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. 2018 Jan 9;8:186. doi: 10.1038/s41598-017-17480-1

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© The Author(s) 2017

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Illustration of the emergent ice rule in magnonic vortex crystals. (a) Schematic representation of the investigated vortex system. The black and white dots represent the polarization of the vortices. The blue arrows illustrate the analogy of two vortices with alternating polarizations to a nanoisland in artificial spin ice systems. (b) Possible configurations of six vortices in a junction depicted by blue arrows. Configurations that obey the ice rule have two arrows pointing towards the junction and one pointing out of it or vice versa. If all three arrows point in or out of the junction the ice rule is broken. (c) Possible polarization configurations of the vortices of a triple, where white and black dots represent a polarization of p = +1 and p = −1, respectively. The exciting magnetic field breaks the symmetry. It is distinguished between four different triple types. For triple type I-III the ice rule is obeyed, for triple type IV the ice rule is broken.