Parameter dependent non-Hermitian quantum systems typically not only possess eigenvalue degeneracies, but also degeneracies of the corresponding eigenfunctions at exceptional points.While the effect of two coalescing eigenfunctions on cyclic parameter variation is well investigated, little attention has hitherto been paid to the effect of more than two coalescing eigenfunctions. Here a characterisation of behaviours of symmetric Hamiltonians with three coalescing eigenfunctions is presented, using perturbation theory for non-Hermitian operators. Two main types of parameter perturbations need to be distinguished, which lead to characteristic eigenvalue and eigenvector patterns under cyclic variation. A physical system is introduced for which both behaviours might be experimentally accessible.

Snowflake growth provides a fascinating example of spontaneous pattern formation in nature. Attempts to understand this phenomenon have led to important insights in non-equilibrium dynamics observed in various active scientific fields, ranging from pattern formation in physical and chemical systems, to self-assembly problems in biology. Yet, very few models currently succeed in reproducing the diversity of snowflake forms in three dimensions, and the link between model parameters and thermodynamic quantities is not established. Here, we report a modified phase field model that describes the subtlety of the ice vapour phase transition, through anisotropic water molecules attachment and condensation, surface diffusion, and strong anisotropic surface tension, that guarantee the anisotropy, faceting and dendritic growth of snowflakes. We demonstrate that this model reproduces the growth dynamics of the most challenging morphologies of snowflakes from the Nakaya diagram. We find that the growth dynamics of snow crystals matches the selection theory, consistently with previous experimental observations. npj Computational Materials (2017) 3:15 ;

Microstructure patterning using the ion beam mixing process results from the competition between thermal diffusion and ballistic disordering induced by impinging ions. Although microstructure patterning under irradiation is now qualitatively understood, so far, no study could quantitatively estimate irradiation conditions leading to patterning. In this work, a new multiscale approach based on phase field was developed to simulate the microstructure evolution, and the occurrence of patterning due to ion irradiation in a silver-copper alloy, from atomic to microstructural scale. For that purpose, an efficient numerical scheme was developed to simulate the microstructure dynamics, within the framework of phase field. Equilibrium parameters of AgCu were computed using a mixed Monte Carlo-molecular dynamics approach. Ballistic effects induced by krypton ion irradiation, and point defect recreation leading to irradiation enhanced diffusion, were estimated using the binary collision approximation framework. As a result, we predicted the range of temperatures and irradiation fluxes leading to the formation of patterned microstructures in the AgCu alloy under krypton ion irradiation. Results were summarized in a pseudo phase diagram in the temperature-flux plane. Our model was in good agreement with a previous diffraction experimental study.