetic region of theuniform magnetic field h , all the concurrences are generated by the random magnetic field andby the thermal fluctuation. In one particular region of h , the next-nearest neighbor concurrenceis generated by the random field but not at finite temperatures. We also find that the qualitativebehavior of the maximum point of the entanglement in the random magnetic field depends onwhether the variance of its distribution function is finite or not. KEYWORDS: entanglement, concurrence, entanglement of formation, thermal entanglement, decoherence,random field, XY model
1. Introduction
We study in the present paper the pairwise entan-glement of the ferromagnetic spin-1/2 XY chain in arandom magnetic field at zero temperature and in theuniform magnetic field at finite temperatures. The en-tanglement
1, 2 is one of the most interesting features ofquantum mechanics. It has the property of non-localityoriginating in the principle of superposition. One typicalexample of the state which has the property of the en-tanglement is the singlet | ψ i = 1 / √ | i − | i ). Themeasurement on one particle of this state affects theother particle immediately, even if two particles are faraway from each other. This non-locality of the entangledstate puzzled many people including Einstein in the earlytimes when the quantum mechanics was born; Einsteinthought that quantum mechanics was an incomplete the-ory because of the non-locality. However, violation of theBell inequality, or more generally the CHSH inequal-ity, showed that the non-locality is a reality. Quantummechanics is now widely accepted, having explained alot of phenomena which were not explained by classicalmechanics.The quantum information processing such as quan-tum teleportation and super dense coding have beenheavily studied recently. The entanglement is vital forimplementation of such techniques. In reality, however,the entanglement may be destroyed by some decoherenceeffects such as the thermal fluctuation and a randommagnetic field. Thus, it is important to know how theentanglement is affected by thermal and impurity dis-turbances. Effects of impurities on the entanglement arealso interesting from the viewpoint of the relation be-tween quantum coherence and impurities.In the present paper, we calculate the entanglementbetween two spins of the ferromagnetic isotropic XY chain in a random magnetic field as well as at finite tem-peratures. To our knowledge, this is the first to study sys-tematically the dependence of the entanglement on therandomness in a spin system. Li et al. studied the de- ∗ Corresponding author. E-mail: [email protected] pendence in the one-electron Anderson model in one di-mension. The conclusion that the entanglement increases due to the randomness in some parameter regions (seebelow) is common to both studies. (After submitting thepresent paper, we noticed a study on the entanglementin random quantum spin- S chains. It, however, is notquite related to the present issue; the study concerns ascaling law of the entanglement entropy in the randomsinglet phase.)There have been a couple of works which studied theeffects of the temperature on the entanglement of spinsystems. Arnesen et al. mentioned that the nearest-neighbor entanglement of the anti-ferromagnetic Heisen-berg chain can be increased by introducing the temper-ature in a uniform magnetic field. Similar work has beendone by Nielsen on the two-spin Heisenberg model. Os-borne and Nielsen studied the nearest-neighbor and thenext-nearest-neighbor entanglement of the anisotropic XY chain and the ferromagnetic transverse Ising chain.Although their main interest is in the entanglementnear the quantum ground-state phase transition, theyalso mentioned calculation of the entanglement at finitetemperatures. Yano and Nishimori
13, 14 also mentioneda finite-temperature calculation of the nearest-neighborentanglement on the anti-ferromagnetic anisotropic XY model. Their results for the nearest-neighbor entangle-ment are almost the same as ours. Our conclusion thatthe entanglement also increases due to the thermal fluc-tuation in some parameter regions (see below) is commonto the above-mentioned studies. None, however, com-pared the entanglement in a random magnetic field andthat at finite temperatures quantitatively in the samemodel.We compute the entanglement between two spins fromthe nearest-neighbor pair to the fifth-neighbor pair. Weuse the concurrence to quantify the pairwise entangle-ment. We find the following:(1) In general, the entanglement is decreased as the ran-domness is increased;(2) In the region of the uniform magnetic field h >
1, the
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Author Name entanglement is increased by the random magneticfield and by the temperature. In yet another region h < /
2, it is increased by the random field but notby the temperature.(3) Qualitative behavior of the maximum point of theentanglement depends on the random magneticfield, in particular, whether the variance of the dis-tribution function is finite or not.(4) The entanglement between two spins at finite tem-peratures becomes weaker than that in the randommagnetic field at zero temperature as the distancebetween the two spins gets greater.The present paper is organized as follows. In §
2, we intro-duce the model and review the outline of computation ofits correlation functions, which are necessary to quantifythe concurrence. In §
3, we calculate the concurrence for:in § XY spin chain in the uniform magnetic fieldat zero temperature; in § XY spin chain in a ran-dom magnetic field at zero temperature; in § XY spin chain in the uniform magnetic field at finite tem-peratures. Finally, we give a summary and discussions in §
2. The model and the entanglement
The Hamiltonian of the XY spin chain in a randommagnetic field is given in the form H = − J N X j =1 (cid:0) σ xj σ xj +1 + σ yj σ yj +1 (cid:1) − N X j =1 ( h + h j ) σ zj , (1)where J ( >
0) is the coupling constant, N is the numberof the spins, σ α ( α = x, y, z ) are the Pauli matrices, h is the uniform magnetic field and { h j } are the randommagnetic field. We impose the periodic boundary condi-tions: σ αN +1 = σ α , ( α = x, y, z ) . (2)Hereafter, the coupling constant J is set to one. The ran-dom magnetic field h j at each site obeys the distributionfunction P q,a ( h j ) ∼ (cid:2) a − (1 − q ) h j (cid:3) − q , (3)where the parameter q determines the type of the distri-bution function and a determines the width of the dis-tribution function. In particular, eq. (3) is reduced to aGaussian distribution function as q →
1. In this case, thescale parameter a is its standard deviation. Equation (3)is also reduced to a Lorentzian distribution function for q = 2. In this case, the scale parameter a is its half widthat half maximum. The variance of the distribution func-tion (3) diverges for q ≥ / q < /
3. Inthe case of the Lorentzian distribution q = 2, Nishimori analytically calculated the average one-point correlationfunction and obtained lower bounds of the average two-point correlation functions. The results in the paper, however, are not used in the present paper, since we takethe random average of the concurrence, which is a non-linear function of the one-point and two-point correlationfunctions. We diagonalize the Hamiltonian (1) as follows. TheHamiltonian can be expressed by the Fermi operators a † and a after the Jordan-Wigner transformation. TheHamiltonian is then reduced to the quadratic form H = N X i,j =1 a † i A ij a j (4)with A = − h − h − · · · ± − − h − h − . . . 00 − . . . . . . ...... . . . · · · . . . − ± · · · − − h − h N , (5)where we dropped a constant term in the Hamilto-nian (4). The signs of the (1 , N ) and ( N,
1) elementsin eq. (5) are negative when the number of the Fermionsin the system is even and positive when odd. The Her-mitian matrix A is diagonalized by a unitary matrix V .We thus have H = N X i =1 ǫ i c † i c i , (6)where the operators c † i and c i are given by c † i = N X l =1 a † l V li , c i = N X l =1 V † il a l (7)and satisfy the anti-commutation relations { c † i , c j } = δ ij and { c i , c j } = 0. In order to quantify the entanglement, we use the con-currence related to the entanglement of formation. The concurrence between the spins at sites i and j iscalculated from the two-site density matrix ρ ij as C i,j = max { , λ − λ − λ − λ } , (8)where { λ i } i =1 are the square roots of the eigenvalues ofthe matrix R = ρ ij ˜ ρ ij in non-ascending order, λ ≥ λ ≥ λ ≥ λ with ˜ ρ ij = (cid:0) σ yi ⊗ σ yj (cid:1) ρ ∗ (cid:0) σ yi ⊗ σ yj (cid:1) . The complexconjugation is taken in the σ z basis. The two-site densitymatrix ρ ij is defined by ρ ij = Tr ˆ ij ρ, (9)where Tr ˆ ij denotes the trace over the degrees of freedomexcept for the sites i and j , and ρ is the density matrixof the whole system: ρ = e − βH /Z .The two-site density matrix can be expanded in termsof the identity matrix and the Pauli matrices as ρ ij = 14 X α,β =0 p αβ σ αi ⊗ σ βj , (10) . Phys. Soc. Jpn. Full Paper
Author Name 3 where σ i denotes the identity operator on the site i , σ i = σ xi , σ i = σ yi and σ i = σ zi . The coefficients p αβ are realnumbers determined by p αβ = Tr (cid:16) σ αi σ βj ρ ij (cid:17) = h σ αi σ βj i . (11)Hence, sixteen coefficients are needed to determine thetwo-site density matrix in general. Thanks to the sym-metry of the Hamiltonian (1), the number of the coeffi-cient is reduced to four; we need h σ zi i , h σ zi i and h σ αi σ αj i ( α = 1 ,
3) only. The others are zero. Hence, the two-sitedensity matrices of the model take the form ρ ij = 14 (cid:16) I ij + h σ zi i σ zi ⊗ I j + h σ zj i I i ⊗ σ zj + X α =1 h σ αi σ αj i σ αi ⊗ σ αj (cid:17) , (12)where σ = σ x , σ = σ y , σ = σ z and h σ i σ j i = h σ i σ j i .The correlation functions are obtained as follows: h σ zi i = G i,i , (13) h σ zi σ zj i = (cid:12)(cid:12)(cid:12)(cid:12) G i,i G i,j G j,i G j,j (cid:12)(cid:12)(cid:12)(cid:12) , (14) h σ xi σ xj i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G i,i +1 G i,i +2 · · · G i,j G i +1 ,i +1 G i +1 ,i +2 · · · G i +1 ,j ... ... . . . ... G j − ,i +1 G j − ,i +2 · · · G j − ,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (15)where G i,j are given as follows in the three cases:i) The XY spin chain in the uniform magnetic fieldat zero temperature. In the thermodynamic limit, { G i,j } are given by G i,i = (cid:26) h > J, − π arccos( − hJ ) for h < J, (16) G i,j = h > J, π l − m sin (cid:2) ( i − j ) arccos( − hJ ) (cid:3) for h < J. (17)ii) The XY spin chain in the uniform magnetic fieldat finite temperatures. In the thermodynamic limit, { G i,j } are given by G i,j = − δ ij + 2 π Z π d φ cos( i − j ) φ − β ( J cos φ + h )) . (18)iii) The XY spin chain in a random magnetic field atzero temperature. In this case, { G i,j } are given by G i,j = 2 N G X l =1 V il V jl − δ ij , (19)where V is the matrix diagonalizing the matrix A in eq. (5) and N G is the number of the Fermions.In the ground state, the Fermions are filled in thelevels with ǫ i < Now that the coefficients in eq. (10) have been ob-tained, we can evaluate the concurrence. We here definethe average concurrence as the random and spatial aver-age: C ( r ) = 1 N N X i =1 [ C i,i + r ] av , (20)where [ · · · ] av denotes the random average, C i,j denotesthe concurrence between the sites i and j , and N is thenumber of the sites. In the absence of the random mag-netic field, the Hamiltonian possesses the translationalinvariance and the averaging is not necessary. We notethat the entanglement of formation after the randomaverage and the spatial average is always greater thanthat obtained by substitution of the average concurrenceinto the relation between the entanglement of formation E and the concurrence CE ( C ) = − √ − C √ − C ! − − √ − C − √ − C ! , (21)since E ( C ) is a concave function of the concurrence C .Hereafter, we simply refer to the average concurrence asthe concurrence.In the case of no random magnetic field, we calculatedthe concurrence rigorously in the thermodynamic limit.In the case with a random magnetic field, we numericallyevaluated the sample average of the concurrence (20). Forall the results below in the random case, the number ofthe sites N is 500 and the number of the samples is 10000.In Fig. 1(a), the next-nearest-neighbor concurrence C (2)is plotted for q = 2 (the Lorentzian distribution) witherror bars at h = 0, 0.5, 1, 1.5, 2, 2.5 and 3, but the er-rors are almost invisible. In Fig. 1(b), the next-nearest-neighbor concurrence C (2) is plotted for q = 2 and forthe system size N = 10, 100, 250 and 500 with the ran-dom average over 10000 samples. The finite-size effectis invisible for N ≥ N = 500 are substantial. Wecalculated the average concurrence C ( r ) for 1 ≤ r ≤
3. Numerical results XY spin chain in a uniform magnetic field at zerotemperature We first study the concurrence of the XY spin chain inthe uniform magnetic field. Figure 2 shows the nearest-neighbor concurrence C (1), the next-nearest-neighborconcurrence C (2), the third-neighbor concurrence C (3)and the fourth-neighbor concurrence C (4). All the con-currences rapidly decrease near h = 1, where the quan-tum phase transition occurs, and vanish in the region h >
1. In the region h >
1, the ground state is given bythe tensor product of the one-spin state | ↑ i as | GS i = | ↑ i| ↑ i| ↑ i · · · , (22) J. Phys. Soc. Jpn.
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Author NameFig. 1. (Color online) (a) The next-nearest-neighbor concurrence C (2) for q = 2 (the Lorentzian distribution) with the scale pa-rameter a = 0 .
3, 0.7, 1. All the lines are plotted as a function ofthe uniform magnetic field h . The random average is taken over10000 samples with the system size 500. (b) The next-nearest-neighbor concurrence C (2) for q = 2. The lines are plotted withthe scale parameter a = 0 . N = 10, 100, 250, 500. where the state | ↑ i i denotes the eigenstate of the ma-trix σ zi satisfying the eigenequation σ zi | ↑ i i = | ↑ i i . Sincethere is no superposition involved, the entanglement van-ishes in h > XY spin chain in a random magnetic field at zerotemperature Next, we study the concurrence in a random magneticfield (in addition to the uniform magnetic field h ) atzero temperature. The random magnetic field obeys thedistribution function (3); we investigate the cases for q =1, 1.35, 5/3, 1.85 and 2.The nearest-neighbor concurrence C (1) in all cases be- Fig. 2. (Color online) The concurrence of the XY spin chain ina uniform magnetic field as a function of the uniform magneticfield h .Fig. 3. (Color online) The nearest-neighbor concurrence C (1) ofthe XY spin chain in a random magnetic field at zero tempera-ture; (a) for q = 1 (the Gaussian distribution); (b) for q = 2 (theLorentzian distribution). All the data are plotted as functions ofthe uniform magnetic field.. Phys. Soc. Jpn. Full Paper
Author Name 5Fig. 4. (Color online) The next-nearest-neighbor concurrence C (2) in a random magnetic field at zero temperature; (a) for q = 1 (theGaussian distribution); (b) for q = 1 .
35; (c) for q = 5 /
3; (d) for q = 1 .
85; (e) for q = 2 (the Lorentzian distribution). All the data areplotted as functions of the uniform magnetic field. haves similarly. In the region h <
1, the nearest-neighborconcurrence in a random magnetic field for each q de-creases as the distribution width a is increased. We hereshow in Fig. 3 only the cases q = 1 and q = 2. The reduc-tion of the nearest-neighbor concurrence C (1) is greateras the scale parameter a is increased. For h >
1, thenearest-neighbor concurrence for all q is increased by therandom magnetic field. That is, the random magneticfield increases the quantum correlation. The reason why the nearest-neighbor concurrence is increased for h > C (2) for each q is decreased for 0 . < h < a is increased. On the other hand, it isincreased for h < / h > a J. Phys. Soc. Jpn.
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Author NameFig. 5. (Color online) Smaller arrows (blue) indicate spins. Biggerarrows (grey) indicate the random magnetic field. (a) The ran-dom magnetic field happens to be almost constant over a spatialregion. (b) The random magnetic field is extremely strong at oneparticular site. is increased. This is in contrast to the finite-temperaturecase in the next subsection, where we show that the next-nearest-neighbor concurrence C (2) for h < / not increased at finite temperatures.The reason of the increase for h > C (1). We have notbeen able to determine decidedly the real reason whythe next-nearest-neighbor concurrence C (2) is increasedfor h < /
2. We, however, can consider some situationswhere the next-nearest-neighbor concurrence is increasedfor h < /
2. Let us first consider the case where the vari-ance of the distribution function is finite, i.e. q < / a over a region of considerable length; see Fig. 5(a).The uniformness then may generate the concurrence inthe region. We can give an argument for this specula-tion. In Fig. 4(a) and (b), the concurrence without theuniform field, h = 0, is generated only for a ≥ .
5. Thisis consistent with the fact that the concurrence withoutthe randomness is zero for h < .
5; if our speculation iscorrect, the concurrence is generated by an “almost uni-form” random field only when the field is greater than0.5.In the case where the variance of the distribution func-tion is infinite, i.e. q ≥ /
3, we could think of a moreplausible situation. In this case, a singularly strong ran-dom field can appear at a site as illustrated in Fig. 5(b).We then can take the Zeeman energy of the site as thenon-perturbation term and calculate the second-orderperturbation of the exchange energy. We may end upwith an effective interaction between the two spins be-side the strong magnetic field. In this situation, the next-nearest-neighbor concurrence may be restored aroundthe strong field.We find that the qualitative behavior of the next-nearest neighbor concurrence C (2) is different depend-ing on whether the variance of the distribution functionis finite or not. The maximum point of the next-nearest-neighbor concurrence for q < /
3, where the variance ofthe distribution function is finite, shifts to the right asthe randomness a is increased as shown in Fig. 4(a) and(b). In contrast, the maximum point of the next-nearest-neighbor concurrence for q ≥ / a is increased. Fig. 6. (Color online) The third neighbor concurrence C (3) in arandom magnetic field at zero temperature; (a) for q = 1 (theGaussian distribution); (e) for q = 2 (the Lorentzian distribu-tion). All the data are plotted as functions of the uniform mag-netic field. The third-neighbor concurrence C (3) and the rest, C (4) and C (5), behave similarly to the next-nearest-neighbor concurrence C (2), only smaller than the next-nearest-neighbor concurrence. The maximum point ofthe third-neighbor concurrence and the rest for the cases q ≤ / XY spin chain in a uniform magnetic field at finitetemperatures Third, we investigate the concurrence of the XY spinchain in the uniform magnetic field at finite tempera-tures; see Fig. 7. The nearest-neighbor concurrence C (1)is decreased for h <
1, whereas it is increased for h > h > h >
1. Wecan hardly see the essential difference between the effectsof the random magnetic field and the temperature on thenearest-neighbor concurrence C (1); compare Fig. 3 andFig. 7(a). . Phys. Soc. Jpn. Full Paper
Author Name 7Fig. 7. (Color online) The concurrence in a uniform field at fi-nite temperatures; (a) The nearest-neighbor concurrence C (1);(b) The next-nearest-neighbor concurrence C (2); (c) The thirdneighbor concurrence C (3). The next-nearest neighbor concurrence C (2) is de-creased for 0 . < h <
1. The increase of the concurrenceappears only for h >
1; the concurrence for h < . C (2) for h < / C (3) and the rest, C (4) and C (5), behave similarly to the next-nearest-neighbor concurrence C (2) except for quantitative differ-ence.Finally, we study the maximum concurrence as a func-tion of the scale parameter a or the temperature kT .The reduction of the maximum concurrence as a func-tion of the scale parameter or the temperature is plot-ted in Fig. 8. The nearest-neighbor concurrence C (1)decreases for the random magnetic field more rapidlythan that at finite temperatures in the plotted ranges. Inthe same plot ranges, however, the third-neighbor con-currence C (3) in the random magnetic field remains fi-nite, whereas the third-neighbor concurrence C (3) at fi-nite temperatures almost vanishes for kT ≥ .
07. Wethus observe that, as the distance between the two spinsincreases, the concurrence becomes considerably weakagainst the thermal fluctuation.
4. Summary and discussions
We have studied the entanglement of the XY spinchain in a random magnetic field at zero temperatureand in a uniform field at finite temperatures. We foundthat: (i) In general, the entanglement is decreased bythe random magnetic field and the temperature; (ii) Theentanglement is increased by the random magnetic fieldand the temperature in some parameter regions. Thatis, quantum correlation can be both increased and de-creased by the disturbances. In particular, we find thatthe next-nearest-neighbor concurrence C (2) for h < / h >
1. The increase of the concurrence for h < / J. Phys. Soc. Jpn.
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Author NameFig. 8. (Color online) (a) The maximum point of the concurrencein a uniform magnetic field at finite temperatures as a function ofthe temperature kT ; (b) The maximum point of the concurrencein a random magnetic field at zero temperature for q = 1 as afunction of the scale parameter a ; (c) The maximum point of theconcurrence in a random magnetic field at zero temperature for q = 2 as a function of the scale parameter a . Acknowledgment
We are grateful to Dr. Akinori Nishino for useful sug-gestions and advice. We acknowledge support by Grant-in-Aid for Scientific Research (No. 17340115) from theMinistry of Education, Culture, Sports, Science andTechnology as well as support by Core Research for Evo-lutional Science and Technology (CREST) of Japan Sci-ence and Technology Agency. The use of facilities at theSupercomputer Center, Institute for Solid State Physics,University of Tokyo is gratefully acknowledged.
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