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SUMMARY:From hyperbolic drum towards hyperbolic topological insulators - T
 omáš Bzdušek (Paul Scherrer Institute)
DTSTART:20211117T141500Z
DTEND:20211117T151500Z
UID:TALK165895@talks.cam.ac.uk
CONTACT:Jan Behrends
DESCRIPTION:Whereas spaces with constant positive curvature (spheres) are 
 naturally realized in the world around us\, and crucially enter\, e.g.\, t
 he description of atomic orbitals (and by extension also of the periodic t
 able of elements and chemistry)\, the situation is markedly different for 
 (hyperbolic) spaces with constant negative curvature. The underlying reaso
 n is captured in Hilbert’s theorem: the hyperbolic plane simply cannot b
 e embedded in the three-dimensional Euclidean space. Nevertheless\, hyperb
 olic lattices can be potentially emulated in metamaterials\, as was demons
 trated in a 2019 experiment by Kollár et al. using coupled coplanar waveg
 uide resonators [1]. This motivated the search for the hyperbolic generali
 zations of the concepts of the Bloch band theory\, which has to a great ex
 tent been recently achieved by Maciejko and Rayan [2]. The most salient fe
 ature of their hyperbolic band theory is the unusually large dimension of 
 the momentum space: the spectrum of particles on a two-dimensional hyperbo
 lic lattice necessitates a characterization with an at least four-dimensio
 nal Brillouin zone. \nIn this seminar I will reflect on our two very recen
 t 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 equilatera
 l triangles meeting at each vertex). We find that the low-energy modes in 
 the spectrum are effectively described by the continuum Laplace-Beltrami o
 perator on a disk\, motivating us to call the setup a “hyperbolic drum
 ”. In particular\, we reveal fingerprints of the negative curvature in b
 oth static (reordering of the Laplacian spectrum) and dynamical (signal pr
 opagation along curved geodesics) experiments. Second\, in a theoretical w
 ork [5]\, we utilize the tools of the hyperbolic band theory to propose co
 ncrete models of hyperbolic Chern and Kane-Mele topological insulators. Th
 ese paradigm models are then used to investigate the bulk-boundary corresp
 ondence of topological invariants computed in the momentum and in the coor
 dinate space. We expect our works to pave the way towards discovering nove
 l models of topological hyperbolic matter. \n \n[1] A. J. Kollár\, M. Fit
 zpark\, and A. A. Houck\, Hyperbolic lattices in circuit quantum electrody
 namics\, Nature 571\, 45—50 (2019) \n[2] J. Maciejko and S. Rayan\, Hype
 rbolic band theory\, Sci. Adv. 7(36)\, eabe9170 (2021)\; J. Maciejko and S
 . Rayan\, Automorphic Bloch theorems for finite hyperbolic lattices\, arXi
 v:2108.09314 (2021) \n[3] I. Boettcher\, A. V. Gorshkov\, A. J. Kollár\, 
 J. Maciejko\, S. Rayan\, and R. Thomale\, Crystallography of Hyperbolic La
 ttices\, arXiv:2105.01087 (2021) \n[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) \n[4] D
 . M. Urwyler\, P. M. Lenggenhager\, T. Neupert\, and T. Bzdušek (in prepa
 ration\, 2022) 
LOCATION:TCM Seminar room\, 530 Mott building
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