1 Introduction

Quantum correlations have been intensively studied in recent decades as a fundamental resource in quantum information theory such as quantum teleportation [1], quantum computation [2], quantum communication [3], and so forth [4, 5]. Quantum correlations reveal non-classical aspects of quantum mechanics, which has no classical counterpart. One kind of well-known quantum correlations is quantum entanglement. Until a few years ago, it was believed that quantum entanglement alone can explain all quantum correlations in a typical system. In this direction, many proper quantifiers, namely concurrence [6, 7], negativity [8, 9], global entanglement [10], Scott measure [11, 12] and many others [13,14,15,16,17,18], have been introduced to characterize the entanglement concept in the bipartite as well as multipartite quantum systems. However, some studies have shown that entanglement cannot grasp all quantum correlations of a system well [19, 20]. Indeed, Ollivier and Zurek showed that the quantum discord (QD) is another promising quantifier which goes beyond entanglement [19]. Strictly speaking, QD reveals the quantumness of correlations even in the separable mixed states [19, 20]. But despite this advantage, an analytical evaluation of this information-theoretic measure remains, generally, very difficult. Alternatively, various metrics based correlations measures such as geometric QD [21], global geometric QD [22], super QD [23], trace distance discord (TDD) [24, 25], and quantum consonance (QC) [26, 27] were proposed.

In recent years, the solid-state systems formed a milestone for several studies in realm of the quantum information theory [28,29,30,31,32,33,34]. In the perspective of the Heisenberg model [35], the dynamics of quantum correlations in a three-qubit with phase decoherence was studied [36]. Moreover, the dynamics of nonlocal correlations in an anisotropic two-qubit Heisenberg XYZ model under the effect of the phase damping was examined [37]. In the intrinsic decoherence framework, the temporal evolution of quantum correlations in a two-qubit Heisenberg spin chain model with the interaction of anti-symmetric and symmetric contributions of spin–orbit coupling  [30, 31] has been also recently studied [38]. Meanwhile, the Heisenberg spin models have been used to characterize the thermal features of the bipartite entropic uncertainty [31, 39]. But excepting some few studies [40, 41], a varieties of Heisenberg spin chains, in various configurations like XXX, XXZ, and XYZ, have been studied without assuming the coupling strength as a function of position. However, in view to expand the effect of coupling distance on quantum correlations, we present in the current paper a study of pairwise Heisenberg quantum correlations in the context of the Heitler–London (HL) approach.

Historically, the first quantum-mechanical description of the chemical bonding is generally credited to W. Heitler and F. London [42], who proposed it just a year after the presentation of Schrödinger equation  [43, 44]. The HL approximation [42] is typically conceived to explain the interaction between a pair of spins, resulting from Coulomb forces between two electrons of the two adjoining atoms [45]. It is an example of the valence bond (VB) method [46], which was introduced to describe the bonding of a hydrogen molecule \(\hbox {H}_{2}\). It is worth mentioning that the VB method views molecules as composed of atomic cores (nuclei plus inner shell electrons) and bonding valence electrons (Fig. 1 scheme is devoted to the procedure of \(\hbox {H}_{2}\) molecule). Heitler and London began with the atomic orbitals toward setting up an approximate solution. They found that the suitable linear combinations of the hydrogen 1s wave functions centered at two atoms \(\mathbf{A}\) and \(\mathbf{B}\) are given by the symmetric \(A_{1} B_{2}+B_{1} A_{2}\) and anti-symmetric \(A_{1} B_{2}-B_{1} A_{2}\) wave functions (Fig. 1). It is assumed that the atomic orbitals A and B are suitably normalized, and therefore, the normalized functions are obtained as \(\psi _{\pm }=(A_{1} B_{2} \pm B_{1} A_{2})/\sqrt{2\left( 1 \pm S^{2}\right) }\) where the amount \(S=\int A_{1} B_{1} d v_{1}\) is the overlap integral [47].

Fig. 1
figure 1

Valence bond treatment of \(\hbox {H}_{2}\) molecule. In the scheme, \(r_{1A(B)}\) and \(r_{2A(B)}\) denote the distance from the first (second) electron labeled as 1(2) to the nuclei labeled as A(B), while \(r_{12}\) is the distance between the electrons 1 and 2. Also, R is the distance between the nuclei A and B

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The concept of exchange coupling of the cluster spins of nonsinglet atoms was proposed in the molecular binding theory of HL and the Heisenberg theory of ferromagnetic [42, 48]. In literature, some few works have been done on HL approach with exchange coupling [45,46,47,48,49,50,51]. On the other hand, so far no work has been published on quantum information that uses HL exchange coupling. So given the obvious importance of this issue, we consider a two-qubit Heisenberg XXX model with HL exchange coupling approach subjected to an external transverse nonuniform magnetic field. The purpose of the present work is to probe the behavior of quantum correlations captured by concurrence, TDD, and QC for our realistic Hamiltonian model at thermal equilibrium. Hence, the layout of the work is structured as follows. In Sect. 2, we briefly present the two-qubit Heisenberg XXX spin system with HL exchange coupling for the quantum system. In Sect. 3, the thermal evolution of mentioned quantifiers in the background of our model has been examined in detail, under the influence of the temperature, the external magnetic field, and the exchange coupling. In Sect. 4, we give the main results with detailed analysis. Finally, some concluding remarks are given in Sect. 5.

2 Hamiltonian model

This section is devoted to describe a system of two nearest neighboring Heisenberg spin model subjected to an external transverse magnetic field. The model of Hamiltonian is given by [52]

$$\begin{aligned} H=\frac{1}{2}\Big [\left( J_x\sigma _{1}^{x}\sigma _{2}^{x}+J_y\sigma _{1}^{y}\sigma _{2}^{y}+ J_z \sigma _{1}^{z}\sigma _{2}^{z}\right) +B\left( \sigma _{1}^{z}-\sigma _{2}^{z}\right) \Big ], \end{aligned}$$
(1)

where \(\sigma _{i=1,2}^{j}\) with \(j \in \{x,y,z\}\) are the 2D-Pauli spin matrices, while B is the nonuniform magnetic field directed along the z-axis. To proceed further, we concentrate in our study on the case of the isotropic XXX configuration of Heisenberg antiferromagnetic spin chain characterized by \(J_{x}=J_{y}=J_{z}=J>0\). Besides, we assume that the exchange spin constant J is identical to HL coupling, namely J(r), given as [47]

$$\begin{aligned} J\equiv J(r)=\left( \frac{56}{45}-\frac{4 \gamma }{15}-\frac{4}{15} \ln r \right) r^3 e^{-2r}+O\left( r^2e^{-2r}\right) , \end{aligned}$$
(2)

where \(\gamma =0.5772\) is Euler’s constant and r refers to the inter-spins relative distance defined as the ratio of the distance between spins R and Bohr radius . Figure 2 displays the plot of the function J(r) in terms of the inter-spins relative distance r.

Fig. 2
figure 2

HL coupling strength J(r) with respect to inter-spins relative distance r

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So, the modified Hamiltonian for XXX configuration with HL coupling in the two-qubit standard computational basis \(\mathcal {B}\equiv \{ \vert 00 \rangle , \vert 01\rangle , \vert 10\rangle , \vert 11\rangle \}\) can be written under the following matrix form

$$\begin{aligned} H=\left( \begin{array}{cccc} \frac{J(r)}{2} &{} 0 &{} 0 &{} 0 \\ 0 &{} -\frac{J(r)}{2}+B &{} J(r) &{} 0 \\ 0 &{} J(r) &{} -\frac{J(r)}{2}-B &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{J(r)}{2} \\ \end{array} \right) . \end{aligned}$$
(3)

The diagonalization of H leads to the following eigenspectrum and the associated eigenvectors

$$\begin{aligned} E_{1,4}= \frac{J(r)}{2}, ~~~~~&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\vert \psi _1\rangle = \vert 00 \rangle ,~\vert \psi _4\rangle =\vert 11 \rangle ,~~~~~~~~~~~~ \end{aligned}$$
(4a)
$$\begin{aligned} E_{2}= -\frac{ J(r)}{2}+\eta ,&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\vert \psi _2\rangle = \frac{1}{\sqrt{1+\xi ^2}}\left( \vert 10 \rangle +\xi \vert 01 \rangle \right) , \end{aligned}$$
(4b)
$$\begin{aligned} E_3= -\frac{ J(r)}{2}-\eta ,&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\vert \psi _3\rangle = \frac{1}{\sqrt{1+\xi ^2}}\left( \vert 01 \rangle -\xi \vert 10 \rangle \right) , \end{aligned}$$
(4c)

where \(\eta :=\sqrt{J^2(r)+B ^2}\) and \(\xi :=\left( B+\eta \right) /J(r)\). The kets \(\vert 0 \rangle \) and \(\vert 1 \rangle \) are taken here, respectively, as the spin-up and spin-down states and \(\left\{ \vert 0\rangle ,\vert 1 \rangle \right\} \) form a computational basis of a qubit.

3 Thermal quantum correlations over HL approximation

To quantify the amount of quantum correlations in a bipartite quantum system, various indicators are suggested in the literature [6,7,8,9,10,11,12,13,14,15,16,17,18,19]. In our study, we employ three main figures of merit, namely concurrence, TDD, and QC. To describe the behavior of the quantum correlations, we calculate firstly the thermal state of our mentioned Hamiltonian which describes a system state in equilibrium at temperature T.

3.1 Thermal state

In order to study the thermal behavior of quantum correlations in a two coupled spin-1/2 Heisenberg model, we will be focused on systems that are in a state of thermal equilibrium. Such state called thermal state and described by the Gibbs density matrix

$$\begin{aligned} \varrho (T)=\frac{1}{\mathcal {Z}}\exp {\left( -\beta H\right) }, \end{aligned}$$
(5)

with \({\mathcal {Z}} = \mathrm{Tr}\left[ \exp \left( -\beta H\right) \right] \) being the partition function and the inverse temperature \(\beta = 1/(k_BT)\) wherein \(k_B\) is the Boltzmann’s constant. Throughout this paper, we use \(k_B=1\) for convenience.

The matrix form of \(\varrho (T)\) can be obtained through the spectral decomposition of the Hamiltonian (3). Accordingly, for the considered two-qubit system, Eq. (5) can be rewritten as

$$\begin{aligned} \varrho (T)=\frac{1}{\mathcal {Z}}\sum _{m=1}^{4}e^{-\beta E_m}|\psi _m\rangle \langle \psi _m|. \end{aligned}$$
(6)

Reporting Eqs. (4a-4c) in Eq. (6), the density matrix describing the suggested system in thermal equilibrium can be written in the standard computational basis \(\mathcal {B}\) as

$$\begin{aligned} \varrho _{T}=\frac{1}{\mathcal {Z}}\left( \begin{array}{cccc} a &{} 0 &{} 0 &{} 0 \\ 0 &{} b^{+} &{} c &{} 0 \\ 0 &{} c &{} b^{-} &{} 0 \\ 0 &{} 0 &{} 0 &{} a \\ \end{array} \right) , \end{aligned}$$
(7)

where

$$\begin{aligned}&a=e^{\frac{-\beta J(r)}{2}},\\&b^{\pm }= e^{\frac{\beta J(r)}{2}}\left( e^{\pm \beta \eta }+\xi ^2 e^{\mp \beta \eta }\right) /(1+\xi ^2),\\&c=-2\xi e^{\frac{\beta J(r)}{2}} \sinh \left( \beta \eta \right) /(1+\xi ^2),\\&\mathcal {Z}=2a+b^{+}+b^{-}. \end{aligned}$$

3.2 Trace distance discord

The concept of quantum correlations beyond entanglement can be quantified by quantum discord [19]. It measures all non-classical correlations in a quantum bipartite system. Let \(\rho ^{ab}\) be a state describing any bipartite system ab in a composite Hilbert space \(\mathcal{H}=\mathcal{H}^a\otimes \mathcal{H}^b\), where \(\mathcal{H}^a\) and \(\mathcal{H}^b\) are the Hilbert spaces corresponding to the subsystems a and b of ab, respectively. The total correlation shared between a and b can be quantified by the quantum mutual information defined as [53],

$$\begin{aligned} \mathcal{I}\left( \rho ^{ab}\right) :=S\left( \rho ^{a}\right) +S\left( \rho ^{b}\right) -S\left( \rho ^{ab}\right) , \end{aligned}$$
(8)

where the quantity \(S(\rho )=-\mathrm{Tr}\left[ \rho \log _2\rho \right] \) stands for the von Neumann entropy and \(\rho ^{a(b)} ={\mathrm{Tr}}_{b(a)} [\rho ^{ab}]\) denotes the marginal state describing the subsystem a(b), which is determined by tracing out the subsystem b(a). Given that a local measurement was performed on the party b, the quantum discord is given by the quantity [19, 20]

$$\begin{aligned} \mathcal{D}\left( \rho ^{ab}\right) := \mathcal{I}\left( \rho ^{ab}\right) -{C C}\left( \rho ^{ab}\right) , \end{aligned}$$
(9)

which is expressed as the difference between the mutual information \(\mathcal{I}\left( \rho ^{ab}\right) \) and classical correlation \( CC\left( \rho ^{ab}\right) \) of the bipartite state \(\rho ^{ab}\), with

$$\begin{aligned} CC\left( \rho ^{ab}\right) =S\left( \rho ^{a}\right) -\min _{\left\{ {\Pi }^{b}_{(j)}\right\} }\sum _j p_jS\left( \rho ^{a|j}\right) . \end{aligned}$$
(10)

We look that the optimization in Eq. (10) is performed over the set \(\displaystyle \left\{ {\Pi }^{b}_{(j)}\right\} \) comprising all one-dimensional orthogonal projectors. The density matrix \(\displaystyle \rho ^{a|j}=p_{j}^{-1}\mathrm{Tr}_{b}\left[ \left( \mathbb {1}^a\otimes \Pi ^{b}_{(j)} \right) \rho ^{ab} \left( \mathbb {1}^a\otimes \Pi ^{b}_{(j)} \right) \right] \) refers to the post-measurement state of a after obtaining outcome j on subsystem b with probability \(p_j=\mathrm{Tr}\left\{ \left( \mathbb {1}^a\otimes \Pi ^{b}_{(j)} \right) \rho ^{ab} \left( \mathbb {1}^a\otimes \Pi ^{b}_{(j)} \right) \right\} \), and \(\mathbb {1}^a\) is the \(2\times 2\) identity operator on the subsystem a. In order to obtain a simple and reliable analytical result of the quantum discord, the best way is to treat this quantifier from a geometric perspective. One of the reliable geometric measures of quantum discord is that based on TDD concept which has been proposed recently [21, 24]. It quantifies the quantum correlation through the nearest Schatten 1-norm distance between the state under study and the quantum-classical state \(\rho ^{qc}=\displaystyle \sum _{k=1}^{2}p_k\rho _{k}^{a}\otimes \vert k\rangle ^{b}\langle k\vert \) with zero discord [54, 55]. For a bipartite state \(\rho ^{ab}\), TDD is defined by [21, 24].

$$\begin{aligned} D_{\text {T}}(\rho ^{ab})=\min _{\rho ^{\text {qc}}\in \Omega _0}\Vert \rho ^{ab}-\rho ^{\text {qc}}\Vert _1, \end{aligned}$$
(11)

where \(\Omega _0\) denotes the set of all zero discord states and \(\Vert .\Vert _1\) is the usual trace norm. An explicit formula of TDD was derived when the shape of considering two-qubit state \(\rho ^{ab}\) is X-structured [25]. It is given by

$$\begin{aligned} D_{\text {T}}(\rho ^{ab})=\sqrt{\frac{\max \{R_{33}^2,R_{22}^2+R_{30}^2\}\cdot R_{11}^2 -\min \{R_{11}^2,R_{33}^2\}\cdot R_{22}^2}{\max \{R_{33}^2,R_{22}^2+R_{30}^2\}-\min \{R_{11}^2,R_{33}^2\}+R_{11}^2-R_{22}^2}}, \end{aligned}$$
(12)

where \({R}_{\mu \nu }=\langle {\sigma _{\mu }^{a} \otimes \sigma _{\nu }^{b}\rangle }\) in Eq. (12) being the components of the correlation matrix occurring after decomposing the state \(\rho ^{ab}\) in the Fano–Bloch representation as [56]

$$\begin{aligned} \rho ^{ab} = \frac{1}{4}\sum _{\mu , \nu =0}^{3} {R}_{\mu \nu } \sigma _{\mu }^{a} \otimes \sigma _{\nu }^{b}. \end{aligned}$$
(13)

In the case of the thermal state \(\varrho _{T}\), which belongs to X-states family, its corresponding TDD can be easily computed. Indeed, by making use of Eqs. (7), (12), and (13), one straightforwardly finds that

(14)

3.3 Concurrence

To quantify the quantum entanglement contained in an arbitrary two-qubit mixed state \(\rho ^{ab}\), shared between two subsystem a and b, we present here the concurrence [6, 7]. It can be evaluated by using the formula

$$\begin{aligned} \mathcal {C}(\rho ^{ab})=\max \Bigl \{0,\lambda _1-\lambda _2-\lambda _3-\lambda _4\Bigr \}, \end{aligned}$$
(15)

where \(\lambda _{i}\) represents the square roots of the eigenvalues of the operator \(\rho ^{ab} \left( \sigma _{y}^{a}\otimes \sigma _{y}^{b}\right) (\varrho ^{ab}){^*} \left( \sigma _{y}^{a}\otimes \sigma _{y}^{b}\right) \) in descending order. Here \((\rho ^{ab}){^*}\) denotes the complex conjugate of the density matrix \(\rho ^{ab}\) in the standard two-qubit computational basis \(\mathcal {B}\), while \(\sigma _{y}^{i=a,b}\) being the well known 2D-Pauli spin matrix for the two level systems. For the two-qubit X-shaped thermal density operator (7), the amount of the thermal entanglement can be straightforwardly evaluated. By using Eqs. (7) and (15), one finds

$$\begin{aligned} \mathcal {C}\left( \varrho _{T}\right) =\frac{2e^{\frac{\beta J(r)}{2}}}{\mathcal {Z}}\max \left\{ 0, \frac{2|\xi |}{1+\xi ^2}\sinh {\left( \beta \eta \right) }-e^{-\beta J(r)}\right\} . \end{aligned}$$
(16)

3.4 Quantum consonance

In the following, we deal with the QC [26] as quantum correlations quantifier. This measure is defined as a sum of the entanglement and some other quantum correlations after omitting, by means of a local unitary operations, the local coherence from the total one [27]. By considering a bipartite quantum state \(\rho ^{ab}\), the QC is defined mathematically according to the following expression

$$\begin{aligned} \mathcal{Q}_C(\rho ^{ab})=\sum _{i j m n}\left| \rho _{i j m n}^{c}\left( 1-\delta _{i m}\right) \left( 1-\delta _{j n}\right) \right| , \end{aligned}$$
(17)

where \(\rho ^{c}=\left( U_{1} \otimes U_{2}\right) \rho ^{ab} \left( U_{1} \otimes U_{2}\right) ^{\dagger }\) stands for the transformed state achieved by making specific local unitary operations \(U_{1}\) and \(U_{2}\) on the original state \(\rho ^{ab}\). Interestingly, QC is simplified for a two-qubit X-state as the sum of off-diagonal terms of the density matrix [31].

The QC for thermal state \(\varrho _{T}\) can be easily computed because it belongs to X-states. By using (7), we indeed find that

(18)

which is identical to the analytical expression of TDD (14). Hence, we restrict in our analysis to compare only the thermal TDD behavior with the thermal entanglement.

4 Results and discussions

In the present section, we examine the behavior of thermal quantum correlations in the isotropic two-qubit XXX Heisenberg model in the HL approach described by the thermal state (7). More precisely, we are concerned in our analysis by studying the quantum correlations quantified by the geometric trace discord and quantum entanglement over the HL coupling relative distance r with the varying parameters; absolute temperature T and nonuniform magnetic field B.

In Fig. 3, we have plotted the TDD (14) and the concurrence (16) as functions of the nonuniform magnetic field B and the inter-spins relative distance r for fixed value of the absolute temperature \(T=0.1\). The results show that the thermal quantum correlations amounts are always zero for \(r\rightarrow 0\) whatever the B values. That is to say that there are no pairwise thermal quantum correlations when the two spins are uncoupled \(J(r)\rightarrow 0\). This happens due to the fact that for \(r\rightarrow 0\), the two-spin system is uncoupled. We notice also that for a fixed value of B, the amounts of quantum correlations become more significant for \(r=1.357\) wherein the exchange coupling constant J(r) is maximal (see Fig. 2). Figure 3 depicts that the quantumness of correlations undergoes an amplitude decreasing with B advancing. Furthermore, it turns out that as HL coupling relative distance increases, both entanglement and geometric quantum discord slowly vanish after a certain range of r. It is interesting to notice that for \(T=0.1\), the entanglement dies at \(r_0\approx 3\) for \(B=0\) while the TDD sustains over the larger ranges of HL coupling relative distance r and weak values of B. The entanglement vanishing moves away from \(r_0\) when the intensity of B increases. This can be perceived by the fact that, for \(\beta =10\), that is \(T=0.1\), and fixed values of B, an entangled-unentangled phase transition takes place for a threshold value \(r_f\) of HL relative distance r.

Fig. 3
figure 3

(Color online) XXX model for fixed temperature \(T=0.1\). TDD (a) and concurrence (b) versus the HL coupling relative distance r and nonuniform magnetic field B

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Fig. 4
figure 4

(Color online) Thermal quantum correlations over the HL coupling relative distance r with fixed \(T=0.1\) and \(B=0.7\) (a) and \(B=5\) (b). Here, dotted red line represents the entanglement and dashed blue line represents the TDD

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According to Eq. (16), this threshold value \(r_f\) can be achieved by resolving the following nonlinear equation

$$\begin{aligned} e^{ 10 J(r_f)}\sinh {\left( {10\eta _f}\right) }=\frac{1+\xi _{f}^{2}}{2|\xi _f|}, \end{aligned}$$
(19)

where \(\eta _f=\sqrt{J^2(r_f)+B ^2}\) and \(\xi _f=\left( B+\eta _f\right) /J(r_f)\). It is worth of mentioning that TDD (QC) reveals the non-classical correlations even in the absence of the entanglement. This implies that the TDD can capture the quantum correlations even in the separable mixed states.

Figure 4 shows that for \(T=0.1\) and \(B=0.7\) (Fig. 4a) and \(B=5\) (Fig. 4b), the TDD and concurrence are considerably matched. This typical behavior emerges due to the link between the quantum entanglement and the quantum coherence concept, quantified here by QC, which is a fundamental manifestation of the quantum superposition principle on which based the quantum entanglement [57].

The thermal behaviors of TDD (QC) and concurrence as functions of the HL coupling distance r and nonuniform magnetic field B are plotted in Figs. 5 and 6, where the temperature is chosen to be \(T=0.3\) and \(T=0.6\), respectively. As can be seen in Fig. 5, the TDD behavior is similar to the previous case (Fig. 3), which means TDD decreases gradually with increasing B; however, the only difference is that its maximum value is less than the previous case. The entanglement behavior here is different from TDD. The amount of the magnetic field in which entanglement reaches its maximum value is larger than the previous case. It is interesting to note that, as the temperature increases, this difference becomes more apparent (see Fig. 6). As seen from Fig. 6, when \(B< 2\), the entanglement is zero while TDD reaches its maximum value in this interval. In other words, although the quantum correlation captured by TDD achieves its maximum in this range, the entanglement does not have any role in this correlation. Moreover, it is observed that the non-classical correlation quantified by TDD decreases whereas entanglement increases to reaches its maximum value to coincide, therefore, with TDD for strong magnetic field values. This indicates that increasing the nonuniform magnetic field plays a positive role in the production of the entanglement in high temperatures.

Fig. 5
figure 5

XXX model for fixed temperature \(T=0.3\). TDD (a) and concurrence (b) versus r and B

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Fig. 6
figure 6

XXX model for fixed temperature \(T=0.6\). TDD (a) and concurrence (b) versus r and B

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Furthermore, by comparing Figs. 3, 4, 5 and 6, it can be found that the behavior of both quantum correlations is the same with varying the HL coupling distance r. One can note that when the HL coupling distance r increases, the quantum correlations after reaching their maximum value then monotonically decrease down to zero. The physical reason for this observation may be rooted in the behavior of HL coupling strength J(r) in terms of the inter-spins relative distance r (see Fig. 2). On the other hand, the correspondence of TDD and thermal entanglement for sufficient magnetic field values may be understood by the fact that the strong nonuniform magnetic field induces entangled phase in the two-qubit (antiferromagnetic XXX model) thermal state.

Fig. 7
figure 7

XXX model for fixed HL relative distance \(r=1.357\). TDD (a) and concurrence (b) versus B and absolute temperature T

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Finally, Fig. 7 shows the thermal evolution of TDD and concurrence over the nonuniform magnetic field B and temperature with fixed HL coupling relative distance r. As can be seen, the behaviors of TDD and entanglement are almost the same although the rate of changes in entanglement is slower than that of TDD. Also, one can observe that for any given magnetic field, the entanglement is equal to zero for temperatures greater than a certain threshold temperature. It is intriguing to note that by increasing B, the critical temperature above that entanglement disappears increases. That is to say, the high value of nonuniform magnetic field plays a crucial role in sustaining entanglement at high temperatures; however, the role of the magnetic field is destructive at low temperatures (see Fig. 3). It seems that the possibility of maintaining entanglement in such a quantum system at high temperatures requires stronger magnetic fields.

5 Concluding remarks

In summary, we have investigated the thermal quantum correlations shared between the bipartite system parts in the ambit of the Heitler–London scheme. We dealt with a system composed of two XXX Heisenberg spins subjected to a transverse external nonuniform magnetic field. For \(T=0.1\), we have found that the amplitude of quantum correlations decrease with nonuniform magnetic field increasing. Furthermore, it is observed that the TDD surpasses the entanglement for small values of B and therefore persists over the large range of HL coupling relative distance r in comparison with entanglement. Besides, the entanglement behavior in terms of B and r detects an entangled-unentangled phase transition. When the temperature increases, TDD retains the same appearance but with a reduced amount. Correspondingly, the thermal entanglement exhibits a completely different behavior to the one of \(T=0.1\), especially for a small range of B. However, there is a kind of correspondence between thermal entanglement and thermal non-classical correlations for a strong value of B. TDD can detect the quantum correlations even in the absence of the entanglement, which is generally an expected behavior of quantum discord. The obtained results confirmed that the suitable strong value of the nonuniform magnetic field helps to sustain entanglement even for higher absolute temperatures. However, the magnetic field has a destructive effect on the entanglement at low temperatures. We mention that the adopted model may be useful for designing quantum wires, data bus, solid-state gates, and quantum processors [41]. Therefore, we think that our observations may provide a new insight into the dynamics of non-classical correlations in Heisenberg spin chains and shed light on the maintenance of quantum correlations within the framework of real quantum systems, especially solid-state ones.

As a prolongation, the present study can be extended to another spin chain configurations such as XXZ and XYZ in the presence of the Dzyaloshinskii–Moriya and Kaplan–Shekhtman–Entin–Wohlman–Aharony interactions [30, 32, 38] to know the behavior of quantum discord and entanglement in the Heitler–London approach. We think this can be useful for quantum information processing.