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Mathematical Physics
Title: Origin of Symmetry Breaking in the Grasshopper Model
(Submitted on 8 Nov 2023 (v1), last revised 22 Mar 2024 (this version, v2))
Abstract: The planar grasshopper problem, originally introduced in (Goulko & Kent 2017 Proc. R. Soc. A 473, 20170494), is a striking example of a model with long-range isotropic interactions whose ground states break rotational symmetry. In this work we analyze and explain the nature of this symmetry breaking with emphasis on the importance of dimensionality. Interestingly, rotational symmetry is recovered in three dimensions for small jumps, which correspond to the non-isotropic cogwheel regime of the two-dimensional problem. We discuss simplified models that reproduce the symmetry properties of the original system in N dimensions. For the full grasshopper model in two dimensions we obtain quantitative predictions for optimal perturbations of the disk. Our analytical results are confirmed by numerical simulations.
Submission history
From: Olga Goulko [view email][v1] Wed, 8 Nov 2023 21:17:28 GMT (21364kb,AD)
[v2] Fri, 22 Mar 2024 01:00:05 GMT (21376kb,AD)
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