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CATEGORIES:Theory of Condensed Matter
SUMMARY:From hyperbolic drum towards hyperbolic topologica
l insulators - Tomáš Bzdušek (Paul Scherrer Instit
ute)
DTSTART;TZID=Europe/London:20211117T141500
DTEND;TZID=Europe/London:20211117T151500
UID:TALK165895AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/165895
DESCRIPTION:Whereas spaces with constant positive curvature (s
pheres) 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 si
tuation is markedly different for (hyperbolic) spa
ces with constant negative curvature. The underlyi
ng reason is captured in Hilbert’s theorem: the hy
perbolic plane simply cannot be embedded in the th
ree-dimensional Euclidean space. Nevertheless\, hy
perbolic lattices can be potentially emulated in m
etamaterials\, as was demonstrated in a 2019 exper
iment by Kollár et al. using coupled coplanar wave
guide resonators [1]. This motivated the search fo
r the hyperbolic generalizations of the concepts o
f the Bloch band theory\, which has to a great ext
ent been recently achieved by Maciejko and Rayan [
2]. The most salient feature of their hyperbolic b
and theory is the unusually large dimension of the
momentum space: the spectrum of particles on a tw
o-dimensional hyperbolic lattice necessitates a ch
aracterization with an at least four-dimensional B
rillouin zone. \nIn this seminar I will reflect on
our two very recent works motivated by these rapi
d developments. First\, in an experimental work [4
]\, we use electric circuits to realize a sample o
f the hyperbolic {3\,7}-tessellation (i.e.\, the r
egular tessellation with seven equilateral triangl
es meeting at each vertex). We find that the low-e
nergy modes in the spectrum are effectively descri
bed by the continuum Laplace-Beltrami operator on
a disk\, motivating us to call the setup a “hyperb
olic drum”. In particular\, we reveal fingerprints
of the negative curvature in both static (reorder
ing of the Laplacian spectrum) and dynamical (sign
al 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 corres
pondence of topological invariants computed in the
momentum and in the coordinate space. We expect o
ur works to pave the way towards discovering novel
models of topological hyperbolic matter. \n \n[1]
A. J. Kollár\, M. Fitzpark\, and A. A. Houck\, Hy
perbolic lattices in circuit quantum electrodynami
cs\, Nature 571\, 45—50 (2019) \n[2] J. Maciejko a
nd 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) \n[3] I. Boett
cher\, A. V. Gorshkov\, A. J. Kollár\, J. Maciejko
\, S. Rayan\, and R. Thomale\, Crystallography of
Hyperbolic Lattices\, 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 (i
n preparation\, 2022)
LOCATION:TCM Seminar room\, 530 Mott building
CONTACT:Jan Behrends
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