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Pairwise thermal entanglement and quantum discord in a three-ligand spin-star structure

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Abstract

In this work, we perform a comparative study between the pairwise thermal entanglement (PWTE) and thermal quantum discord (TQD) to detect quantum phase transitions (QPT)s in a three-ligand spin-star structure whose magnetic interactions are described by different model Hamiltonians such as pure Dzyaloshinskii–Moriya (DM) interaction, anisotropic Heisenberg model (XXZ), and XXZ model with the different components of the DM interaction. Representing the system’s energy spectrum, we also focus on the critical points of QPTs where the ground-state level crossing happens in such models. Taking advantage of the concurrence as a measure of the PWTE, we found that while the ligand–ligand concurrence in all models is sensitive to the ground-state level crossing, the concurrence between the central qubit and a ligand cannot exhibit a QPT. In contrast, the TQD between any two arbitrary qubits can be a signature of a QPT in a large range of temperature. However, depending on the model studied, the behavior of the TQD at the critical point will be different. In addition, the TQD behaves quite differently than the concurrence. Moreover, in order to confirm the numerical results, we analytically study the entanglement behavior at the low-temperature limit as well as the high-temperature regime. We realized that, at the low-temperature limit, the maximum value of the concurrence is approximately equal to 0.33, independent of the model studied. On the other hand, at high-temperature regime, the concurrence is suppressed down to zero rapidly beyond a critical value of temperature. The dependence of the critical temperature on the DM interaction and the anisotropy parameter is obtained explicitly. Finally we show that there is a perfect agreement between the analytical results and the numerical predictions.

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Acknowledgements

This work is based upon research supported by Shahid Bahonar University Foundation. The author’s special thanks should be given to Ali Davody for his assistance in mathematical manipulations and reviewing the manuscript. The author would also like to thank S. Mahdavifar, M. H. Alizadeh, H. Maleki, and Mrs. F. Mohtashamian for their help in improving the presentation. The author thanks all scientists who helped and communicated for clarifying some details and approaches in the past years in special, A. Honecker and M. Troyer.

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  1. Department of Physics, Shahid Bahonar University of Kerman, Kerman, Iran

    Mostafa Motamedifar

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Correspondence to Mostafa Motamedifar.

Appendices

Appendix 1

The eigenvalues and eigenvectors of the Hamiltonian written in Eq. 13 (DM-3LSSS) are as

$$\begin{aligned} E_1= & {} -D~~,E_2=D,~~E_{3,4}=-\frac{\sqrt{3}D}{2},~~E_{5,6}=\frac{\sqrt{3}D}{2},~~ E_{7,8}=-\frac{D}{2},\\ E_{9,10}= & {} \frac{D}{2},~~E_{11,12,13,14,15,16}=0\\ |\psi _1\rangle= & {} \frac{1}{\sqrt{6}}\Big [i|0011\rangle +i|0101\rangle +i|0110\rangle +|1001\rangle +|1010\rangle +|1100\rangle \Big ],\\ |\psi _2\rangle= & {} \frac{1}{\sqrt{6}}\Big [-i|0011\rangle -i|0101\rangle -i|0110\rangle +|1001\rangle +|1010\rangle +|1100\rangle \Big ],\\ |\psi _3\rangle= & {} \frac{1}{\sqrt{6}}\Big [i\sqrt{3} |0111\rangle +|1011\rangle +|1101\rangle +|1110\rangle \Big ],\\ |\psi _4\rangle= & {} \frac{1}{\sqrt{6}}\Big [i|0001\rangle +i|0010\rangle +i|0100\rangle +\sqrt{3}|1000\rangle \Big ],\\ |\psi _5\rangle= & {} \frac{1}{\sqrt{6}}\Big [-i\sqrt{3} |0111\rangle +|1011\rangle +|1101\rangle +|1110\rangle \Big ],\\ |\psi _6\rangle= & {} \frac{1}{\sqrt{6}}\Big [-i|0001\rangle -i|0010\rangle -i|0100\rangle +\sqrt{3}|1000\rangle \Big ],\\ |\psi _7\rangle= & {} \frac{1}{ 2}\Big [-i|0011\rangle +i|0110\rangle -|1001\rangle +|1100\rangle \Big ],\\ |\psi _8\rangle= & {} \frac{1}{2\sqrt{3}}\Big [i|0011\rangle -2i|0101\rangle +i|0110\rangle -|1001\rangle +2|1010\rangle -|1100\rangle \Big ],\\ |\psi _9\rangle= & {} \frac{1}{ 2}\Big [i|0011\rangle -i|0110\rangle -|1001\rangle +|1100\rangle \Big ],\\ |\psi _{10}\rangle= & {} \frac{1}{2\sqrt{3}}\Big [-i|0011\rangle +2i|0101\rangle -i|0110\rangle -|1001\rangle +2|1010\rangle -|1100\rangle \Big ],\\ |\psi _{11}\rangle= & {} |1111\rangle ,\\ |\psi _{12}\rangle= & {} \frac{1}{\sqrt{2}}\Big [-|1011\rangle +|1110\rangle \Big ],\\ |\psi _{13}\rangle= & {} \frac{1}{\sqrt{6}}\Big [-|1011\rangle +2|1101\rangle -|1110\rangle \Big ],\\ |\psi _{14}\rangle= & {} \frac{1}{\sqrt{2}}\Big [-|0001\rangle +|0100\rangle \Big ],\\ |\psi _{15}\rangle= & {} \frac{1}{\sqrt{6}}\Big [-|0001\rangle +2|0010\rangle -|0100\rangle \Big ],\\ |\psi _{16}\rangle= & {} |0000\rangle , \end{aligned}$$

Appendix 2

The eigenvalues and eigenvectors of the Hamiltonian written in Eq. 15 (XXZ-3LSSS) are as

$$\begin{aligned} E_1= & {} \frac{1}{4} (-\varDelta -4)~~,E_{2,3}=\frac{1}{4} (-\varDelta -2),~~E_{4,5}= \frac{2-\varDelta }{4},~~E_{6}=\frac{4-\varDelta }{4}, \\ E_{7,8,9,10}= & {} \frac{\varDelta }{4},~~ E_{11,12}=\frac{3\varDelta }{4}, E_{13,14}=\frac{1}{4} \left( -2 \sqrt{\varDelta ^2+3}-\varDelta \right) , \\ E_{15,16}= & {} \frac{1}{4} \left( 2 \sqrt{\varDelta ^2+3}-\varDelta \right) \end{aligned}$$
$$\begin{aligned} |\psi _1\rangle= & {} \frac{1}{\sqrt{6}}\Big [-|0011\rangle -|0101\rangle -|0110\rangle +|1001 \rangle +|1010\rangle +|1100\rangle \Big ],\\ |\psi _2\rangle= & {} \frac{1}{ 2}\Big [|0011\rangle -|0110\rangle -|1001\rangle +|1100 \rangle \Big ],\\ |\psi _3\rangle= & {} \frac{1}{ 2}\Big [|0101\rangle -|0110\rangle -|1001\rangle +|1010\rangle \Big ],\\ |\psi _4\rangle= & {} \frac{1}{ 2}\Big [-|0011\rangle +|0110\rangle -|1001\rangle +| 1100\rangle \Big ],\\ |\psi _5\rangle= & {} \frac{1}{ 2}\Big [-|0101\rangle +|0110\rangle -|1001\rangle +| 1010\rangle \Big ],\\ |\psi _6\rangle= & {} \frac{1}{\sqrt{6}}\Big [|0011\rangle +|0101\rangle +|0110\rangle + |1001\rangle +|1010\rangle +|1100\rangle \Big ],\\ |\psi _{7}\rangle= & {} \frac{1}{\sqrt{2}}\Big [-|1011\rangle +|1110\rangle \Big ],\\ |\psi _{8}\rangle= & {} \frac{1}{\sqrt{2}}\Big [-|1011\rangle +|1101\rangle \Big ],\\ |\psi _{9}\rangle= & {} \frac{1}{\sqrt{2}}\Big [-|0001\rangle +|0100\rangle \Big ],\\ |\psi _{10}\rangle= & {} \frac{1}{\sqrt{2}}\Big [-|0001\rangle +|0010\rangle \Big ],\\ |\psi _{11}\rangle= & {} |1111\rangle ,\\ |\psi _{12}\rangle= & {} |0000\rangle ,\\ |\psi _{13}\rangle= & {} \frac{1}{\sqrt{3+\zeta _{13}^2}}\Big [-\zeta _{13} |0111\rangle +|1011\rangle +|1101\rangle +|1110\rangle \Big ],\\ |\psi _{14}\rangle= & {} \frac{\zeta _{14}}{\sqrt{1+3\zeta _{14}^2}}\Big [-|0001 \rangle -|0010\rangle -|0100\rangle +\frac{1}{\zeta _{14}}|1000\rangle \Big ],\\ |\psi _{15}\rangle= & {} \frac{1}{\sqrt{3+\zeta _{15}^2}}\Big [\zeta _{15} |0111\rangle +| 1011\rangle +|1101\rangle +|1110\rangle \Big ],\\ |\psi _{16}\rangle= & {} \frac{\zeta _{16}}{\sqrt{1+3\zeta _{16}^2}}\Big [|0001\rangle +| 0010\rangle +|0100\rangle +\frac{1}{\zeta _{16}}|1000\rangle \Big ],\\ \zeta _{13}= & {} \sqrt{\varDelta ^2+3}+\varDelta ,~~~\zeta _{14}=\frac{1}{\sqrt{\varDelta ^2+3}+ \varDelta },\\ \zeta _{15}= & {} \sqrt{\varDelta ^2+3}-\varDelta ,~~~\zeta _{16}=\frac{1}{\sqrt{\varDelta ^2+3}- \varDelta }, \end{aligned}$$

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Motamedifar, M. Pairwise thermal entanglement and quantum discord in a three-ligand spin-star structure. Quantum Inf Process 16, 162 (2017). https://doi.org/10.1007/s11128-017-1611-1

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