Probing quantum spin glass like system with a double quantum dot

Journal of Magnetism and Magnetic Materials 407 (2016) 314–320

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm

Probing quantum spin glass like system with a double quantum dot C.Y. Koh a,n, L.C. Kwek a,b,c a b c

National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Singapore Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore Institute of Advanced Studies (IAS), Nanyang Technological University, 60 Nanyang View, Singapore 639673, Singapore

art ic l e i nf o

a b s t r a c t

Article history: Received 17 April 2014 Received in revised form 1 February 2016 Accepted 7 February 2016 Available online 12 February 2016

We study the ground state properties of a 4-qubit spin glass like (SGL) chain with probes at the end of the chain and compare our results with the non-spin glass like (NSGL) case. The SGL is modeled as a spin chain with nearest-neighbor couplings, taking on normal variates with mean J and variance Δ2. The entanglement between the probes is used to detect any discontinuity in the ground state energy spectrum. For the NSGL case, it was found that the concurrence of the probes exhibits sharp transitions whenever there are abrupt changes in the energy spectrum. In particular, for the 4-qubit case, there is a sudden change in the ground state energy at an external magnetic field B of around 0.66 (resulting in a drop in concurrence of the probes) and 1.7 (manifest as a spike). The latter spike persists for finite temperature case. For the SGL sample with sufficiently large Δ, however, the spike is absent. Thus, an absence in the spike could act as a possible signature of the presence of SGL effects. Moreover, the sudden drop in concurrence at B ≈ 0.66 does not disappear but gets smeared with increasing Δ. However, this drop can be accentuated with a smaller probe coupling. The finite temperature case is also briefly discussed. & 2016 Elsevier B.V. All rights reserved.

Keywords: Spin glass Spin chain Entanglement

1. Introduction Phase transitions in crystalline or ordered solids have been studied extensively and it is now quite well understood. However, this is not the case for disordered solids. One such disordered solid example is the spin glass, which remains as one of the most challenging and interesting problems in condensed matter physics. A spin glass consists of spins or magnetic moments arranged in randomness either through their positions or the different in signs between the neighboring couplings [1–7]. The neighboring couplings refer to ferromagnetic and antiferromagnetic. The disorder in site or coupling causes frustration in a spin glass. At certain critical temperature or sometimes called freezing temperature, this random collection of spins will be frozen in place. It is because of this frozen state transition that the frustration arises. Due to this frustration among the neighboring spins, many possible ground state configurations resulted. This produced a complex and rugged free energy landscape. Spin glasses are magnetic alloys comprising x concentration of magnetic impurities occupying random sites in a non-magnetic host metal [6]. By controlling the concentration and distribution of n Corresponding author at: NIE, NTU, Physics, BLK 164C RIVERVALE CRESCENT #08-276, Singapore 543164, Singapore. Tel.: þ6593852854. E-mail address: [email protected] (C.Y. Koh).

http://dx.doi.org/10.1016/j.jmmm.2016.02.023 0304-8853/& 2016 Elsevier B.V. All rights reserved.

such impurities randomly on the host, it is possible to observe the spin glass transition at low temperature. Examples of such impurities include Manganese (Mn), Iron (Fe) and Europium (Eu). Magnetic alloys which are originally studied are Cu1
x Mnx [8] and Au1
x Fex [9]. These magnetic alloys are considered as canonical spin glasses. Other alloys which are studied as a spin glass include Eux Sr1
x S [10] and La1
x Gdx Al2 [11]. Theories like the EdwardsAnderson (EA) model [12] which only allows the spins to interact via nearest-neighbor couplings with no long range order and Sherrington–Kirkpatrick (SK) model [13] for which every spin couples equally with every other spin are used in an attempt to explain mainly the cusp in the magnetic susceptibility. Essentially, the EA model replaces the site disorder and Ruderman–Kittel– Kasuya–Yosida (RKKY) distribution [14–16] with a random set of bonds, for instance, based on a gaussian distribution. An order parameter q is used to characterize the spin glass phase. The original EA equations are not simple to solve and are only soluble in the limits T → 0 and T → Tf . At these limits, the EA equations showed an asymmetric cusp in the magnetic susceptibility and specific heat. In contrast, the results by Fischer [17] showed that the theoretical specific heat is different from the experimental result except for the low temperature linear dependence when 1 using spin S = 2 . Although the SK model did exhibit a cusp in the magnetic susceptibility and specific heat, the entropy S goes to negative limit and the free energy is maximum with respect to the order parameter q. In addition, when q¼ 0, the spin glass state has

C.Y. Koh, L.C. Kwek / Journal of Magnetism and Magnetic Materials 407 (2016) 314–320

lower free energy than it is when q ≠ 0. All these are in contrast with what a second order transition should exhibit. Due to this instability in SK solution, Almeida and Thouless (AT) [18] did a detailed analysis on it and showed that the solution is unstable at low temperature. They showed the stability limits of the SK solution with the AT line dividing the unstable and stable areas in the spin glass phase diagram. Primarily, the instability is due to the fact that SK model treats all the replicas as indistinguishable – a replica symmetric solution. In order to solve this problem, Parisi [19–23] came out with a replica symmetry breaking (RSB) scheme which removes the unphysical negative entropy. It was found to be at least marginally stable. In spite of some success in using these models to describe some behaviors of the spin glass, they are not able to fully account for all the experimental results. This may due to the fact that these theories are classical in nature and did not consider the quantization of the spins of the impurities [24]. Experimentally, the measurement of the alternating-current (a.c.) susceptibility on a spin glass exhibits a cusp at certain freezing temperature Tf at low applied magnetic field. The sharp cusp in the magnetic susceptibility suggests that the alloy undergoes a phase transition [25]. Indeed, for alloys like gold–iron [9] and copper– manganese [8], these spin glasses show a cusp at certain critical temperature. It has been shown that the produced cusp is sensitive to the applied magnetic field and with just 100 G of magnetic field, a broader maxima is produced [9,26,27]. Besides been field dependent, certain spin glasses are also found to be frequency dependent [8,24]. Other measurement like finding the specific heat of Au0.92 Fe0.08 [28] and CuMn [29] were studied. In recent years, LiHoxY1
x F4 which can be described with a quantum Ising spin glass model has been studied experimentally and numerically [30– 35]. For an x concentration of ≤0.25, it is believed that there is a spin glass phase. As the concentration is diluted, it is still an open question of whether a spin glass or an antiglass spin phase exists. In quantum information theory, the study of entanglement in spin chain is believed to be important [36–43] as it is regarded as a key resource in applications like quantum key distribution, quantum teleportation, quantum dense coding, entanglement swapping and others [41]. Quantum correlation have been used in many-body systems to explore the phase transition at zero and finite temperature [43]. In the past, different order parameters and excitation spectrum are been used to characterize the phase transition of a many-body system. The synergy between quantum information and condensed matter physics has provided many new insights and directions. Specifically, a thorough analysis of the entanglement in the quantum critical models have been done [44– 46]. With the use of entanglement from quantum information theory, the study of the ground state for many-body systems reveals new and interesting properties. The appearance of abrupt changes in the entanglement in the quantum phase transition is seen as a sensitive probe in determining the phase transition in a spin chain when one external qubit is coupled to both sides of the chain. This approach has been applied to the XY Heisenberg model with the use of nonlocal probe qubits to detect the quantum phase transition [47–49]. The entire spin chain, including the nonlocal probe qubits can be engineered experimentally using quantum dots with one or more electrons [50–54]. Recently, Shim et al. [55] have studied the quantum phase transition by using a double quantum dot coupled locally to a XXX Heisenberg model spin chain as an alternative and efficient probe for detection of a quantum phase transition. Although much studies have been done on understanding quantum dots and using them as a probe to detect phase transition through computation of the entanglement between probes, very little work is found on understanding how the entanglement of the SGL chain behaves at low temperature when it is coupled to two external probes. Hence, the focus of this paper is to investigate the entanglement of a SGL XXX Heisenberg

315

model by using nonlocal probe coupled to both ends of the SGL chain. With this motivation, we numerically investigate a quantum SGL chain consisting of 4 qubits described by the XXX Heisenberg model. Each end of the SGL chain is then coupled to a qubit or quantum dot which serves as a probe. Using this model, we examine how the entanglement of the probes that are attached to the SGL chain changes by varying the external magnetic field and the standard deviation of the random coupling (between the SGL sites). The paper is organized as follows. We begin in Section 2 by defining the Hamiltonian of a 4-qubit system coupled to a probe qubit on each end of the spin chain. The XXX Heisenberg model spin chain is essentially modeled as a SGL exhibiting the usual characteristics of disorder and frustration. We then discuss the entanglement (concurrence) between the 2 probes which is considered as nonlocal, after tracing out the SGL chain as a function of the external applied magnetic field and the standard deviation in the coupling numerically. These results are presented and discussed in Section 3. In Section 4, we summarize our results.

2. Theoretical formulation In this section, we present the Hamiltonian model used in our study. After formulating the Hamiltonian for the SGL chain, we coupled both ends of the SGL chain with a probe (qubit). In order to look at the quantum correlation between the two probes, we trace out the SGL chain which can be in general make up of n sites and compute with suitable entanglement measure to observe the ground state properties of the SGL chain. The XXX Heisenberg SGL chain is described by n

Hn =

∑ Ji (Siα Siα+ 1)

(1)

i=1

Siα

= α σ 2 i

where Ji are the random variables and denotes the Pauli = matrices (α = x, y, z ) of the ith spin [43] which is subject to the open boundary condition Siα+ 1 ≠ S1α . In this case, the ith spin represents the individual site number from the SGL chain. The exchange energies Ji are quenched random variables with a prob2 2 1 ability distribution P (Ji ) = e−J /2Δ where Δ is the standard 2πΔ

deviation for the distribution. By applying a uniform external magnetic field on each qubit in the SGL chain, the Hamiltonian for the SGL chain (Hsgl ) model is given by n

Hsgl =

n

∑ Ji (Siα Siα+ 1) + B ∑ Siz i=1

i=1

(2)

where B is the external magnetic field applied transversely across the individual spin site Siz . By coupling a single probe (qubit) at each end of the SGL chain, n

Hsglp =

n

∑ Ji (Siα Siα+ 1) + B ∑ Siz + Jp (S1α S pα1 + SNα S pα2 ) i=1

i=1

(3)

where S pα1 and S pα2 are the spin matrices (α = x, y, z ) for the probes at the start and end of the chain respectively. Experimentally, the probe could be implemented with quantum dots. The probe coupling Jp describes the bond between the probes (qp1 and qp2) and end sites of the SGL chain (q1 and qn) respectively. The SGL chain is described with the qubit sites as q1, q2, …qn − 1, qn . The model is illustrated in Fig. 1. In order to measure the quantum entanglement between the two nonlocal probes, we adopt a bipartite measure for a two-level system called concurrence [56,57]. It is a measure of the non-separability of two-qubit density matrix with a value of zero for a

316

C.Y. Koh, L.C. Kwek / Journal of Magnetism and Magnetic Materials 407 (2016) 314–320

Fig. 1. The simplified diagram illustrates how the SGL chain in the form of XXX Heisenberg model (q1, q2…qn − 1, qn ) interact with the 2 probes (qp1 and qp2). The yellow arrows denote the applied magnetic field (B) on the individual site of the SGL chain. The probe coupling Jp – between q1 and qp1; qn and qp2. The random coupling Ji connects the sites (q1, q2, …qn − 1, qn ) of the SGL chain. The cylindrical box measured the entanglement (concurrence) between the two probes. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

separable state and one for a maximally entangled state. The concurrence for a bipartite two-level system with ϱ as the reduced density matrix of the 2-qubit probes is defined as

C (ϱ) = max (0, λ1 − λ2 − λ3 − λ 4 )

(4)

where λn (n = 1, 2, 3, 4) are the square roots of the eigenvalues arranged in decreasing order. The eigenvalues are obtained from the matrix ϱϱ˜ . The eigenvalues λn are real and non-negative. Here, ϱ˜ is defined as the “spin-flipped” state, expressed as

ϱ˜ = (σy ⊗ σy )ϱ⁎ (σy ⊗ σy )

(5)

where sy is the Pauli matrix expressed as

⎛ 0 − i⎞ ⎟ σy = ⎜ ⎝i 0⎠

(6)

in computational basis. Also,

ϱϱ˜ = ϱ(σy ⊗ σy ϱ⁎σy ⊗ σy )

(7)

where ϱ⁎ is the complex conjugate of the ϱ. To describe a thermal equilibrium state, the system state is described by Gibb's density operator ϱ = e−βHsglp /Z , where Z = tr (e−βHsglp ) is the partition function, Hsglp the Hamiltonian of the spin chain and the probes and 1 β = kT with k as Boltzmann's constant and T the temperature. By generating random normal variates chosen from N (μ, Δ2 ) and taking mean μ to be one, we computed the expected mean and standard deviation Δ of the concurrence for each magnetic field B from 0 to 2. This method of sampling is applied to the model by varying Δ from 0.1 to 0.9 and probe coupling Jp from 0 to 1 with an increment of 0.01. With reference to the data gathered in the various 3-dimensional plots for the average concurrence, the differences in the concurrence between NSGL and SGL are computed. Based on Fig. 2, the maximum concurrence in the spike for two specific values Jp ¼0.02 and 0.1 is carefully extracted over the range of Δ = 0–0.07 with mean J ¼1. All results mentioned are discussed and presented as plots in Section 3. 3. Results and discussion In this section, we present the results and discuss the entanglement (concurrence) between the probes of a 4-qubit SGL

Fig. 2. (a)–(c) show the 3-dimensional plots of average concurrence with varying applied magnetic field B and probe coupling Jp. In (a), the J coupling is equal to 1 with standard deviation Δ = 0 . In (b) and (c), the J coupling is equal to 1 with standard deviation Δ = 0.1 and 0.3 respectively.

chain with varying applied magnetic field B and standard deviation Δ in the J coupling. These results are compared with the case where J coupling is a constant (i.e. no fluctuation in J). In general, as the probes are swept from left to right, for B¼
2 to 2, the concurrence shows a symmetrical pattern. Hence, we will only limit our results and discussion to B¼ 0–2. Fig. 2(a) shows how the concurrence changes with magnetic field B and probe coupling Jp for a XXX Heisenberg model (J ¼1). The inset in (a) shows the many crossings of the energy spectrum for B from 0 to 2 with Jp ¼0.02 (a full discussion can be found in [55]). In comparison, (b) and (c) show how the average concurrence changes with B and Jp when mean J ¼1 and standard deviation Δ = 0.1 and 0.3 respectively. Essentially, these 3 plots show the difference between a NSGL and a SGL case when Δ is change from 0 to 0.1 and 0.3. With

C.Y. Koh, L.C. Kwek / Journal of Magnetism and Magnetic Materials 407 (2016) 314–320

1.0

1.0

0.6 0.4

0.8

0.25 0.20 0.15 0.10 0.05 0.00 0.0

0.5

1.0

1.5

Concurrence

Difference in Concurrence

0.8 Concurrence

317

2.0

B

0.2

0.5

1.0 B

(a)

0.4 0.2

0.0 0.0

0.6

1.5

0.0 0.0 (a)

2.0

0.5

1.0 B

1.5

2.0

0.5

1.0 B

1.5

2.0

0.0 1.0

Concurrence

0.8 0.6 0.4

Chain Energy

Difference in Concurrence

0.5 0.06 0.04 0.02 0.00 0.02 0.0

0.5

1.0

1.5

2.0

1.0 1.5 2.0 2.5

B

3.0

0.2

0.0 (b)

0.0 0.0

(b)

0.5

1.0 B

1.5

2.0

Fig. 3. (a) shows the plot of the average concurrence for Δ = 0.1 with Jp ¼0.1 (red circle) and Jp ¼0.02 (blue square) with magnetic field B. These two average concurrence curves are compared with the red solid line ( Δ = 0 , Jp ¼ 0.1) and blue dotted line ( Δ = 0 , Jp ¼ 0.02). The inset in (a) shows the difference in concurrence between the two average concurrence curves with magnetic field B and Δ = 0.1. (b) shows the plot of the average concurrence for Δ = 0.7 with Jp ¼0.1 (red circle) and Jp ¼ 0.02 (blue square) with magnetic field B. These two average concurrence curves are compared with the red solid line ( Δ = 0 , Jp ¼0.1) and blue dotted line ( Δ = 0 , Jp ¼0.02). The inset in (a) shows the difference in concurrence between the two average concurrence curves with magnetic field B and Δ = 0.7. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

fluctuation in J, we observe that the sharp transition for JP ¼0.02 in (a) changes to a smoother transition in (b). For Jp ¼0.02, the plateau in (a) which shows maximum concurrence decreases as J fluctuates with Δ. However, the concurrence stretches further beyond the point where the concurrence drops to 0 for J ¼1. This is also true when Δ is changed from 0.1 to 0.7 as illustrated in Fig. 3. In Fig. 3(a), there is a difference in the average concurrence when different Jp is used between the probe and the end of the SGL chain ( Δ = 0.1). This significant difference can be observed from the inset in Fig. 3(a) where the differences between the two average concurrence is taken. By comparing the average concurrence with the blue dotted line (Jp ¼0.02) and red solid line (Jp ¼ 0.1) respectively, it shows there is a smooth transition at the plateau from a NSGL ( Δ = 0) and SGL ( Δ = 0.1) in Fig. 3(a). When the standard deviation Δ is increased from 0.1 to 0.7, there is no significant difference between the two average concurrence curves in Fig. 3(b) which is also reflected in the inset when the differences are taken. The two curves also further show that by increasing the fluctuation in J, i.e. an increase in Δ, the concurrence decreases at low B and reaches beyond the magnetic field where the concurrence reaches 0 for both blue dotted line and red solid line ( Δ = 0).

Fig. 4. (a) shows the concurrence plot for Δ = 0 (black solid line) and Δ = 0.1 (blue circle) with J¼ 1. (b) shows the chain energy for J¼ 1 and Δ = 0.1 (broad spectrum), the mean chain energy (blue circle) and Δ = 0 (yellow solid line). The two red dotted vertical lines show the two level crossing at different magnetic field B. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

By finding the energy spectrum of the XXX model without the two probes for J ¼1 and Δ ≠ 0, the chain energy for a SGL chain is plotted in alignment with the average concurrence detected when the two probes is applied as shown in Fig. 4. Fig. 4(a) and (b) shows how the concurrence for J ¼1 with Δ = 0 and 0.1 coincides with the energy level crossings for both SGL and NSGL cases with Jp = 0.02 respectively. As shown in Fig. 4(b), the first and second level crossing is at B ≈ 0.66 and B ≈ 1.7 respectively. The broad energy spectrum shows the ground state energy fluctuates as Δ = 0.1 and J ¼1. With Δ = 0 and J¼ 1 (NSGL), the ground state energy (yellow solid line) is aligned with the mean ground state energy (blue circle) for Δ = 0.1. As presented in Fig. 5(a), taking the differences between the average concurrence (for Δ = 0.1) and the concurrence for Δ = 0, the phase transition can only be differentiated for small Jp between SGL and NSGL case when using the probes. When Δ increases to 0.3, we are able to distinguish based on the larger differences in concurrence shown in Fig. 5(b) as compared to Fig. 5(a). By increasing Δ, Jp can also be increased for detecting the phase transition between NSGL and SGL case. This is unlike the case in Fig. 5 (a) where the differences can only be detected for small Jp. In general, the observed concurrence between the probes is dependent on the selection of Jp used in the detection of phase transition. By increasing Δ, the differences in the concurrence will get larger and spread further out from the spike observed in Fig. 5(a). At high temperature, we may expect the plots of the average concurrence with Δ = 0.1 and 0.7 to behave in a rather similar pattern as shown in Fig. 3(a) and (b) for low temperature. This is

318

C.Y. Koh, L.C. Kwek / Journal of Magnetism and Magnetic Materials 407 (2016) 314–320

Concurrence

0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0 B

1.5

2.0

Fig. 6. The plot shows the comparison between the 2 average concurrence curves [red circle ( Δ = 0.1) and blue square ( Δ = 0.01)] and the black solid curve for Δ = 0 . All 3 plots are computed at a finite temperature of β = 104 . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Average Concurrence

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.00

Fig. 5. (a) shows the 3-dimensional plot of the difference in concurrence (between Δ = 0 and 0.1) with magnetic field B and probe coupling Jp. (b) shows the 3-dimensional plot of the difference in concurrence (between Δ = 0 and 0.3) with magnetic field B and probe coupling Jp.

because the fluctuation in J can be in some sense presume to be equivalent to thermal fluctuation at finite temperature. Surprisingly, this is not the case as presented in Fig. 6. The plot shows that the pattern of the average concurrence (red circle) with Δ = 0.1 is distinctively different from the one in Fig. 3. By comparing Δ = 0.1 (SG) with Δ = 0 (NSG), the peak of the concurrence smoothen when there is a fluctuation in J for Δ⪢0. This is in contrast to no fluctuation where there is a sharp transition. At low B, the average concurrence for Δ = 0.1 closely align with the solid curve ( Δ = 0). However, we can observe a difference at the transition point as the average concurrence ( Δ = 0.1) deviates from the sharp peak (black solid curve) which occurs at B ≈ 0.62. In addition, the spike at B ≈ 1.7 is also washed out for Δ = 0.1. For Δ ≤ 0.01, no difference can be observed for the first spike when compare to concurrence for Δ = 0 as shown in Fig. 6. However, the complete disappearance of the second small spike maybe used as a possible signature when the SGL effects is present. With reference to Fig. 2(a), a small ridge at B ≈ 1.7 stretches along at small Jp for Δ = 0 (NSGL) is observed. At Δ = 0.1, the ridge is washed out due to the fluctuation in J. At Δ < 0.1, the shape of the ridge is slowly smoothen out as Δ approaches 0.1. The average concurrence of the spike's peak for Jp ¼0.02 (red circle) is compared with Jp ¼0.1 (blue square) over a range of different Δ. This is illustrated in Fig. 7. For Jp ¼0.02 and 0.1,

0.01

0.02 0.03 0.04 0.05 Standard Deviation

0.06

0.07

Fig. 7. A comparison of the spike in average concurrence for Jp ¼ 0.02 (red circle) and 0.1 (blue square) over a range of Δ from 0 to 0.07. Two power law curves are fitted to the respective average concurrence values. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

the power law curves that describe the average concurrence are 5.216 0.03316 C = 6497x + 77.75 and C = 18.43x + 0.2401 respectively. In general, the distinctive feature of the ridge will be washed out significantly for Δ > 0.02 (standard deviation at half the maximum average concurrence). Another level crossing or a sudden drop in concurrence of the probes is shown at B ≈ 0.66 and the average concurrence for Δ = 0 to 0.07 are plotted for both Jp ¼ 0.02 and 0.1 respectively. For Jp ¼0.02 (red circle) in Fig. 8, the points are fitted with two different straight lines. For the first 4 points, the straight line is described by C = 4.059x + 0.2595 and the next 4 points by C = 1.025x + 0.3558. For Jp ¼0.1 (blue square), all the points are fitted with a straight line described by C = 0.691x + 0.2328 where C and x denote the unknown variables for average concurrence and standard deviation respectively.

4. Conclusion In this paper, we have examined the ground state properties of a 4-qubit spin glass like chain by connecting the ends to probes and compare the results to a non-spin glass like case. With an increase in the standard deviation of the J coupling, the

C.Y. Koh, L.C. Kwek / Journal of Magnetism and Magnetic Materials 407 (2016) 314–320

319

Average Concurrence

References

0.40 0.35 0.30 0.25 0.00

0.01

0.02 0.03 0.04 0.05 Standard Deviation

0.06

0.07

Fig. 8. A comparison of the drop in average concurrence for Jp ¼0.02 (red circle) and 0.1 (blue square) over a range of Δ from 0 to 0.07. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

concurrence is increased beyond the point where the concurrence of NSGL case is zero. However, there is a general reduction in the entanglement between the probes at low magnetic field when randomness and frustration are introduced into the spin chain. In general, for small Δ, we are able to distinguish the differences in entanglement (concurrence) between a NSGL and SGL case when a small Jp value is used. It is important to note that by using a large Jp, no entanglement will be detected. It is only when Δ increases that a larger Jp can be used. The average concurrence detected by the probes at low temperature looks different from the one at finite temperature. Hence, the thermal fluctuation gives rise to a different entanglement landscape as compare to one at low temperature where only quantum fluctuation due to the magnetic field B exists. The complete disappearance of the spike for the average concurrence at B ≈ 1.7 is a distinctive feature which can be used by one to differentiate between a NSGL and SGL case. Moreover, the sudden drop in concurrence at B ≈ 0.66 gets smeared with increasing Δ. Nevertheless, this smearing can be made more noticeable with a smaller probe coupling. Although this study only investigates the entanglement between two probes of a spin glass like chain with n ¼4 qubits, it is believed that the results is general and is similar for n is even and >4 . We focus here on low n, since any experimental simulation of such spin glass like system, for instance with ion trap architecture, is currently only feasible with small number of qubits [58–61]. The numerical simulations grow exponentially with the number of spin sites. With n¼ 4 and the number of iteration ¼100 for varying Jp and B, it already takes approximately 2 days for a 4-core processor to perform parallel processing to compute a single result. Moreover, the demand for computational resources increases as the number of qubits to be considered increases. With sufficient computational power, more studies can be carried out for n > 4 qubits. Even at n ¼4 qubits, the results can cast light on the possible effects of spin glass like effects in quantum simulators [38– 42] and engineering of quantum dots [50–54]. Future work could be carried out to explore the dynamics of the entanglement (concurrence) between the probes for finite spin chain with the other kinds of Heisenberg model.

Acknowledgment This work is supported by the National Research Foundation and Ministry of Education, Singapore.

[1] K. Binder, A.P. Young, Spin glasses: experimental facts, theoretical concepts and open questions, Rev. Mod. Phys. 58 (4) (1986) 801. [2] J.A. Mydosh, G.J. Nieuwenhuys, Dilute transition metal alloys; spin glasses, In: Handbook of Ferromagnetic Materials, vol. 1, Elsevier, North-Holland, Amsterdam, 1980, pp. 71–182. [3] J.A. Mydosh, Spin glasses—recent experiments and systems, J. Magn. Magn. Mater. 7 (1) (1978) 237–248. [4] K.H. Fischer, Spin glasses (i), Phys. Status Solidi B 116 (2) (1983) 357–414. [5] K.H. Fischer, H. Konrad, J.A. Hertz, Spin Glasses, vol. 1, Cambridge University Press, Great Britain, Cambridge, 1993. [6] J.A. Mydosh, T.W. Barrett, Spin Glasses: An Experimental Introduction, vol. 125, Taylor & Francis, London, 1993. [7] A. Theumann, V. Gusmão, Quantum Ising spin-glass, Phys. Lett. A 105 (6) (1984) 311–314. [8] C.A.M. Mulder, A.J.V. Duyneveldt, J.A. Mydosh, Susceptibility of the cu mn spin-glass: frequency and field dependences, Phys. Rev. B 23 (3) (1981) 1384. [9] V. Cannella, J.A. Mydosh, Magnetic ordering in gold–iron alloys, Phys. Rev. B 6 (11) (1972) 4220. [10] H. Maletta, W. Felsch, Insulating spin-glass system Eux Sr1
x S, Phys. Rev. B 20 (3) (1979) 1245. [11] J. Aarts, W. Felsch, H.V. Löhneysen, F. Steglich, Frequency dependence of the freezing temperature in spin glasses: a comparative study of (La, Gd) B6, (Y, Gd) Al2 and (La, Gd) Al2, Z. Phys. B: Condens. Matter 40 (1) (1980) 127–132. [12] S.F. Edwards, P.W. Anderson, Theory of spin glasses, J. Phys. F 5 (1975) 965. [13] D. Sherrington, S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett. 35 (26) (1975) 1792. [14] T. Kasuya, A theory of metallic ferro- and antiferromagnetism on Zener's model, Prog. Theor. Phys. 16 (1) (1956) 45–57. [15] K. Yosida, Magnetic properties of Cu–Mn alloys, Phys. Rev. 106 (5) (1957) 893. [16] M.A. Ruderman, C. Kittel, Indirect exchange coupling of nuclear magnetic moments by conduction electrons, Phys. Rev. 96 (1) (1954) 99. [17] K.H. Fischer, Static properties of spin glasses, Phys. Rev. Lett. 34 (23) (1975) 1438. [18] J.R.L. de Almeida, D.J. Thouless, Stability of the Sherrington–Kirkpatrick solution of a spin glass model, J. Phys. A 11 (5) (1978) 983. [19] G. Parisi, A sequence of approximated solutions to the SK model for spin glasses, J. Phys. A 13 (4) (1980) L115. [20] G. Parisi, Infinite number of order parameters for spin-glasses, Phys. Rev. Lett. 43 (23) (1979) 1754. [21] G. Parisi, Magnetic properties of spin glasses in a new mean field theory, J. Phys. A 13 (5) (1980) 1887. [22] G. Parisi, The magnetic properties of the Sherrington–Kirkpatrick model for spin glasses: theory versus monte carlo simulations, Philos. Mag. B 41 (6) (1980) 677–680. [23] G. Parisi, The order parameter for spin glasses: a function on the interval 0–1, J. Phys. A 13 (3) (1980) 1101. [24] P.J. Ford, Spin glasses, Contemp. Phys. 23 (2) (1982) 141–168. [25] C.Y. Huang, Some experimental aspects of spin glasses: a review, J. Magn. Magn. Mater. 51 (1) (1985) 1–74. [26] J.L. Tholence, R. Tournier, Susceptibility and remanent magnetization of a spin glass, J. Phys. Colloq. 35 (C4) (1974) C4–229. [27] O.S. Lutes, J.L. Schmidt, Low-temperature magnetic transitions in dilute Aubased alloys with Cr, and Fe, Phys. Rev. 134 (3A) (1964) A676. [28] L.E. Wenger, P.H. Keesom, Magnetic ordering of Au0.92 Fe0.08: a calorimetric investigation, Phys. Rev. B 11 (9) (1975) 3497. [29] L.E. Wenger, P.H. Keesom, Calorimetric investigation of a spin-glass alloy: CuMn, Phys. Rev. B 13 (9) (1976) 4053. [30] C. Ancona-Torres, D.M. Silevitch, G. Aeppli, T.F. Rosenbaum, Quantum and classical glass transitions in LiHox Y1
x F4, Phys. Rev. Lett. 101 (5) (2008) 057201. [31] J.A. Quilliam, S. Meng, C.G.A. Mugford, J.B. Kycia, Evidence of spin glass dynamics in dilute LiHox Y1
x F4, Phys. Rev. Lett. 101 (18) (2008) 187204. [32] A. Biltmo, P. Henelius, Phase diagram of the dilute magnet LiHox Y1
x F4, Phys. Rev. B 76 (5) (2007) 054423. [33] H.M. Rønnow, R. Parthasarathy, J. Jensen, G. Aeppli, T.F. Rosenbaum, D. F. McMorrow, Quantum phase transition of a magnet in a spin bath, Science 308 (5720) (2005) 389–392. [34] A. Biltmo, P. Henelius, Low-temperature properties of the dilute dipolar magnet LiHox Y1
x F4, Phys. Rev. B 78 (5) (2008) 054437. [35] K.-M. Tam, M.J.P. Gingras, Spin-glass transition at nonzero temperature in a disordered dipolar ising system: the case of LiHox Y1
x F4, Phys. Rev. Lett. 103 (8) (2009) 087202. [36] C.Y. Koh, L.C. Kwek, S.T. Wang, Y.Q. Chong, Entanglement and discord in spin glass, Laser Phys. 23 (2) (2013) 025202. [37] C.Y. Koh, L.C. Kwek, Entanglement of a 2-qubit system coupled to a bath of quantum spin glass, Physica A 403 (2014) 54–64. [38] O. Gühne, G. Tóth, Entanglement detection, Phys. Rep. 474 (1) (2009) 1–75. [39] G. Tóth, Entanglement witnesses in spin models, Phys. Rev. A 71 (1) (2005) 010301. [40] L.-A. Wu, S. Bandyopadhyay, M.S. Sarandy, D.A. Lidar, Entanglement observables and witnesses for interacting quantum spin systems, Phys. Rev. A 72 (3) (2005) 032309. [41] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum

320

C.Y. Koh, L.C. Kwek / Journal of Magnetism and Magnetic Materials 407 (2016) 314–320

entanglement, Rev. Mod. Phys. 81 (2) (2009) 865–942. [42] M.B. Plenio, S. Virmani, Introduction to entanglement measures, Quantum Inf. Comput. 7 (2007) 1–51. [43] L. Amico, R. Fazio, A. Osterloh, V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517. [44] T.J. Osborne, M.A. Nielsen, Entanglement in a simple quantum phase transition, Phys. Rev. A 66 (3) (2002) 032110. [45] A. Osterloh, L. Amico, G. Falci, R. Fazio, Scaling of entanglement close to a quantum phase transition, Nature 416 (6881) (2002) 608–610. [46] G. Vidal, J.I. Latorre, E. Rico, A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (22) (2003) 227902. [47] X.X. Yi, H.T. Cui, L.C. Wang, Entanglement induced in spin-1/ 2 particles by a spin chain near its critical points, Phys. Rev. A 74 (5) (2006) 054102. [48] Z.G. Yuan, P. Zhang, S.S. Li, Disentanglement of two qubits coupled to an XY spin chain: role of quantum phase transition, Phys. Rev. A 76 (4) (2007) 042118. [49] Q. Ai, S. Tao, G. L, C.P. Sun, Induced entanglement enhanced by quantum criticality, Phys. Rev. A 78 (2) (2008) 022327. [50] D. Loss, D.P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57 (1) (1998) 120. [51] L. Gaudreau, G. Granger, A. Kam, G.C. Aers, S.A. Studenikin, P. Zawadzki, M. Pioro-Ladriere, Z.R. Wasilewski, A.S. Sachrajda, Coherent control of threespin states in a triple quantum dot, Nat. Phys. 8 (1) (2011) 54–58. [52] J.R. Petta, A.C. Johnson, J.M. Taylor, E.A. Laird, A. Yacoby, M.D. Lukin, C. M. Marcus, M.P. Hanson, A.C. Gossard, Coherent manipulation of coupled electron spins in semiconductor quantum dots, Science 309 (5744) (2005)

2180–2184. [53] B.M. Maune, M.G. Borselli, B. Huang, T.D. Ladd, P.W. Deelman, K.S. Holabird, A. A. Kiselev, I. Alvarado-Rodriguez, R.S. Ross, A.E. Schmitz, M. Sokolich, C. A. Watson, M.F. Gyure, A.T. Hunter, Coherent singlet-triplet oscillations in a silicon-based double quantum dot, Nature 481 (7381) (2012) 344–347. [54] M.D. Shulman, O.E. Dial, S.P. Harvey, H. Bluhm, V. Umansky, A. Yacoby, Demonstration of entanglement of electrostatically coupled singlet-triplet qubits, Science 336 (6078) (2012) 202–205. [55] Y.-P. Shim, S. Oh, J. Fei, X. Hu, M. Friesen, Probing quantum phase transitions in a spin chain with a double quantum dot, Phys. Rev. B 87 (15) (2013) 155405. [56] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998) 2245. [57] W.K. Wootters, Entanglement of formation and concurrence, Quantum Inf. Comput. 1 (1) (2001) 27–44. [58] X.S. Ma, B. Dakic, W. Naylor, A. Zeilinger, P. Walther, Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems, Nat. Phys. 7 (5) (2011) 399–405. [59] K. Kim, M.S. Chang, S. Korenblit, R. Islam, E.E. Edwards, J.K. Freericks, G.-D. Lin, L.M. Duan, C. Monroe, Quantum simulation of frustrated ising spins with trapped ions, Nature 465 (2010) 590. [60] A. Friedenauer, H. Schmitz, J.T. Glueckert, D. Porras, T. Schaetz, Simulating a quantum magnet with trapped ions, Nat. Phys. 4 (10) (2008) 757–761. [61] K. Kim, M.S. Chang, R. Islam, S. Korenblit, L.M. Duan, C. Monroe, Entanglement and tunable spin-spin couplings between trapped ions using multiple transverse modes, Phys. Rev. Lett. 103 (12) (2009) 120502.