Quantum discord in the transverse field XY chains with three-spin interaction

Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Q2 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Contents lists available at ScienceDirect

Physica B journal homepage: www.elsevier.com/locate/physb

Quantum discord in the transverse field XY chains with three-spin interaction Shuguo Lei a,b, Peiqing Tong a,c,n a

Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, PR China College of Science, Nanjing Tech University, Nanjing 211816, PR China c Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, PR China b

art ic l e i nf o

a b s t r a c t

Article history: Received 1 September 2014 Received in revised form 20 November 2014 Accepted 1 January 2015

The ground state quantum discord in the transverse field anisotropic and isotropic XY chains with XZY– YZX type three-spin interaction has been studied. The three-spin interaction induces new gapless quantum phases in the transverse field anisotropic XY chain besides the ferromagnetic and paramagnetic phases. It is found that the first-order derivative of the quantum discord at the Ising type transition between the gaped phases has a logarithmic divergence scaling with the system size. However, the firstorder derivative of the quantum discord at the quantum phase transitions between the gaped and gapless phases does not increase with the system size. For the transverse field isotropic XY chain, the first-order derivative of the quantum discord at the quantum phase transitions between the gapless phases has a similar behavior with that between the gaped and gapless phases. & 2015 Published by Elsevier B.V.

Keywords: Quantum discord Quantum phase transition Three-spin interaction The quantum spin chains

1. Introduction One of the most fascinating characters of a quantum composite system is the nonclassical correlations between its subsystems. Quantum discord (QD) introduced in Refs. [1,2] measures the pure quantumness of the correlations in the state and has attracted much attention in quantum information science [3]. It has been proposed that the QD is the resource responsible for the speed-up in the deterministic quantum computation with one quantum bit [4,5]. In the field of condensed matter physics, a quantum phase transition (QPT) [6–9] is a fundamental change in the ground state of a quantum many-body system as some parameters entering its Hamiltonian are varied. Recently, there has been a lot of work on the relation between the QD and the QPTs of one-dimensional quantum spin models [10–18]. Refs. [10–15] studied the QD between two spins of a quantum spin- 1 chain in the thermo2

dynamical limit as well as finite size, concluding that the nonanalyticities in QD can be used to identify the QPTs of a many-body system with great success even at finite temperature [16]. For example, in the transverse field Ising chain (TFIC) [10,11] and the transverse field XY chain [12,18], the first-order derivative of the ground state QD between the nearest neighbor sites with respect n Corresponding author at: Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, PR China. E-mail address: [email protected] (P. Tong).

to the external field at the critical points of the QPTs shows logarithmic scaling behaviors with the system size. Refs. [17,18] investigated the ground state QD in the transverse field anisotropic XY chains with XZX þ YZY type three-spin interaction. It is found that this type three-spin interaction in the transverse field XY chain only shifts the values of the external fields [19] of the QPTs between the gaped phases, thus the critical behavior of QD is the same as that at QPTs between the gaped quantum phases (referred as the Ising type QPTs in the following) in the TFIC and the transverse field XY chain. However, it has also shown that the extended spin chain models with other multi-spin interaction may lead to new types of quantum phases and QPTs. For instance, the inclusion of XZY–YZX type three-spin interaction in the transverse field anisotropic XY chain can bring about extra gapless chiral phases and therefore gives rise to the QPTs between the gaped and gapless quantum phases [20–23]. Hence there are two kinds of QPTs, one is between gaped quantum phases and the other is between gaped and gapless quantum phases. Although these QPTs are of continuous transitions, the critical behaviors of the von Neumann entropy and fidelity [24] at the QPTs between the gaped and gapless phases are quite different from that between the gaped phases [21]. The QD is a quantum correlation measurement that quantifies the total amount of quantumness of a state, including the nonlocal part termed by entanglement [10]. The property of the QD at the QPTs between the gaped and gapless phases is unclear. In this paper we study the QD in transverse field anisotropic XY chain with XZY-YZX type three-spin interaction. It is found that the first-order

http://dx.doi.org/10.1016/j.physb.2015.01.031 0921-4526/& 2015 Published by Elsevier B.V.

Please cite this article as: S. Lei, P. Tong, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.031i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

S. Lei, P. Tong / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

derivative of the QD at the QPTs between the gaped and gapless phases does not increase with the system size, and is different from the logarithmic scaling behaviors of the QD at the Ising type QPTs. In addition, the transverse field isotropic XY chains with XZY– YZX type three-spin interaction has two gapless quantum phases. The properties of the QD at the QPTs between the gapless quantum phases are also studied in this paper. It is found that the firstorder derivative of the QD at the QPTs between the gapless phases has a similar behavior with that between the gaped and gapless phases. The organization of the rest of the paper is as follows. In Section 2 we review the concept of the QD from the point view of the information theory. The ground state QD between the nearest neighbor spins in the transverse field anisotropic and isotropic XY chains is studied in Sections 3 and 4, respectively. Some concluding remarks are presented in the finial section.

2. The quantum discord The total common correlations of a quantum system comprising of two subsystems A and B are measured by quantum mutual information written as

I (ρ AB ) = S (ρ A ) + S (ρ B ) − S (ρ AB ),

(1)

where ρ A (ρB ) is the reduced density operator of the subsystems A (B), ρ AB is the density operator of the system as a whole, and S (ρ) = − tr(ρ log2ρ) is the von Neumann entropy of the system. The total amount of correlations can be split into the classical and quantum parts [1,2]. The classical correlations between the parts A and B are considered as the maximal amount of information that can be extracted about one system, for instance A, by performing a measurement on the other system B. It is a natural generation of classical mutual information and defined as

C (ρ AB ) = sup{B k } [S (ρ A ) − S (ρ A | {Bk })], where

S (ρ A | {Bk }) =

∑k pk S (ρ Ak )

is

(2) the

quantum

conditional

entropy with ρ Ak = trB (ρk ), ρk = (1/pk )(I ⊗ Bk ) ρ AB (I ⊗ Bk ), pk = trAB ((I ⊗ Bk ) ρ (I ⊗ Bk )) and the maximum is taken over all local projective measurements {Bk } performed only on subsystem B. The difference between the total correlations and classical correlations defines the so-called quantum discord

D (ρ AB ) = I (ρ AB ) − C (ρ AB ),

external field, γ measures the anisotropy in the XY-plane and α is the strength of XZY–YZX type three-spin interaction. Without loss of generality α is set as 0.5 throughout this section of the paper. By successive application of the Jordan–Wigner and Bogoliubov transformations, the Hamiltonian finally takes in the diagonal form (see the Supplemental Material)

H=

⎛

∑ ε k ⎜⎜ηk† ηk k

⎝

−

1 ⎞⎟ , 2 ⎟⎠

(5)

where

εk =

α sin 2k + 2

(γ sin k)2 + (h + cos k)2

is the single particle excitation energy of the spinless Bogoliubov quasi-particle associated with wave vector k. The energy spectrum εk is not always greater than zero, thus the ground state of the system corresponds to the configuration where all the states with εk < 0 are filled while the states with εk ≥ 0 are empty. Then the ground state |GS〉 yields

ηk |GS〉 = 0

if ε k ≥ 0,

ηk† |GS〉

if ε k < 0.

=0

(6)

The ground state phase diagram of the transverse field anisotropic XY chain for α ¼0.5 is shown in Fig. 1 [21]. Comparing with the transverse filed anisotropic XY chain without three-spin interaction, in case of γ < α , a new gapless phase III with nonzero chirality order is induced besides the usual gaped ferromagnetic (FM) phase I and the paramagnetic (PM) phase II. In the chiral phase, the fidelity is zero and the von Neumann entropy between a block of L contiguous spins and the rest of the chain is propor1 tional to the block size SL ∼ 3 log2L . The QPTs between the gaped and/or gapless quantum phases are of second-order. The critical behaviors of the fidelity and the von Neumann entropy at the QPTs from FM to PM phase are the same as that at the Ising type QPTs in the transverse field anisotropic chain without three-spin interaction. However, the behaviors of the fidelity and the von Neumann entropy at the QPTs between the gaped and gapless phases are similar to that at the anisotropy type transitions in the transverse field anisotropic chain [21]. All the information needed for quantifying the QD is derived from the reduced two-body density matrix ρm, n , obtained from the ground-state wave function in which all the spins except those at positions m and n have been traced out. Due to the symmetry of

(3)

which measures the amount of quantumness in the state. A nonzero D implies that the entire information about A cannot be extracted by local measurement on B.

3. The transverse field anisotropic XY chain 3.1. The model and the phase diagram In this section we study the ground state QD in the transverse field anisotropic XY chain with three-spin interaction described by the Hamiltonian [21,20]: N

H=

∑ − [(1 + γ) Slx Slx+ 1 + (1 − γ) Sly Sly+ 1 + hSlz ] l= 1

− α (Slx− 1Slz Sly+ 1 − Sly− 1Slz Slx+ 1),

(4)

where S's are the spin 1/2 operators and x,y,z stand for the spin direction and subscript the lattice index, h is the transverse

Fig. 1. The phase diagram of the transverse field anisotropic XY chain for α ¼0.5. The vertical bold line hc ¼ 1 in the top of the figure denotes an Ising transition between the FM phase I and the PM phase II, the part III in bottom stands for the chiral phase. For γ ¼0.3, the critical chiral phase lies between hc1 = 0.55445 and hc2 = 1.06518.

Please cite this article as: S. Lei, P. Tong, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.031i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

S. Lei, P. Tong / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

3

the Hamiltonian (4), the reduced two-spin density matrix of the model in natural basis {| ↑ ↑ 〉, | ↑ ↓ 〉, | ↓ ↑ 〉, | ↓ ↓ 〉} has the form [9,25,26]

ρm, n

⎛ a11 0 0 a14 ⎞ ⎜ ⎟ ⎜ 0 a22 a23 0 ⎟ =⎜ ⎟, ⁎ ⎜ 0 a23 a 33 0 ⎟ ⎜ ⁎ ⎟ 0 a 44 ⎠ ⎝a14 0

(7)

with the elements given in terms of two-spin correlation functions:

a11 =

1 4

+ M z + 〈Smz Snz 〉,

a22 =

1 4 1 4

− 〈Smz Snz 〉 = a 33,

a 44 = a14 =

− M z + 〈Smz Snz 〉, Smx Snx

−

Smy Sny

− i(

Smx Sny

+

Smy Snx

),

a23 = Smx Snx + Smy Sny + i ( Smx Sny − Smy Snx ),

(8)

where M z = 〈Smz 〉 is the magnetization. By means of the Wick theorem, the ground state spin correlation functions Smα Snβ (α , β = x, y, z) can be expressed as an expansion of pfaffians [27–29] with the elementary contraction (see the Supplemental Material):

〈Bm A n 〉 =

1 N

∑

cos[k (m − n) − 2θ k ] Sgn( − ε k ),

k

〈Bm Bn 〉 = − δmn −

2i N

∑

sin k (m − n) Θ ( − ε k ),

(9)

k

where Sgn(x) and Θ (x) is the Sign function and Heaviside step function, respectively, operators Am , Bm are introduced from Jor† † + cm, Bm = cm − cm , θ k ′s are the dan–Wigner fermions by Am = cm Bogoliubov transformation coefficients associated with Jordan– Wigner fermions and can be written explicitly as (see the Supplemental Material)

tan θ k =

γ sin k (γ sin k)2 + (h + cos k)2 − (h + cos k)

. (10)

In what follows we concentrate on the correlations between the nearest neighbor spins. By straightforward calculation it is found that the nearest neighbor spin correlation function has the form 〈Smx Smy + 1〉 = − 〈Smy Smx + 1〉 = − i〈Bm Bm + 1〉/4 . So element a14 for the nearest neighbor spins is always a real number and a23 may be a complex number depending on which phase the system is in. For the calculation of the QD, we follow the approach in Ref. [11]. Introducing a set of projectors for an arbitrary local measurement on part B given by {Bk = VΠk V †} , where {Πk = |k〉〈k| : k = 0, 1} is the set of projectors on the computational basis and V ∈ U (2) has the form

⎛ θ ⎜ cos 2 V=⎜ θ ⎜⎜ sin eiϕ ⎝ 2

θ −iϕ ⎞ e ⎟ 2 ⎟, θ ⎟ ⎟ − cos 2 ⎠

sin

(11)

where 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π can be interpreted as the azimuthal and polar angles, respectively, of a qubit over the Bloch sphere. The classical correlations of Eq. (2) thus are reduced to a general numerically optimization problem over two variables θ and ϕ. In this paper the QD given by Eq. (3) is directly calculated by minimizing over all angles θ and ϕ.

Fig. 2. The QD between the nearest neighbor spins of the anisotropic XY chain as functions of external field h with and without three-spin interaction, the black solid for α ¼ 0.0, red dashed for α ¼0.5, in all cases γ ¼0.3, N¼ 16 384. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

3.2. The ground state quantum discord Now we study the ground state QD in the transverse field anisotropic XY chain with three-spin interaction. Fig. 2 shows the QD between the nearest neighbor spins as functions of external field h in the anisotropic XY chain with and without three-spin interaction for anisotropy parameter γ ¼0.3. It is seen that the QD is influenced by the three-spin interaction only in the gapless chiral phase between hc1 and hc2 for the given anisotropy parameter. Comparing with the QD in the chain without three-site interaction, the three-site interaction first increases then decreases and finally enhances the QD again with the increase of the external field in this phase. The QD in the gapless phase is changed for the following reasons. In the gapless phase, the contraction of 〈Bm Bn 〉 in Eq. (9) for the ground state is not δ i, j and leads to different values of spin correlation functions compared with that in the chain without three-site interaction. Therefore, the density matrix ρm, m + 1 of the nearest neighbor spins determined by these spin correlation functions and then the classical as well as quantum correlations therein are different. For γ > α (not shown in the figure), the QD is in agreement with that in the TFIC [10,11] and the anisotropic XY chain [12,18] without the three-spin interaction. Fig. 3 shows the first-order derivative of the QD with respect to external field h for different system size N, and γ = 0.7( > α). It is seen that the derivative has a pronounced minimum at hmin, which approaches the critical point hc ¼1 in the thermodynamic limit. The values of the derivatives d (QD)/dh at hmin increase with the system size. The numerical results show that the derivative d (QD)/dh taken at hmin has a logarithmic divergence with the system size and can be fitted by d (QD)/dh|hmin = a + b log N . The insert in the figure shows this logarithmic divergence with a¼  0.00118 and b ¼  0.14 for γ ¼0.7. It is also found that the coefficients a and b are dependent on the anisotropy parameter γ for γ > α [18]. This is consistent with the critical behavior of QD in Refs. [10–12,18] for the TFIC and the transverse field anisotropic XY chain without the three-site interaction. Therefore, the derivative of the QD at the QPTs from the gaped FM phase to gaped PM phase in this model is similar to that at the Ising type QPTs in the model without three-site interaction. Fig. 4 shows the same as that in Fig. 3 but for γ = 0.3( < α). Under the given parameters, the model undergoes in succession two QPTs at hc1 and hc2, respectively (see Fig. 1). The first is between the gaped FM and gapless chiral phases and the second is

Please cite this article as: S. Lei, P. Tong, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.031i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 Q3 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

S. Lei, P. Tong / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

the isotropic version, i.e., γ ¼0 of the model in the previous section [22,23]. By mapping the spin operators to spinless fermions via successive application of the Jordan–Wigner and Fourier transformations, the Hamiltonian are transformed to the diagonal form (see the Supplemental Material)

H=

⎛

∑ ϵ k ⎜⎜ck† ck k

⎝

−

1 ⎞⎟ , 2 ⎟⎠

(12)

where ck† (ck ) is fermion creation (annihilation) operator in momentum space and

⎛ ⎞ α ϵ k = − ⎜h + cos k − sin 2k⎟ ⎝ ⎠ 2 Fig. 3. The first-order derivatives of the QD between the nearest neighbor spins in the transverse field anisotropic XY chain with respect to external field h for γ ¼0.7, the black solid line for N ¼ 1000 and the red dashed line for N¼ 9000. Insert shows the logarithmic divergence of d (QD)/dh taken at the critical point fitted by d (QD)/dh|hmin = − 0.00118 − 0.14 log N . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.

between the gapless chiral and gaped PM phases. It is seen that the first-order derivative of QD both shows discontinuity at two critical points. Different from the finite-size logarithmic scaling of the first-order derivative of QD in the Ising type QPTs, the extrema of the derivatives of the QD has no sharp increase with the system size. Therefore, no finite-size scaling behavior is found in our numerical results in case of γ < α . This is similar with the behaviors of fidelity and fidelity susceptibility at the critical points with the same parameters reported in Ref. [21]. In summary, the presence of the three-spin interaction only influences the QD in the gapless chiral phase. The singularity of the QD close to the critical points indicates that it is feasible to identify the QPTs between the gaped and gapless quantum phases. However, the scaling behavior of the derivative of the QD at the QPTs between the gaped and gapless phases is different from that at the QPTs between the gaped phases.

4. The transverse field isotropic XY chain 4.1. The model and the phase diagram

is the energy dispersion. The QPTs of the transverse field isotropic XY chain with threespin interaction have been studied in Refs. [22,23] and the phase diagram of the model is shown in Fig. 5. Different from the transverse field anisotropic XY chain in previous section, there are two gapless quantum phases I and II besides the gaped phase III in this model, and accordingly there are QPTs between these gapless phases. The two gapless phases are classified by the number of the Fermi zero points (i.e., zero solutions of energy spectrum) and are characterized by different decay of spin–spin correlations. The red dotted lines stand for the QPT between two gapless phases I and II, and the black solid lines denote the QPTs between the gapless phase I and the gaped phase III. The QPTs between the gaped and gapless phases or between the gapless phases are also of second order. The properties of QD in the QPTs between the gapless phases in are seldom considered with the exception of Ref. [34], in which the authors limited to the QD in the model with no fields. In this section, we extend the study of Ref. [34] to any transverse fields. Since the equivalence of the spin–spin correlation functions in X and Y directions in this case, the elements ρ14 and ρ41 of the reduced two-spin density matrix of the transverse field isotropic XY chain vanish. The reduced density matrix equation (7) of the chain in this section for spins at sites m and n now has a similar but simpler form

Now we turn to investigate the QD in the model that has QPTs between the gapless phases. The model studied in this section is

Fig. 4. The first-order derivatives of the QD between the nearest neighbor spins of the transverse field anisotropic XY chain with respect to external field h for γ = 0.3( < α), N = 16 384 .

Fig. 5. The phase diagram of the transverse field isotropic XY chain. The red dotted lines stand for the quantum transition between two gapless phases I and II. The black solid lines denote the quantum transition from gapless phase I to gaped phase III. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.

Please cite this article as: S. Lei, P. Tong, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.031i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

S. Lei, P. Tong / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

1

a11 = ( 2 + M z)2 − |gmn |2 , a22 =

1 4

− (M z)2 + |gmn |2 = a 33, 1

a 44 = ( 2 − M z)2 − |gmn |2 , a14 = 0, a23 = gnm ,

(13)

where the magnetization

M z = 〈Smz 〉 =

1 N

∑k

Θ ( − ϵk) −

1 , 2

and the two-point correlations † gmn = 〈cm cn 〉 =

1 N

∑ eik (m − n) Θ ( − ϵ k ).

(14)

k z

In case of h¼ 0, the magnetization M ¼ 0 and the gmn is always a real number, the bipartite state expressed in this reduced density matrix becomes a Bell-diagonal form [30] thus the QD therein can be analytically evaluated [31–33]. Otherwise, the QD will be determined numerically by minimization over all the possible von Neumann projection measurements based on the reduced density matrix. 4.2. The ground state quantum discord In what follows, we first present our numerical results of the ground state QD in the transverse field isotropic XY chain as functions of the three-spin interaction strength α. Fig. 6 shows the dependence of QD on α for different transverse fields: (a) for h¼0, 0.50, 0.80, 0.90,0.99 and (b) for h¼1.0, 1.20, 1.50, respectively. It is seen that for a small nonzero transverse field, the QD decreases with the increase of the three-spin interaction strength α. In the limit case h¼ 0, the QD first keeps constant before the critical point αc = 1, then decreases monotonously with α. This monotonous behavior is similar to that in Ref. [34]. However, for somewhat larger fields, the QD first increases then decreases with the three-spin interaction. In case of even more larger fields h > 1.0, the QD in the gaped phase III reduces to zero. In order to understand the zero QD in the gaped phase, it is noted that in the gaped phase III the energy spectrum εk < 0 (upper part of phase III in Fig. 5) or εk > 0 (lower part of phase III in Fig. 5) for every k. Therefore in the gaped phase, for example, the upper part of phase III, the two-point correlations gmn defined in Eq. (14) is zero for any sites m and n, and the magnetization Mz ¼0.5. According to Eq. (13), all the elements of two-body

5

reduced density matrix of the model are zero with the only exception of a11 = 1, which means that all the spins are fully polarized due to the external field, and certainly there have not been any quantum correlations between any two sites. Secondly, similar to the case of transverse field anisotropic XY chain, we focus on the relation between the QD and the QPTs of the isotropic case. Along the horizontal direction of the phase diagram in Fig. 5, for h¼0 and h ¼0.5, with the increase of α, the system undergoes a single QPT at αc = 1.0 and αc = 2.2845, respectively, between the gapless phases I and II; for h¼1.2, the system first undergoes a QPT between the gaped phase III and the gapless phase I at αc1 = 0.7639, then undergoes a QPT between two gapless phases at αc2 = 3.7404 . It is reported in Ref. [35] that for h ¼0 there are two non-analytical points of entanglement quantified by concurrence [36] but only one of them corresponds a real QPT and the other comes from the positivity requirement of the definition of concurrence, while there are only one non-analyticity of QD (black solid line in Fig. 6) corresponding to the QPT at αc = 1 [34]. However, in case of nonzero transverse field, it is clearly seen from Fig. 6 that there is a good correspondence between the non-analyticity of the QD and the QPTs of the model, since the number of the cusps in the curves is the same as the number of QPTs undergone in the model. Fig. 7 shows the first-order derivative of QD between the nearest neighbor spins with respect to three-spin interaction strength α: (a) for h ¼0.5 and (b) for h¼ 1.2. The derivative of QD shows discontinuity at the critical points. We have examined the derivative of QD of the system for different sizes from several to tens of thousands. The values of the derivatives close to the critical points of different system size are almost the same. Therefore similar to the derivative of QD at the critical points of the QPTs between the gaped and gapless phases in the transverse field anisotropic XY chain with XZY–YZX type three-spin interaction, no finite-size scaling behavior is found at the QPTs between the gapless phases in the isotropic chain. So it can be concluded that the non-analyticity of the QD at the QPTs involved with the gapless quantum phases is different from that between the gaped phases.

5. Conclusion In summary, the ground state QD between the nearest neighbor spins of the transverse field anisotropic and isotropic XY chains with XZY–YZX type three-spin interaction has been studied. In the anisotropic chain, the QD is influenced by the three-spin

Fig. 6. The QD between the nearest neighbor spins in the transverse field isotropic XY chain as functions of α for different transverse fields: (a) for h ¼ 0, 0.50, 0.80, 0.90, 0.99 and (b) for h ¼ 1.0, 1.20, 1.50, in all cases, N ¼65 536.

Please cite this article as: S. Lei, P. Tong, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.031i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

S. Lei, P. Tong / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Fig. 7. The derivative of the QD between the nearest neighbor spins with respect to three-spin interaction strength α for different fields: (a) for h ¼ 0.5 and (b) for h ¼ 1.2, in all cases N ¼ 65 536.

interaction only in the gapless chiral phase associated with the partly negative energy spectrum for γ < α . Outside the gapless phase, the QD is the same as that in transverse field anisotropic XY chain without three-spin interaction. In the isotropic chain, the QD in the gaped quantum phases reduces to zero due to the full polarized state in case of the large enough external fields. The nonanalyticity of the QD can be used to identify the QPTs, whether the QPTs are between gaped and gapless phases or between gapless phases. In the QPTs of both cases, no finite-size effect is found in the derivative of the QD near the critical points. On the contrary, the derivative of QD is logarithmic divergent with the system size at the QPTs between the gaped quantum phases for a finite system.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 11175087). We would like to thank Ming Zhong and Xiaoxian Liu for useful discussions. We acknowledge Xiaoming Lu, Yichen Huang and Wenlong You for helpful suggestions about the evaluation of QD.

Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.physb.2015.01.031.

References [1] H. Olliver, W.H. Zurek, Phys. Rev. Lett. 88 (2001) 017901; W.H. Zurek, Rev. Mod. Phys. 75 (2003) 715. [2] L. Henderson, V. Vedral, J. Phys. A: Math. Gen. 34 (2001) 6899; V. Vedral, Phys. Rev. Lett. 90 (2003) 050401. [3] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81 (2009) 865. [4] A. Datta, A. Shaji, C.M. Caves, Phys. Rev. Lett. 100 (2008) 050502. [5] B.P. Lanyon, M. Barbieri, M.P. Almeida, A.G. White, Phys. Rev. Lett. 101 (2008)

200501. [6] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, UK, 1999. [7] B.K. Chakrabarti, A. Dutta, P. Sen, Quantum Ising Phases and transitions in transverse Ising models, Springer, Germany, Berlin, Heidelberg, New York, 1996. [8] M.A. Continentino, Quantum Scaling in Many-Body Systems, World Scientific, Singapore, 2001. [9] A. Osterloh, L. Amico, G. Falci, R. Fazio, Nature (London) 416 (2002) 608. [10] R. Dillenschneider, Phys. Rev. B 78 (2008) 224413. [11] M.S. Sarandy, Phys. Rev. A 80 (2009) 022108. [12] J. Maziero, H.C. Guzman, L.C. Céleri, M.S. Sarandy, R.M. Serra, Phys. Rev. A 82 (2010) 012106; J. Maziero, L.C. Céleri, R.M. Serra, M.S. Sarandy, Phys. Lett. A 376 (2012) 1540. [13] Y. Yao, H.-W. Li, C.-M. Zhang, Z.-Q. Yin, W. Chen, G.-C. Guo, Z.-F. Han, Phys. Rev. A 86 (2012) 042102. [14] B. Çakmak, G. Karpat, Z. Gedik, Phys. Lett. A 376 (2012) 2982. [15] S. Campbell, J. Richens, N.L. Gullo, T. Busch, Phys. Rev. A 88 (2013) 062305. [16] T. Werlang, G. Rigolin, Phys. Rev. A 81 (2010) 044101; T. Werlang, C. Trippe, G.A.P. Ribeiro, G. Rigolin, Phys. Rev. Lett. 105 (2010) 095702; T. Werlang, G.A.P. Ribeiro, G. Rigolin, Phys. Rev. A 83 (2011) 062334. [17] Y.-C. Li, H.-Q. Lin, Phys. Rev. A 83 (2011) 052323. [18] F. Altintas, R. Eryigit, Ann. Phys. 327 (2012) 3084. [19] W.W. Cheng, J.-M. Liu, Phys. Rev. A 81 (2011) 044304. [20] D. Gottlieb, J. Rössler, Phys. Rev. B 60 (1999) 9232. [21] X. Liu, M. Zhing, H. Xu, P. Tong, J. Stat. Mech.: Theor. Exp. 01 (2012) P01003. [22] P. Lou, W.-C. Wu, M.-C. Chang, Phys. Rev. B 70 (2004) 064405. [23] M. Topilko, T. Krokhmalskii, O. Derzhko, V. Ohanyan, Eur. Phys. J. B 85 (2012) 278. [24] P. Zanardi, N. Paunković, Phys. Rev. E 74 (2006) 031123. [25] T.J. Osborne, M.A. Nielsen, Phys. Rev. A 66 (2002) 032110. [26] S.-J. Gu, G.-S. Tian, H.-Q. Lin, Phys. Rev. 71 (2005) 052322. [27] L. Amico, A. Osterloh, Phys. Rev. A 69 (2004) 022304. [28] G. Sadiek, B. Alkurtass, O. Aldossary, Phys. Rev. A 82 (2010) 052337. [29] Y. Holovatch, Order, Disorder and Criticality, World Scientific, Singapore, 2003. [30] B. Li, Z.-X. Wang, S.-M. Feil, Phys. Rev. A 83 (2011) 022321. [31] S. Luo, Phys. Rev. A 77 (2008) 042303. [32] M. Ali, A.R.P. Rau, G. Alber, Phys. Rev. A 81 (2010) 042105. [33] D. Girolami, G. Adesso, Phys. Rev. A 83 (2011) 052108. [34] B.Q. Liu, B. Shao, J. Zou, Commun. Theor. Phys. 56 (2011) 46; W.W. Cheng, C.J. Shan, Y.B. Sheng, L.Y. Gong, S.M. Zhao, Physica B 407 (2012) 3671. [35] M.-F. Yang, Phys. Rev. A 71 (2005) 030302 (R). [36] S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022; W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245; K.M. O'Connor, W.K. Wooters, Phys. Rev. A 63 (2001) 052302.

Please cite this article as: S. Lei, P. Tong, Physica B (2015), http://dx.doi.org/10.1016/j.physb.2015.01.031i

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132