Introduction

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Many-body systems have been attracting growing interest in semiconductor physics for several decades, particularly when interested in, quantum phase transition (QPT) [1–7] and technology of quantum system with highest entangled and strong correlated sates [8–15]. Several theoretical works have been done in this aim using different scenarios, among which the study of two potential quantifiers of quantum coherence in a physical state characterized by the ℓ1-norm of the system density matrix and the quantum relative entropy with respect to a diagonal matrix [16]. The dynamic of information in quantum systems, have been linked to quantum coherence trough quantum correlations measurements parameters by making used of establishment of an hier archical and explicit relationship between quantum coherence, quantum discord and entanglement between quantum states of a multi-particle system [7,17,18]. However, due to the elastic dimension of the Hilbert space of many-body systems, the analytical and exact study of quantum correlations by the ordinary mathematical approaches become complex. That is, the quantum renormalization group (QRG) approaches has been proposed as well-done approach [19–21]. For example, the Kadanoff QRG of spin squeezing approaches (which consists of dividing spin chain into an intra-block (HB) and inter-block (HBB) components) have been used as an indicator of QPT[22]. The non-analytic behavior at the critical points (translating a transition from the spin liquid phase to the N´eel phase) depending on the anisotropy of the spin chain have been obtained as a signature of quantum phase transition [18,21,22] leading also to quantum coherence lost, which becomes prominent with the dimension of the Hilbert space defined by different iterations of the QRG. on the overview of the considered method, some authors have investigate quantum coherence dynamics in an Ising transverse field model (ITF) and in an Heisenberg XXZ model, with and without Dzyaloshinskii Moriya (DM) interaction on a periodic spin chain of N lattice sites. They showed that the spin compression parameter obtained after each step of the QRG widened when the scale of the system becomes larger; Similarly, as a function of the QRG iteration number, the different ground state (GS) spin compression parameters exhibited an abrupt change in concavity with a discontinuity in the behavior of the first derivatives of these parameters, characterizing QPT signature [18,21,22].

In the light of the work done by joya in 2021 and Balazadeh in 2018[18,21], our objective in this paper is to evaluate the contribution of the electric polarization and of the Modulated Dzyaloshinskii-Moriya Interaction (DMI)on the non-analytical dynamics of and entangled spin states in Heisenberg spin-1/2 xxz chain using QRG method. Our consideration aims to maintain quantum correlations and entanglement as long as possible.

In order to improve the transport properties in spin chains, the problem of coupling between states of the system is growing, particularly when interested to quantum coherence or quantum phase transition. Among other coupling possibilities, the spinorbit coupling [23,24] particularly in magnetic thin layers, between spins, between orbital moments and degrees of freedom related to the structure of materials as well as the interactions between systems and external excitation field [25,26]show better parameters. An interest is also often placed on the types of interface and surface states as well as on the exotic spin states [28]. The study of DM interaction in thin film systems has shown an even greater interest in spin electronics, which has motivated the exploitation of the strength of its intensity for the development of even more functional materials [29,30]. However, this study only concerned systems whose interactions are limited to the first neighbors, but with vector spin variables on each site of the network. This is how the modulation effects coming from the competition between the usual Heisenberg exchange, scalar in nature, and an additional exchange associated with the vector products of DM interactions have been taken into account [30,32,33].

Beyond taking into account the modulated DM interaction in spin chains, the interest of which has already been demonstrated, we wish to go further by considering in our work the effect of a polarization induced by the electric field on the dynamics of the bipartite and tripartite states of a 1/2 − XXZ Heisenberg renormalised spin chain.

To achieve the goal of this work, the paper will be organized as follows: the second part will present the quantum model as well as the method used for the calculation of the descriptive parameters of the system. The third part will present the analytical an numerical results with the different interpretations; the work will end with a conclusion and perspectives.

XXZ Model and Application of Quantum Renormalization Group Approach

The model used for this purpose is that of a long and periodic chain of 1/2-XXZ Heisenberg model in the presence of both modulated Dzyaloshinskii-Moriya interaction and constant external magnetic field, under the influence of an external electrical polarization ( see Eq. (1), with h = 1 for simplicity):

is the XXZ part hamiltonian, J_i=x,y,z > 0 the antiferromagnetic coupling term between the spins along the X, Y and Z directions.

The Hamiltonian of the Dzyaloshinskii-Moriya interaction modulated by the spinchain position dependence is [32]:

The contribution of the bias field is given by:

where →E = (0,E, 0) is a constant electric field oriented along →, e_y and

the polarization vector, with Sⁱ_n the spin operator. The model described in Eq. (1) is spin-position dependent according to

the external magnetic field h(n) and the Dzyaloshinskii Moriya interaction coefficient d(n) which are the function of the spin position in the chain defined through < n>:

With the help of the Bloch relation and the RG approaches, the characteristic Hamiltonian of the spin chain in Eq. (1) can be transformed into the following form [7,18]:

the Hamiltonian of the inter-block i.e that reflecting the interaction between the block l and the block l + 1 , With l the rank of the block; where σ²_n in is the Pauli matrice, △ = J_z/jthe anisotropy factor and ξ = E JN the electric polarization factor. ˜h(n) = 2h(n) J and D(n) = d(n) J are the new quantities of both modulated external magnetic field and DM Interaction coefficient.

In order to evaluate the fundamental parameters of the system in application of the GR-principle, the operator of projection ζ is constructed by the lower eigenvectors, that is to say, those which derive the energy from its fundamental state:

the kets |⇑il and |⇓il are the new states in l −th bloc of spin defined in the renormalized Hilbert space obeying the same properties as those of the spin-1/2 ; ∅0| (∅′0|) are eigen vectors of the fundamental state. The energy of the fundamental state. The energy of the fundamental state

from Eq. (10) of the l-th block is doubly degenerate so that

where |↑i and |↓i are eigenstates of the Pauli matrix σ^z. The parameters α , β , γ that verify the property α2+β2 + γ2 = 1 , can be defined by the following relations.

with

Let ζ† be the transpose complex conjugate of the projection operator ζ , the effective Hamiltonian of the system built around the total Hamiltonian of the system Eq. (8) and which acts on the renormalized subspace through the relation

where p is the order of the renormalization, D^(p+1)(n) = D^(p)(n) = D(n) and ε^(p+1) = ε^(p) = ε . From the first renormalisation i.e p = 1 the hamiltonian Eq. (16) gives:

iteration relations between the parameters of the original system and those of the first step of the RG.

Here, the dynamics of Bipartite and tripartite quantum information resources under the effect of the anisotropy parameters △', the modulated Dzyaloshinskii interaction D′(n), the electric polarization coefficient ε′ and of the modulated magnetic field ˜h′(n) is measured trough quantum coherence and the Monogamy of entanglement, as well as Polygamy of entanglement, by using the ℓ1 -norm of density matrix approximation.

Considering one of the degenerate ground states of the system obtained in Eq. (13)), the initial density matrix ρ₀¹²³ = |∅₀ )(∅_0| of an uncorrelated states of three spins located in a block defined :

In the ℓ₁-norm approximation [16] the closed form of the quantum coherence measurment factor is given by

This factor is the sum of the absolute values of the off-diagonal elements of the density matrix. |i) and |j)are the basis vectors of Hilbert space. For an initial states, Eq. (19)) can be rewrite:

with ρ_i,j the density matrix of the initial states define by:

where i, j, k represents the positions of the spins in the block.

Depending on different and possible permutations of the spin positions in the chain, only the formation of three bipartite states C_l1 (ρ₀^ij) and one tripartite states are possible(with i ≠ j ≠ k and can occupy only the following distinguish positions 1, 2, 3).

Dynamics of Bipartite States

The density matrix that characterizes the collective behavior of the two spins between the three considered spin at different positions i, j, k in the block is derived by partial tracing on the third as ρ^ij0 = T_rk (ρ^ijk 0) . Thus, for the 1st and 2nd spin of one bloc, the density matrix(eqn.(22))

is the bipartite density matrix for spins at positions 1, 2. Considering Eq. (22)) into Eq. (21)), the quantization parameters of bipartite quantum coherence is obtained:

Similarly, the tripartite quantum coherence quantization

Figure 1

by considering Eq. (18)) into Eq. (20)) as,

Such as it has been demonstrated in [21] , it has also been found here that the nearest neighbors are strongly correlated than the next nearest neighbors in a block. This correlations strend is the expression of the position dependency of bipartite quantum coherence and entanglement within a block.

To further elucidate the effect of the staggered part of the modulated DM interaction and the Electric polarisation on quantum coherence for different number of spins, we plot the ℓ1-norm of the nearest and next-neighbors (see Fig. 4 and Fig. 5) for different

QRG iterations respectively in Fig. 1 to Fig. 2 and in (Fig. 4) to (Fig. 5). The number N of total spins in the system at pth iteration is given by N = ^3p+1. dC_l1 (ρ₀¹²) depending on the anisotropy , for different iterations of QRG (with J = 1). Compared to case (b), case (a) exhibits modulated behavior in the absence of the influence of DM interactions. At low temperature and in the limit of the small number of spins (i.e for the first two iterations of the QRG), the bipartite quantum coherence decreases monotonically with the anisotropy (Fig. 1) and adopts a non-analytical behavior when much more of iterations are considered.

The transition from analytical behavior to non-analytical behavior of coherence is a particular characteristic of entanglement [22] which also reflects a signature of quantum phase transitions occurring at a critical value of the anisotropy Δc = 1 (Fig. 1.(Ca)) [21]. The corresponding quantum transition point becomes dynamic according to the variation of DM interactions coefficient and the staggered part of DM. Increasing

the value of the staggered part of the DM parameter, shifts the QPT to the right in a coherent manner both in the case of even sites (Fig. 1.(Cb)) and for odd sites (Fig. 1.(Cc)) according to the law

However, the black color curve obtained for odd sites (Fig. 1.(Cc)) in the absence of renormalization is shifted from the critical point, which reflects a delay in the entanglement which will be immediately corrected by the electric polarization for electric field values E ≥ 2 (Fig. 1.(Cd)). It emerges from this observation that the electric polarization reinforces the quantum correlations in the direction of displacement of the information.

In the thermodynamic limit (temperature close to absolute zero) and according to Bethe’s Ansatz theory, Δc establishes the boundary between the phase of maximally correlated spins (liquid phase of spins) and the phase of completely uncorrelated spins (N´eel phase) [1]. The case of the even sites (Fig. 2.(Ca)) contrary to that of odd sites (Fig. 2.(Cb)) presents a liquid phase of predominant spins and whose correlation time will be more important depending on the value of the staggered parts d1 of the DM and d1 of the magnetic field. A phase alternation is observed when the dynamics of the chain depends on the site of occupation of the spin. The non-analytical nature of quantum coherence can be visualized from the divergence of its first derivative with respect to Δ as shown in (Fig. 3). The degree of non-analyticity can be estimated from the increasing depth of the minimum C and its variation as a function of the number of iterations. Each curve of the derivative reveals a minimum which becomes singular as one approaches the critical point. This singular behavior of quantum coherence is increasingly noticeable as the number of spins in the system increases.

Dynamics of Quantum Correlations: Monogamy of Entanglemet

The monogamy of entanglement is based on the condition that if two parts of a multipartite system are maximally entangled then no entanglement can be transferred to the rest of the parts of the system. It has already been the subject of several researches as a parameter for measuring quantum correlations [35–37], through the following formula:

where C represents the concurrency (the measurement parameter of quantum entanglement). The bipartite concurrency of state with a single partition of the density matrix is a function of the eigenvalues of the transfer matrix ρ defined by:

where λ1 ≥ λ2 ≥ λ3 ≥ λ4 are eigenvalues of the reduced density matrix R defined by:

with ρ∗ the complex conjugate of ρ . In the case of a pure state with N partitions, the concurrency is said to be multipartite and defined by:

concurrency is said to be multipartite and defined by:

where m = 2^(N−1) − 1 . represents the dimension of the Hilbert space of the system, ρi the partial density matrix obtained by the partial trace of the density matrix ρ . In application of the relation Eq. (29)) the expressions of the competitions obtained respectively between the 1^st and the 2^nd spin of a block, between the 1st and the 3^rd spin of a block and between the 2nd and the 3rd spin of a block are identical to those of the coherences given by Eq. (23)) and Eq. (27)) i.e. C_l1 (ρ₀¹²) = C_l1 (ρ₀¹²) = C_l1 (ρ₀²³) = C_l1(ρ₀²³) = and C_l1 (ρ₀¹³) = C_l1 (ρ₀¹³) for the bipartite cases. On the other hand, by making an extension of Eq. (28)) for the study of tripartite entanglement, it is found in this case that d = 8 and N = 3 then m = 3 ; leading to the following expression of concurrency:

where the term Σ³_i=1Tr(ρi)2 represents the sum of the traces of the squares of the reduced matrices defined by the relations Eq. (22)), hence the following relation of the tripartite entanglement reads:

From the definition of monogamy of entanglement

the following relation of the monogamy of entanglement can be obtained:

where C_l1( ρ₀^1,23) is the l1-norm of the consistency between qubit 1 and the bipartite block{23} , with

the partial density matrix of 8×8 order. The monogamy of entanglement presents an analytical behavior which evolves towards a non-analytical behavior after a change of state around a value of the anisotropy C = 1 with the increase in the number of iterations of the QRG (Fig. 7(Ma)). This change of behavior reflects a quantum phase transition around the quantum transition point (QPT) defined by the critical value of the anisotropy Δ_C = 1. The QPT moves with the variation of the critical value of the anisotropy due to the variation of the DM interaction coefficient by its staggered part d0 , as shown in (Fig. 7,(Mb)-(Md)). However, the monogamy of the entanglement displays a polygamous nature for the first three steps of the QRG and a monogamous nature after more iterations also depending on the value of the DM interaction.

In fact, for d0 ≥ 0 , the dynamics of QPT is such that the length and coherence time becomes large according to the value of DM interaction parameter. This coherence time will be longer stable with the increases of the value of the magnetic field intensity by its staggered part. The long coherence time (length) is the consequence of monogamous quantum nature which is very important in quantum information technology, for example in the field of telecommunications for designing the channels of information transport with less risk of loss of information over reasonable distances.

Conclusion

We have studied the dynamics of quantum coherence of a renormalized and polarized 1/2−XXZ Heisenberg spin chain by an electric field, in the presence of the modulated DM interaction and of the magnetic field. We used as a theoretical approach, the ℓ1-norm method associated with the density matrix to measure the impact of the polarization factor, of the modulated part of DM interaction and of the magnetic field, on the dynamics of the quantum transition point of bipartite and tripartite states through the quantum coherence factor. The results obtained show how much the dynamics of the critical point, depends considerably on the parity of the system defined by the staggered part of the DM interaction (d1) and of the applied magnetic field (h1). This staggered part that helps to improve the value of the intensity of DM and of the magnetic field, generates equally a phase alternation between the liquid phase of the spins towards the Ne'el phase and vice versa. This setting help to stabilize the system in a phase and contributes not only to the variation of the quantum transition point, but even more to increase the entanglement delay in the systems. The polarization due to the electric field, reinforces the quantum correlations by an arrangement of the spins along the direction of the polarization wave vector, which is of very great interest in the control of the quantum coherence during the transport of the information by spin chains.

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