Interplay between Landau-Zener transition dynamic and quantum phase transition in dissipative spin chain with Dzyaloshinsky-Moriya interaction

Accepted Manuscript Interplay between Landau-Zener transition dynamic and quantum phase transition in dissipative spin chain with Dzyaloshinsky-Moriya interaction J.T. Diffo, M.E. Ateuafack, G.C. Fouokeng, L.C. Fai, M. Tchoffo PII:

S0749-6036(17)31088-1

DOI:

10.1016/j.spmi.2017.06.044

Reference:

YSPMI 5093

To appear in:

Superlattices and Microstructures

Received Date: 5 May 2017 Revised Date:

14 June 2017

Accepted Date: 14 June 2017

Please cite this article as: J.T. Diffo, M.E. Ateuafack, G.C. Fouokeng, L.C. Fai, M. Tchoffo, Interplay between Landau-Zener transition dynamic and quantum phase transition in dissipative spin chain with Dzyaloshinsky-Moriya interaction, Superlattices and Microstructures (2017), doi: 10.1016/ j.spmi.2017.06.044. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Superlattices and Microstructures 00 (2017) 1–10

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Interplay between Landau-Zener transition dynamic and quantum phase transition in dissipative spin chain with Dzyaloshinsky-Moriya interaction J.T. Diffoa,b , M.E. Ateuafacka,∗, G.C. Fouokenga , L.C. Faia , M. Tchoffoa and Multilayer Structures Laboratory, Department of Physics, Faculty of Science, University of Dschang, Cameroon of Physics, Higher Teachers’ Training College, The University of Maroua, PO BOX 55 Maroua, Cameroon

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Abstract

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The paper investigates the transition dynamic of a two-level system coupled uniformly to a general XY dissipative spin-chain environment with the Dzyaloshinsky-Moriya interaction. The collective effect of the spin-chain environment is studied and we obtained the exact expression of the final occupation of the system. It is observed that the increase of the decay parameter favors a shortcut to adiabaticity. In the absence of decay, the transition probability oscillations are modulated in time by the renormalized Landau-Zener (LZ) parameter that we derived. We found that in the vicinity of the critical point of the environment, depending on the strength of the system-environment coupling, the decay of the population transfer can be tailored by tuning the other parameters of the model. The critical point and thus the occurrence of the phase transition are observed to be independent of these parameters. It is shown that this quantum state transition is related to the occurrence of the quantum phase transitions in the chain. The adiabatic change for the magnetization observed in some magnetic molecules may be due to the number of spin involved in the chain, the anisotropy parameter of the chain or the external magnetic field strength. c 2017 Elsevier Ltd. All rights reserved

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Keywords: decoherence, qubit, spin chain, phase transition, LZ transition PACS: 03.67.Mn, 03.65.Yz, 75.10.Pq, 64.60.av

1. Introduction

With the advances in nanotechnology enabling controlled miniaturization and optimization of future nanodevices, qubit-spin systems modeled by avoided level crossings are gaining increased attention in the scientific community. It has created opportunities for the fabrication of new nanoscale systems as witness by the emerging fields such as coherent electronics, spintronics and quantum computation. In these excited domains, scientists consider realistic quantum physical systems where the main source of decoherence stems from the coupling to the environment [1]. These systems allow access to many different aspects of Landau-Zener (LZ) tunneling [2] including decoherence, relaxation and tunneling [3–5]. However, the qubit implementation (engineering application) requires its isolation from its environment. Several theoretical proposals have been developed to minimize the coupling between the qubit and its surrounding environment [6–11]. Understanding and controlling this coupling are therefore a major subject in the field of quantum ∗ Corresponding

author Email addresses: [email protected] (J.T. Diffo), [email protected] (M.E. Ateuafack), [email protected] (L.C. Fai)

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information processing [2, 12–17]. Nevertheless, it is not only the coupling strength but also the dynamic of the environment that governs the quantum coherence. This situation has been revisited many times and extended to a broader class of systems. The quantity of primary interest is the probability that finally the system ends up in one or the other of the two states. The general expressions of this transition probability have been derived by several authors in various contests. For instance a universal aspect of dissipative LZ transitions for both a harmonic-oscillator bath [5], a spin bath or baths of nonlinear oscillators [3] and even for baths consisting of a set of spins interacting with each other by an arbitrary anisotropic interaction [18] have been presented. On the other hand, Wang and co-authors showed that the LZ transitions are determined by the spin chains’ magnetic moments and their variance considering a time-dependent two-state system coupled to a spin chain in transverse fields [19]. But the magnetic moments of the chains contain the information of quantum phase transitions suggesting a new way to study quantum phase as well as LZ transitions. Recently, quantum interference effects of different tunneling paths have been observed in Mn12 [20, 21]. This nanomagnet and other magnetic molecules have been used to study explicit real-time quantum dynamics such as tunneling, relaxation and quantum (de)coherence of the magnetization [22–34]. As the magnetization dynamics of these molecules is determined by the (tiny) level repulsions, the adiabatic changes of the magnetization at the resonant fields can be well understood in terms of the LZ transitions [35–38]. In fact, disregarding the single-ion anisotropy, the dominant contribution to the magnetic anisotropy due to neighbor spin-spin interactions [39–43] is given by the Dzyaloshinsky-Moriya (DM) interaction which can substantially change energy-level crossings into energy-level repulsions [44]. Thus a realistic Hamiltonian model should minimally contain (strong) Heisenberg interactions, the single-ion anisotropy, the DM interaction and a coupling to the applied magnetic field [32, 45–49]. In Ref. [44], the authors demonstrated that the effect of the DM interaction on the magnetization dynamics strongly depends on the direction of the applied magnetic field. But this directional dependence has not been observed in experiments on V15 nanomagnet [44]. Therefore the DM interaction might not be the main mechanism responsible for the adiabatic change for the magnetization in magnetic molecules. As another source of level repulsion, De Raedt et al. showed that coupling the system to a non-hermitian Hamiltonian produces an effective level repulsion for the total Hamiltonian [50]. Non-hermitian Hamiltonians with complex eigen-values are finding numerous applications in modern physics [51, 52] and particularly in the theory of open quantum systems [53–58]. It has been demonstrated that the interaction between the electron spin and the nuclear bath is described by the Fermi contact hyperfine interaction S~ · ~h, where S~ and ~h are respectively the spin vector and the field generated by the nuclear spins at the position of the electron. But, considering the effect of the external magnetic field sufficiently large, the transverse terms of the hyperfine interaction responsible for the electron-nuclear-spin flip-flops has been neglected in the previous works. In this paper, we will investigate the non-adiabatic transition probabilities of the open-multi-level LZ model, consisting of a single qubit LZ model coupled to a dissipative spin chain. We will also consider decaying spin orbit coupling effects that induce DM interaction with dissipative sites. The non-hermitian Hamiltonian of the system is treated with a look in Ref. [19]. In Ref. [50], the authors derived the transition probability considering only the contribution of each site of the environment (non-collective effect). A relevant aspect of the approach developed in this work is the study of the collective effect of the environment on the quantum system. The chronological evolution of the transition probability expressions are derived for this problem applying the rotation operator approach [59]. As we shall show you, the environment’s properties are reflected in the dynamic of the systems’ LZ transition probabilities. These properties may serve as control parameters of the quantum system. Such a model improves the understanding of the electron-environment coupling system as an important key for future quantum computers using electron spin as a qubit. The paper is organized as follows: Section 2 reviews the LZ qubit coupled to an environment consisting of a XY spin chain in a transverse magnetic field with dissipation. The model therefore allows us to pinpoint how the original LZ qubit is modified by its coupling to a stationary spin bath. This is archived by deriving and analyzing the survival transition probability of the qubit in section 3. The special case of a non-decaying spin chain is considered in section 4 where the expression of the renormalized LZ gap is obtained. Section 5 is the conclusion.

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We consider the LZ model coupled to a XY spin chain in a transverse magnetic field with dissipation and governed by the following non-hermitian Hamiltonian: ! N   y 4g x x 1−γ y y J X 1+γ z z open − + x z z y 0 σ j σ j+1 + σ j σ j+1 + D x σ j σ j+1 − σ j σ j+1 +iδ σ j σ j + λσ j + σ σj . HLZ = ξ t σz + J σ x − 2 j=1 2 2 J (1) Here, HLZ = ξ tσz + J 0 σ x (2)

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is the original qubit LZ model consisting of two diabatic states with a fixed tunneling matrix element J 0 and linearly fluctuating with time with a speed v = ξ/2. The bath (spins chain) Hamiltonian is described by ! N   J X 1+γ z z 1−γ y y HB = − (3) σ j σ j+1 + σ j σ j+1 + D x σyj σzj+1 − σzj σyj+1 + iδσ−j σ+j + λσ xj . 2 j=1 2 2 The spin chain bath in a transverse field has nearest neighbor interactions with periodic boundary . condition, σN+1 = y α th ± z σ1 . σ j (α = x, y, z) stands for the Pauli operators at the j site of the lattice and σn = σn ± iσn 2 are the spin-spin raising and lowering operators. J and δ represent respectively the strength of the spin coupling and the spontaneous decay parameter whereas the real λ represents the external magnetic field strength and D x , the DM interaction strength. The number of spins in the chain is denoted by N and γ measures the anisotropy in XY spin chain. The XY Hamiltonian will turn into the transverse Ising chain for γ = 0, and the XX chain for γ = 1. The qubit-bath interaction Hamiltonian is N X σ xj (4) HS B = −gσ x

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where g represents the qubit-bath coupling constant. In a static transverse field, the spin chain can be mapped after a Jordan-Wigner transformation σ xj = 1 − 2C +j C j ,  Q  Q

y σ J = i C +j − C j 1 − 2Cm+ Cm and σzj = C +j + C j 1 − 2Cm+ Cm onto a fermionic problem. In this new reprem< j m< j n  o P sentation, this part of the Hamiltonian can be diagonalized using the Fourier transformation C j = √1N Ck exp i 2 πN j k − π4 , k

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and a suitable canonical (Bogoliubov) transformation Ck = cos (θk /2) bk + sin (θ−k /2) b+−k . Here, sin θk = Jωγk sin 2πk N ,   J ˜ 2πk 2πk ˜ and, cos θk = ωk λ + 2D x sin N − cos N with θk being a complex angle and λ = λ + iδ. This procedure leads to the following diagonalized expression of the Hamiltonian (1): ! !# X  1 X" 2πk 2πk 1 open + 0 x + J cos − 2D x sin + iJδ − 2ωk bk bk − (5) HLZ = ξ tσz + J σ x − 2gσ 1 − 2Ck Ck − 2 k N N 2 k where

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is the complex energy spectrum. Hereafter, we set D ≡ D x . It is convenient to continue the calculation in the interaction picture in order to calculate the probability of the open (t) = H0 (t) + Hint where qubit state flips due to the LZ sweep. We may divide the Hamiltonian into two parts as HLZ " ! !# 1 1X 2πk 2πk H0 (t) = ξ tσz − J cos − 2D sin + i J δ − 2ωk b+k bk − (7) 2 k N N 2 is the qubit-flip free Hamiltonian and, Hint = σ x

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open (t) = the qubit-flip interaction. The effective Hamiltonian in the interaction picture may be obtained through H˜ LZ n Rt o + U0 (t) Hint U0 (t) where the unitary operator U0 (t) is defined such that from U0 (t) = exp −i −∞ H0 (τ) dτ , we have:    X      ξt2  + + (9) i −ω0 + ωk bk bk + b−k b−k  U0 (t) = exp  + it |↑i h↑| + cc.     2 k>0

According to the procedure elaborated above, the system’s interaction Hamiltonian takes the form n o n o h i Hint = exp ξ it2 |↑i h↓| + exp −ξit2 |↓i h↑| ⊗ W (t) where

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      X  X    X   + θ θ k k  2 + 2 + + 1 − 2cos bk bk − 2sin b−k b−k + sin θk bk bk exp 2it W (t) = J − g ωk . ωk  + b−k b−k exp −2it 2 2 0

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3. Non-adiabatic transition in a decaying spin chain with DM interaction In this section, we consider that at time t0 = −∞ the two-level system is prepared in itsground state |0i and the Q XY spin chain system starts in its ground state, i.e., |0i = cos θ2k |0ik |0i−k − sin θ2k |1ik |1i−k such that, bk |0i = 0. k>0

We now begin to calculate the survival probability of the initial state |0i at time t → ∞, i.e., P↑→↑ (∞) = system ato any time is expressed by the following relation |ψ (t)iI = U˜ (t, t0 ) |ψ (t0 )iI |hψ (t0 ) | ψ (∞)i|2 . The state n Roft the open ˜ (t, t0 ) = T exp −i H˜ (t0 ) dt0 is a unitary time evolution operator given such that U˜ (t1 , t2 ) U˜ (t2 , t3 ) = where U t0 LZ ∞ P U˜ (t1 , t3 ) and U˜ (t, t) = 1. The elaborated procedure (the interaction picture) requires that U˜ (t, t0 ) = (−i)2l S l , where l=0

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E ˜ (∞) can be and with the help of the perturbation series, the state of the system describing the wave function h↑| Ψ expressed as:   !2k Z∞ Z∞ Z∞ Z∞ k ∞   X Y  X   1  2  2 0 (t2l − t2l−1 ) ⊗W (t2l − t2l−1 ) (13) ω dt1 dt2 dt3 · · · dt2l exp  iξ t − t + h↑| S |↑i = k 2l 2l−1     i 0 k=0

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      2    X X  X   0   2 X 2          W (t2l − t2l−1 ) =  J − g cos θk  exp  i (t2l − t2l−1 ) ωk0  + g sin θk  exp  −i (t2l − t2l−1 ) ωk0  .         0 0 k

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2  X ˜λ + 2D sin 2πk − cos 2πk sin2 2πk  N N N 2 2  . + g ˜ γ   2  2 2πk 2 ˜ 2πk 2πk 2πk 2 λ + 2D sin N − cos 2πk + γ2 sin2 2πk k λ˜ + 2D sin N − cos N + γ sin N N N (15) 4

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Figure 1. Derivative of the survival transition probability ∂P/∂λ as the function of λ/J in the environment of XY spin chain for different values of the spin-bath coupling strength (a), decay parameter (b), DM interaction strength (c) and anisotropy parameter (d). We set N = 200 and ∆ = .075 in the numerical simulation.

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. Here ∆ = J 02 2ξ stands for the LZ parameter and g˜ = g/2ξ is the renormalized qubit-bath coupling strength. The survival transition probability reads: 2 P↑→↑ (∞) = |h↑| S |↑i|2 = exp {−π– λ (δ)} . (16)

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This result tells us that besides the parameters considered in [19, 50], both the DM interaction strength and the non-hermitian character of the Hamiltonian also impact the asymptotic LZ transition probability (see Figs 1-3). Independently of the chosen environmental parameter, figure 1 shows an abrupt change of the derivative of the LZ transition probability around the point λc = 1. This quantum critical phenomenon emphasizes on the fact that far before and after this critical point, the system configuration changes from paramagnetic to ferromagnetic state [19, 50]. Thus, the magnetic strength determines the quantum phase transition and hence the quantum configuration of the system. After an exceedingly long time, the diabatic transition probability sharply increases before the critical point and moderately after this point and later turns to a constant with the increase of the magnetic strength (see figure 2). These observations attest that the critical point of the environment is reflected in the LZ transition probability. It is observed from figure 2 that the increase of the decay parameter, the anisotropy parameter and the DM interaction strength decreases the probability of the system to remain in its initial state after an exceedingly long time: The LZ probability is greater for a system with hermitian energy operator than for the system with non-hermitian energy operator. The LZ diabatic probability is higher considering the one dimensional spin chain than that of the system in the transverse spin chain. The strengthening of neighbor spin-spin interaction considerably decreases the LZ probability. However, raising the strength of the spin-bath coupling enhances this transition probability. Figure 3 depicts the probability of the whole system to remain in the ground state at the end of the evolution. It is observed that increasing magnetic field strength λ essentially enhances the probability of remaining in the ground state. Similar tendency is observed in figure 3b even for moderate dissipation (small value of δ). From figures 3c and 3d, it is obvious that the vanishing rate of the excited state population decreases as the DM interaction and anisotropy parameters increase respectively. This phenomenon indicates that, the decay of the population transfer due to interaction with spin bath can be modified by tuning the DM interaction. The plot in figure 3c also indicates that the 5

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Figure 2. Survival transition probability P↑→↑ (∞) as the function of λ/J in the environment of XY spin chain for different values of the spin-bath coupling strength (a), decay parameter (b), DM interaction strength (c) and anisotropy parameter (d). We set N = 200 and ∆ = .075 in the numerical simulation.

Figure 3. Survival transition probability P↑→↑ (∞) as the function of λ/J in the environment of XY spin chain and spin-bath coupling strength (a), decay parameter (b), DM interaction strength (c) and anisotropy parameter (d). We set N = 200 and ∆ = .075 in the numerical simulation.

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effects of increasing the intensity of the DM interaction on the decay of the population transfer are not remarkable for strong magnetic field. We hope that this method will be potentially very useful for studying quantum phase transition in more complicated nanomagnets materials. In the next section, we choose the special case where there is no dissipation in the system and study the collective effect of the XY spin chain environment on the dynamic of the LZ (two-level) system.

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4. Renormalization of the LZ gap and transition dynamic in zero decay spin chain

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The result obtained in Eq. (16) cannot be written in the form of the standard LZ transition probability due to the complex form of λ – (δ) or more globally to the non-hermitian form of the model described in section 2. Assume dissipation of the system vanishes such that the model described in Eq. (6) becomes hermitian; therefore the parameter –λ (δ) → –λ (0) becomes real and both the survival P0 (∞) and adiabatic P1 (∞) probabilities can now be easily evaluated as the total wave function is conserved: P0 (∞) = exp {−2π– λ (0)} ; P1 (∞) = 1 − exp {−2π– λ (0)} ,

(17)

where the renormalized form of the LZ parameter reads  2   2πk 2πk X X λ + 2D sin N − cos N sin2 2πk   √ N 2 2  –λ (0) =  ∆ − g˜ . + g ˜ γ q   2  2  2πk 2 2πk 2πk 2πk 2 λ + 2D sin N − cos 2πk + γ2 sin2 2πk k k λ + 2D sin N − cos N + γ sin N N N (18) In accordance with Ref. [19], Eq. (17) can be written in function of the expectation value of the magnetic moment M and its variance ∆M at the ground state. Figure 4 shows that both the expectation value of the magnetic moment and its variance depend on the strength of the transverse field. It follows that the information behind the quantum transition phase may be reflected in the LZ transition probability. The changes in the LZ transition probability are only due to the expectation value of the 7

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D = 0.2J

0.7

γ=1

Pd (τ )

0.8

D = 0.8J

0.6

γ=2

0.6 0.5

0.5

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−5

τ

1

0.4

0.4 −20

8

0.3

−15

−10

−5

0

5

τ

10

15

0.2 −20

20

−15

−10

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0

τ

Figure 5. Temporal evolution of the non-adiabatic LZ transition probability in a XY spin chain for N = 200, λ = 0.5J, ∆ = 1, γ = 0.8, D = 0.2J, g˜ = 0.5.

and

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magnetic moment of the spin chain environment when 0 ≤ λ/J ≤ 1. However, when λ/J ≥ 1, both the expectation value of the magnetic moment and its variance contribute to the LZ transition probability although the contribution of the latter appears to be dominant. The LZ problem in spin chain is a diffusion problem in the constrained sense that changes in the qubits’ states will occur only during a finite time interval around the crossing point t = 0. This implies that the qubit flip occurs for small |t|. In this light, our model behaves like the standard LZ model with the renormalized parameter λ – (0). Therefore the time dependent diabatic and adiabatic transition probabilities describing the complete dynamic of the system are respectively [59]: n o 2  D−iλ–(0) τ exp i 5π 4 Pd (τ) = (19) n o 2 n o 2   5π 5π D−iλ–(0) τ exp i 4 + –λ (0) D−i–λ(0)−1 τ exp i 4 n o 2  –λ (0) D−iλ(0)−1 τ exp i 5π 4 Pa (τ) = (20) n o 2 n o 2   5π 5π D−iλ–(0) τ exp i 4 + –λ (0) D−iλ–(0)−1 τ exp i 4 √ where we introduce the dimensionless time τ = t ξ and the Weber parabolic cylinder function Dn (z). Figure 5 depicts the time dependent LZ probability of the whole system to remain in the ground state in zero decay approximation. As in figures 1 to 4, all the properties of the environment are reflected in the transition dynamic. The probability of the system to remain in the ground state is enhanced by the increase of the coupling strength g˜ or the magnetic field strength λ. The same effect is observed when the DM interaction or the anisotropy parameter γ decreases. In the absence of decay, the transition probability oscillations are modulated in time by the renormalized LZ parameter –λ. These oscillations arise due to interference of the energy levels and are not altered with the increase of g˜ and D, providing the stoutness of the system against dephasing and decoherence.

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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We investigate the dynamic of a two-level system coupled uniformly to a general XY dissipative spin-chain environment with the DM interaction. We obtained exact expression of the final occupation of the system and the time evolution of transition dynamic is derived in zero decay approximation using the rotation operator approach. It is found that even small dissipation essentially increases the number of the population transfer. This amplification of the population transfer due to the non-hermitian character of the system (introduced by the decay parameter) can be modified by tuning the parameters of the model. This tendency is well observed in figures 2 and 3 attesting that shortcut to adiabaticity can be tailored and may probe new way of controlling decoherence in nanomagnets materials [60–62]. By renormalizing the LZ parameter in the chain, we suggest a rather exciting relationship between time dependent LZ transitions and the environments’ properties. Therefore the results may provide a new way to study the decoherence phenomenon as well as LZ transition dynamic. The transition probability for an adiabatic transition in zero decay approximation mimics the standard LZ result irrespective of the bath coupling strength and the DM interaction. The adiabatic change for the magnetization observed in some magnetic molecules may be due to the number of spin involved in the chain, the anisotropy parameter or the external magnetic field strength. It is also found that the LZ-dynamics could be manipulated via the coupling spin chain, the driving field, the DM strength and the velocity at which the qubit levels cross. The critical point and thus the occurrence of the phase transition is observed to be independent of these parameters. This result will be potentially very useful for studying quantum phase in other nanomagnets materials coupled to a dissipative Ising models with frustrated interactions. The system studied in this paper reveals the feasibility of both non-adiabatic and adiabatic sweeps and can be exploited for quantum state preparation in quantum computing task. The effect of a time dependent DM interaction may reveal interesting physical features.

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ACCEPTED MANUSCRIPT Highlights for the paper entitled Interplay between Landau-Zener transition dynamic and quantum phase transition in dissipative spin chain with Dzyaloshinsky-Moriya interaction 

The magnetic strength determines the quantum phase transition and hence the

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quantum configuration of the system.

The adiabatic change for the magnetization observed in some magnetic molecules may be due to the number of spin involved in the chain, the anisotropy parameter of the chain or the external magnetic field strength.

The critical point and thus the occurrence of the phase transition are observed to be independent of the parameters of the model.

In the vicinity of the critical point of the environment, depending on the strength of the

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environment coupling, the decay of the population transfer can be compensated by tuning the parameters of the model.

In the absence of decay, the transition probability oscillations are modulated in time

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by the renormalized LZ parameter that we derived.

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