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Non-local spin entanglement in a fermionic chain

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Abstract

An effective two-spin density matrix (TSDM) for a pair of spin-1/2 degree of freedom, residing at a distance of R in a spinfull Fermi sea, can be obtained from the two-electron density matrix following the framework prescribed in Oh and Kim (Phys Rev A 69:054305, 2004. https://doi.org/10.1103/PhysRevA.69.054305). We note that the single-spin density matrix (SSDM) obtained from this TSDM for generic spin-degenerate systems of free fermions is always pinned to the maximally mixed state, i.e. \((1/2) \ {\mathbb {I}}\), independent of the distance R, while the TSDM confirms to the form for the set of maximally entangled mixed state (the so-called X-state) at finite R. The X-state reduces to a pure state (a singlet) in the \(R\rightarrow 0\) limit, while it saturates to an X-state with the largest allowed value of von-Neumann entropy of \(2 \ln 2\) as \(R\rightarrow \infty \) independent of the value of chemical potential. However, once an external magnetic field is applied to lift the spin-degeneracy, we find that the von-Neumann entropy of SSDM becomes a function of the distance R between the two spins. We also show that the von-Neumann entropy of TSDM in the \(R\rightarrow \infty \) limit becomes a function of the chemical potential and it saturates to \(2 \ln 2\) only when the band in completely filled unlike the spin-degenerate case. Finally, we extend our study to include spin–orbit coupling and show that it does effect these asymptotic results. Our findings are in sharp contrast to previous works, which were based on continuum models owing to physics which stem from the lattice model.

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Data availability

The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

A.V.V. acknowledges the Council of Scientific and Industrial Research (CSIR), Govt. of India, for financial support. S.D. would like to acknowledge the MATRICS grant (Grant No. MTR/ 2019/001 043) from the Science and Engineering Research Board (SERB) for funding.

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Authors and Affiliations

  1. Institute of Physics Sachivalaya Marg, Bhubaneswar, 751005, India

    Sayan Jana & Arijit Saha

  2. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai, 400094, India

    Sayan Jana & Arijit Saha

  3. School of Mechanical Engineering, Tel Aviv University, 69978, Tel Aviv, Israel

    Sayan Jana

  4. Indian Institute of Science Education and Research Kolkata, Mohanpur, Nadia, West Bengal, 741246, India

    Anant V. Varma & Sourin Das

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  1. Sayan Jana
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The first two authors, S.J. and A.V.V., have contributed equally to this work.

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Correspondence to Sayan Jana.

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Jana, S., Varma, A.V., Saha, A. et al. Non-local spin entanglement in a fermionic chain. Quantum Inf Process 21, 374 (2022). https://doi.org/10.1007/s11128-022-03718-z

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