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Some Aspects of Affleck–Kennedy–Lieb–Tasaki Models: Tensor Network, Physical Properties, Spectral Gap, Deformation, and Quantum Computation

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Entanglement in Spin Chains

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Abstract

Affleck, Kennedy, Lieb, and Tasaki constructed a spin-1 model that is isotropic in spins and possesses a provable finite gap above the ground state more than three decades ago. They also constructed models in two dimensions. Their construction has impacted subsequent research that is still active. In this review article, we review some selected progresses, such as magnetic ordering of the AKLT models, emerging phases under deforming the AKLT Hamiltonians, symmetry-protected topological order in several AKLT models, their spectral gap, and applications for quantum computation.

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Notes

  1. 1.

    This operation is in fact the starting point of both proofs [36] and [32].

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Acknowledgements

T.-C.W. acknowledges support from National Science Foundation under Grants No. PHY 1314748, No. PHY 1620252, and No. PHY 1915165 on subjects related to AKLT models. R.R. is supported by the Canada First Research Excellence Fund, Quantum Materials, and Future Technologies Program.

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Authors and Affiliations

  1. C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook, NY, USA

    Tzu-Chieh Wei

  2. Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada

    Robert Raussendorf & Ian Affleck

  3. Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, BC, Canada

    Robert Raussendorf & Ian Affleck

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  1. Tzu-Chieh Wei
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  2. Robert Raussendorf
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  3. Ian Affleck
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Correspondence to Ian Affleck .

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Editors and Affiliations

  1. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, China

    Abolfazl Bayat

  2. Department of Physics and Astronomy, University College London, London, UK

    Sougato Bose

  3. Department of Physics, University of Gothenburg, Göteborg, Sweden

    Henrik Johannesson

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Wei, TC., Raussendorf, R., Affleck, I. (2022). Some Aspects of Affleck–Kennedy–Lieb–Tasaki Models: Tensor Network, Physical Properties, Spectral Gap, Deformation, and Quantum Computation. In: Bayat, A., Bose, S., Johannesson, H. (eds) Entanglement in Spin Chains. Quantum Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-031-03998-0_5

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