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Quantum coherence and its distribution in the extended Ising chain

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Abstract

We investigate the two-spin coherence and its distribution in an extended Ising model with topological characterization. Some interesting discrepancies among the basis-dependent coherences under the different bases are found with the help of the three-spin interaction and magnetic field. The basis-independent coherence and its distribution (the collective and localized coherence) for the spin-pairs with different site distances are studied. We find that the collective coherence possesses the long-range property due to the contribution of classical correlation. We demonstrate that the first-order derivatives of the two-spin coherences can correctly characterize all of the topological quantum phase transitions regardless of the basis transformations, the site distances, or the types of coherence.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under the Grant Nos. 61575055, 11874132, and 11675046. We thank Dr. Biao-Liang Ye for helpful suggestions.

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Authors and Affiliations

  1. Key Laboratory for Photonic and Electronic Bandgap Materials, Ministry of Education, School of Physics and Electronic Engineering, Harbin Normal University, Harbin, 150025, China

    Shaoying Yin, Yueyuan Wang, Lin Li & Wenjun Sun

  2. School of Physics, Harbin Institute of Technology, Harbin, 150001, China

    Jie Song & Shutian Liu

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  1. Shaoying Yin
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  2. Jie Song
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Appendix: TQPTs driven by the magnetic field

Appendix: TQPTs driven by the magnetic field

In this Appendix, by means of the energy spectra of the ground state, winding parameter, and characteristic equations, we analyze the topological characterization of the extended Ising model as the magnetic field is the driving parameter. Then, we consider the parameters of the extended Ising model as \(N=201\), \(\gamma =1\), \(\delta = 1\), and \(\alpha =1.5\). The critical points of TQPTs can be derived from the characteristic equation \(1.5\xi ^2 + \xi -\lambda = 0\), where \(\xi = \exp \left( \frac{i2\pi k}{N}\right) \) and \(|\xi | = 1\), and it can be numerically verified that \(\lambda _{c1}=-1.5\), \(\lambda _{c2}=0.5\), and \(\lambda _{c3}=2.5\) satisfy the characteristic equation [28, 64,65,66]. The energy spectra as a function of \(\lambda \) are shown in Fig. 6a, which also manifest the locations of the critical point again. Moreover, the trajectories of winding vectors are plotted in Fig. 6b with \(\lambda =-2, -1, 2\), and 3. The winding numbers can be obtained in the auxiliary \(y-z\) plane according to Eq. (17), and it changes from 0 to 2 at \(\lambda _{c1}=-1.5\), from 2 to 1 at \(\lambda _{c2}=0.5\), and from 1 to 0 at \(\lambda _{c3}=2.5\).

Fig. 6
figure 6

a Energy spectra as a function of \(\lambda \). b Trajectories of winding vectors in y–z plane with parameters \(\lambda =-2, -1, 2\), and 3, the winding numbers are 0, 2, 1, and 0, respectively. The other parameters are set to \(N=201\), \(\gamma =1\), \(\delta =1\), and \(\alpha = 1.5\)

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Yin, S., Song, J., Wang, Y. et al. Quantum coherence and its distribution in the extended Ising chain. Quantum Inf Process 20, 326 (2021). https://doi.org/10.1007/s11128-021-03266-y

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