Aharonov-Bohm cages, flat bands, and gap labeling in hyperbolic tilings
- URL: http://arxiv.org/abs/2206.04543v3
- Date: Wed, 12 Oct 2022 06:59:19 GMT
- Title: Aharonov-Bohm cages, flat bands, and gap labeling in hyperbolic tilings
- Authors: R. Mosseri, R. Vogeler, J. Vidal
- Abstract summary: We show that Aharonov-Bohm cages are also found in an infinite set of hyperbolic dice tilings defined on a negatively curved hyperbolic plane.
We also consider the energy spectrum of hyperbolic kagome tilings, which displays interesting features.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Aharonov-Bohm caging is a localization mechanism stemming from the
competition between the geometry and the magnetic field. Originally described
for a tight-binding model in the dice lattice, this destructive interference
phenomenon prevents any wavepacket spreading away from a strictly confined
region. Accordingly, for the peculiar values of the field responsible for this
effect, the energy spectrum consists of a discrete set of highly degenerate
flat bands. In the present work, we show that Aharonov-Bohm cages are also
found in an infinite set of hyperbolic dice tilings defined on a negatively
curved hyperbolic plane. We detail the construction of these tilings and
compute their Hofstadter butterflies by considering periodic boundary
conditions on high-genus surfaces. As recently observed for some regular
hyperbolic tilings, these butterflies do not manifest the self-similar
structure of their Euclidean counterparts but still contain some gaps. We also
consider the energy spectrum of hyperbolic kagome tilings (which are the dual
of hyperbolic dice tilings), which displays interesting features, such as
highly degenerate states arising for some particular values of the magnetic
field. For these two families of hyperbolic tilings, we compute the Chern
number in the main gaps of the Hofstadter butterfly and propose a gap labeling
inspired by the Euclidean case. Finally, we also study the triangular Husimi
cactus, which is a limiting case in the family of hyperbolic kagome tilings,
and we derive an exact expression for its spectrum versus magnetic flux.
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