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Dynamics of the Bipartite and Multipartite Entanglement in the Heisenberg Chain with Dzyaloshinskii-Moriya Interaction

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Abstract

We study and compare the dynamics of the bipartite entanglement (BE) and multipartite entanglement (ME) in the Heisenberg chain with Dzyaloshinskii-Moriya (DM) interaction. We have shown that the DM interaction can induce both entanglement sudden death and entanglement sudden birth in the BE, while no such phenomena occurs in the ME features of the systems. The robustness dynamic of ME in the time domain would be interesting to those are working in the quantum information and quantum computing communities.

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Acknowledgments

It is our pleasure to thank Saeed Mahdavifar for his very suggestive comments and interesting discussion.

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Authors and Affiliations

  1. Department of Physics, College of Science Yadegar-e- Imam Khomeini (RAH) Shahre-rey Branch, Islamic Azad University, Tehran, Iran

    M. R. Soltani

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  1. M. R. Soltani
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Correspondence to M. R. Soltani.

Appendix: Time evelotion operators matrix elements

Appendix: Time evelotion operators matrix elements

$$\begin{array}{@{}rcl@{}} u_{11}&=&e^{-\frac{i}{2}({\Delta}+3h)}\\ u_{22}&=&\frac{1}{2}e^{-\frac{iht}{2}}+\frac{1}{2x}e^{-i(\frac{2h-{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\\ u_{23}&=&-\frac{2i(J+iD)}{x}e^{-i(\frac{2h-{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{25}&=&\chi\left( -\frac{1}{2}e^{-\frac{iht}{2}}+\frac{1}{2x}e^{-i(\frac{2h-{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\right)\\ u_{32}&=&\frac{2i(J-iD)}{x}e^{-i(\frac{2h-{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{33}&=&\frac{1}{x}e^{-i(\frac{2h-{\Delta}}{4})t}\left( x\cos\frac{xt}{4}+i{\Delta}\sin\frac{xt}{4}\right)\\ u_{35}&=&-\frac{2i(J+iD)}{x}e^{-i(\frac{2h-{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{44}&=&\frac{1}{2}e^{-\frac{iht}{2}}+\frac{1}{2x}e^{i(\frac{2h+{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\\ u_{46}&=&-\frac{2i(J+iD)}{x}e^{i(\frac{2h+{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{47}&=&\chi\left( -\frac{1}{2}e^{\frac{iht}{2}}+\frac{1}{2x}e^{i(\frac{2h+{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\right)\\ u_{52}&=&\chi^{\ast}\left( -\frac{1}{2}e^{\frac{-iht}{2}}+\frac{1}{2x}e^{-i(\frac{2h-{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\right)\\ u_{53}&=&\frac{2i(J-iD)}{x}e^{-i(\frac{2h-{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{55}&=&\frac{1}{2}e^{-\frac{iht}{2}}+\frac{1}{2x}e^{-i(\frac{2h-{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\\ u_{64}&=&\frac{2i(J-iD)}{x}e^{i(\frac{2h+{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{66}&=&\frac{1}{x}e^{i(\frac{2h+{\Delta}}{4})t}\left( x\cos\frac{xt}{4}+i{\Delta}\sin\frac{xt}{4}\right)\\ u_{67}&=&-\frac{2i(J+iD)}{x}e^{i(\frac{2h+{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{74}&=&\chi^{\ast}\left( -\frac{1}{2}e^{\frac{iht}{2}}+\frac{1}{2x}e^{i(\frac{2h+{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\right)\\ u_{76}&=&-\frac{2i(J-iD)}{x}e^{i(\frac{2h+{\Delta}}{4})t}\sin\frac{xt}{4}\\ u_{77}&=&-\frac{1}{2}e^{\frac{iht}{2}}+\frac{1}{2x}e^{i(\frac{2h+{\Delta}}{4})t}\left( x\cos\frac{xt}{4}-i{\Delta}\sin\frac{xt}{4}\right)\\ u_{88}&=&e^{\frac{i}{2}({\Delta}-3h)} \end{array} $$

with \(x=\sqrt {{\Delta }^{2}+8(D^{2}+J^{2})}\) and \(\chi =\frac {J+iD}{J-iD}\).

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Soltani, M.R. Dynamics of the Bipartite and Multipartite Entanglement in the Heisenberg Chain with Dzyaloshinskii-Moriya Interaction. J Supercond Nov Magn 29, 847–854 (2016). https://doi.org/10.1007/s10948-015-3359-0

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