Exact holographic tensor networks for the Motzkin spin chain

Rafael N. Alexander1,2, Glen Evenbly3, and Israel Klich4

1Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia
2Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA
3School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
4Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA

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The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network's utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice. We discuss connections to renormalization and holography.

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[1] Thomas Schuster, Bryce Kobrin, Ping Gao, Iris Cong, Emil T. Khabiboulline, Norbert M. Linke, Mikhail D. Lukin, Christopher Monroe, Beni Yoshida, and Norman Y. Yao, "Many-Body Quantum Teleportation via Operator Spreading in the Traversable Wormhole Protocol", Physical Review X 12 3, 031013 (2022).

[2] Glen Evenbly, "Number-state preserving tensor networks as classifiers for supervised learning", Frontiers in Physics 10, 858388 (2022).

[3] Zhao Zhang and Israel Klich, "Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models", SciPost Physics 15 2, 044 (2023).

[4] Cristian Voinea, Songyang Pu, Ammar Kirmani, Pouyan Ghaemi, Armin Rahmani, and Zlatko Papić, "Deformed Fredkin model for the ν=5/2 Moore-Read state on thin cylinders", Physical Review Research 6 1, 013105 (2024).

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[6] Jeevan Chandra and Thomas Hartman, "Toward random tensor networks and holographic codes in CFT", Journal of High Energy Physics 2023 5, 109 (2023).

[7] Rafael N. Alexander, Amr Ahmadain, Zhao Zhang, and Israel Klich, "Exact rainbow tensor networks for the colorful Motzkin and Fredkin spin chains", Physical Review B 100 21, 214430 (2019).

[8] Glen Evenbly, "Number-State Preserving Tensor Networks as Classifiers for Supervised Learning", arXiv:1905.06352, (2019).

[9] Jacob Miller, Guillaume Rabusseau, and John Terilla, "Tensor Networks for Probabilistic Sequence Modeling", arXiv:2003.01039, (2020).

[10] A. Ahmadain and I. Klich, "Emergent geometry and path integral optimization for a Lifshitz action", Physical Review D 103 10, 105013 (2021).

[11] Fumihiko Sugino, "Highly Entangled Spin Chains and 2D Quantum Gravity", Symmetry 12 6, 916 (2020).

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