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From hyperbolic drum towards hyperbolic topological insulators

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Whereas spaces with constant positive curvature (spheres) are naturally realized in the world around us, and crucially enter, e.g., the description of atomic orbitals (and by extension also of the periodic table of elements and chemistry), the situation is markedly different for (hyperbolic) spaces with constant negative curvature. The underlying reason is captured in Hilbert’s theorem: the hyperbolic plane simply cannot be embedded in the three-dimensional Euclidean space. Nevertheless, hyperbolic lattices can be potentially emulated in metamaterials, as was demonstrated in a 2019 experiment by Kollár et al. using coupled coplanar waveguide resonators [1]. This motivated the search for the hyperbolic generalizations of the concepts of the Bloch band theory, which has to a great extent been recently achieved by Maciejko and Rayan [2]. The most salient feature of their hyperbolic band theory is the unusually large dimension of the momentum space: the spectrum of particles on a two-dimensional hyperbolic lattice necessitates a characterization with an at least four-dimensional Brillouin zone. In this seminar I will reflect on our two very recent works motivated by these rapid developments. First, in an experimental work [4], we use electric circuits to realize a sample of the hyperbolic {3,7}-tessellation (i.e., the regular tessellation with seven equilateral triangles meeting at each vertex). We find that the low-energy modes in the spectrum are effectively described by the continuum Laplace-Beltrami operator on a disk, motivating us to call the setup a “hyperbolic drum”. In particular, we reveal fingerprints of the negative curvature in both static (reordering of the Laplacian spectrum) and dynamical (signal propagation along curved geodesics) experiments. Second, in a theoretical work [5], we utilize the tools of the hyperbolic band theory to propose concrete models of hyperbolic Chern and Kane-Mele topological insulators. These paradigm models are then used to investigate the bulk-boundary correspondence of topological invariants computed in the momentum and in the coordinate space. We expect our works to pave the way towards discovering novel models of topological hyperbolic matter.

[1] A. J. Kollár, M. Fitzpark, and A. A. Houck, Hyperbolic lattices in circuit quantum electrodynamics, Nature 571, 45—50 (2019) [2] J. Maciejko and S. Rayan, Hyperbolic band theory, Sci. Adv. 7(36), eabe9170 (2021); J. Maciejko and S. Rayan, Automorphic Bloch theorems for finite hyperbolic lattices, arXiv:2108.09314 (2021) [3] I. Boettcher, A. V. Gorshkov, A. J. Kollár, J. Maciejko, S. Rayan, and R. Thomale, Crystallography of Hyperbolic Lattices, arXiv:2105.01087 (2021) [3] P. M. Lenggenhager, A. Stegmaier, L. K. Upreti, T. Neupert, R. Thomale, T. Bzdušek, et al., Electric-circuit realization of a hyperbolic drum, arXiv:2109.01148 (2021) [4] D. M. Urwyler, P. M. Lenggenhager, T. Neupert, and T. Bzdušek (in preparation, 2022)

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