Hyperbolic Lattices

Curved spaces are usually associated with high-energy physics and cosmology. However, the possibility of tabletop experiments emulating such phenomena and the interest in synthetic matter in curved spaces have elevated curved spaces to relevance in condensed matter physics as well. Spaces with negative curvature are particularly difficult to realize experimentally. One way to achieve it, is by discretizing the space and implementing the resulting lattice. In this project, we study various aspects of such hyperbolic lattices, both experimentally and theoretically.

Hyperbolic lattices can be interpreted as tilings of a plane with constant negative curvature. On these lattices one can define tight-binding models of different degrees of complexity. Starting with a simple nearest-neighbor hopping model (with equal and real hopping amplitudes), we demonstrate that such lattices can be realized in electric circuits and that signatures of the negative curvature can be experimentally obtained. Next, we show theoretically that topological models analogous to the Haldane and the Kane-Mele model can be defined on hyperbolic lattices and we study bulk invariants and the bulk-boundary correspondence. Finally, we make progress on developing a band-theoretic description of hyperbolic lattices based on the recent extension of Bloch’s theorem to hyperbolic lattices.

Publications

Simulating Holographic Conformal Field Theories on Hyperbolic Lattices

S. Dey, A. Chen, P. Basteiro, A. Fritzsche, M. Greiter, M. Kaminski, P. M. Lenggenhager, R. Meyer, R. Sorbello, A. Stegmaier, R. Thomale, J. Erdmenger, and I. Boettcher, arXiv:2404.03062 (2024).

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@article{Dey:2024,
    title               = {Simulating Holographic Conformal Field Theories on Hyperbolic Lattices}, 
    author              = {Santanu Dey and Anffany Chen and Pablo Basteiro and Alexander Fritzsche and Martin Greiter and Matthias Kaminski and Patrick M. Lenggenhager and Rene Meyer and Riccardo Sorbello and Alexander Stegmaier and Ronny Thomale and Johanna Erdmenger and Igor Boettcher},
    journal             = {arXiv preprint},
    year                = {2024},
    doi                 = {10.48550/arXiv.2404.03062},
    url                 = {https://arxiv.org/abs/2404.03062}
    eprint              = {2404.03062},
    eprintType          = {arXiv},
    archivePrefix       = {arXiv}
}

Non-Abelian hyperbolic band theory from supercells

P. M. Lenggenhager, J. Maciejko, and T. Bzdušek, Phys. Rev. Lett. 131, 226401 (2023).

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@article{Lenggenhager:2023,
  title               = {Non-{A}belian hyperbolic band theory from supercells}, 
  author              = {Lenggenhager, Patrick M. and Maciejko, Joseph and Bzdu\v{s}ek, Tom\'{a}\v{s}},
  journal             = {Phys. Rev. Lett.},
  volume              = {131},
  issue               = {22},
  pages               = {226401},
  numpages            = {7},
  year                = {2023},
  month               = {Dec},
  publisher           = {American Physical Society},
  doi                 = {10.1103/PhysRevLett.131.226401},
  url                 = {https://link.aps.org/doi/10.1103/PhysRevLett.131.226401}
}

Symmetry and topology of hyperbolic Haldane models

A. Chen, Y. Guan, P. M. Lenggenhager, J. Maciejko, I. Boettcher, and T. Bzdušek, Phys. Rev. B 108, 085114 (2023).

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@article{Chen:2023,
  title               = {Symmetry and topology of hyperbolic {H}aldane models}, 
  author              = {Chen, Anffany and Guan, Yifei and Lenggenhager, Patrick M. and Maciejko, Joseph and Boettcher, Igor and Bzdu\v{s}ek, Tom\'{a}\v{s}},
  journal             = {Phys. Rev. B},
  volume              = {108},
  issue               = {8},
  pages               = {085114},
  numpages            = {42},
  year                = {2023},
  month               = {8},
  publisher           = {American Physical Society},
  doi                 = {10.1103/PhysRevB.108.085114}
}

Hyperbolic non-Abelian semimetal

T. Tummuru, A. Chen, P. M. Lenggenhager, T. Neupert, J. Maciejko, and T. Bzdušek, arXiv:2307.09876 (2023).

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@article{Tummuru:2023,
  title               = {Hyperbolic non-Abelian semimetal}, 
  author              = {Tarun Tummuru and Anffany Chen and Patrick M. Lenggenhager and Titus Neupert and Joseph Maciejko and Tom\'{a}\v{s} Bzdu\v{s}ek},
  journal             = {arXiv preprint},
  year                = {2023},
  doi                 = {10.48550/arXiv.2307.09876},
  url                 = {https://arxiv.org/abs/2307.09876}
  eprint              = {2307.09876},
  eprintType          = {arXiv},
  archivePrefix       = {arXiv}
}

Hyperbolic topological band insulators

D. M. Urwyler, P. M. Lenggenhager, I. Boettcher, R. Thomale, T. Neupert, and T. Bzdušek, Phys. Rev. Lett. 129, 246402 (2022).

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@article{Urwyler:2022,
  title               = {Hyperbolic topological band insulators}, 
  author              = {Urwyler, David M. and Lenggenhager, Patrick M. and Boettcher, Igor and Thomale, Ronny and Neupert, Titus and Bzdu\v{s}ek, Tom\'{a}\v{s}},
  journal             = {Phys. Rev. Lett.},
  volume              = {129},
  issue               = {24},
  pages               = {246402},
  numpages            = {7},
  year                = {2022},
  month               = {Dec},
  publisher           = {American Physical Society},
  doi                 = {10.1103/PhysRevLett.129.246402},
  url                 = {https://link.aps.org/doi/10.1103/PhysRevLett.129.246402}
}

Simulating hyperbolic space on a circuit board

P. M. Lenggenhager, A. Stegmaier, L. K. Upreti, T. Hofmann, T. Helbig, A. Vollhardt, M. Greiter, C. H. Lee, S. Imhof, H. Brand, T. Kießling, I. Boettcher, T. Neupert, R. Thomale, and T. Bzdušek, Nat. Commun. 13, 4373 (2022).

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@article{Lenggenhager:2022a,
  title               = {Simulating hyperbolic space on a circuit board}, 
  author              = {Lenggenhager, Patrick M. and Stegmaier, Alexander and Upreti, Lavi K. and Hofmann, Tobias and Helbig, Tobias and Vollhardt, Achim and Greiter, Martin and Lee, Ching Hua and Imhof, Stefan and Brand, Hauke and Kie{\ss}ling, Tobias and Boettcher, Igor and Neupert, Titus and Thomale, Ronny and Bzdu\v{s}ek, Tom\'{a}\v{s}},
  journal             = {Nature Communications},
  volume              = {13},
  pages               = {4373},
  issn                = {2041-1723},
  number              = {1},
  year                = {2022},
  doi                 = {10.1038/s41467-022-32042-4},
  url                 = {https://doi.org/10.1038/s41467-022-32042-4},
}

Project members: P. M. Lenggenhager, A. Stegmaier, D. M. Urwyler, A. Chen, T. Tummuru, T. Neupert, R. Thomale, J. Maciejko, T. Bzdušek, et. al