Disconnected glass-glass transitions and diffusion anomalies in a model with two repulsive length scales

Building on mode-coupling-theory calculations, we report a novel scenario for multiple glass transitions in a purely repulsive spherical potential: the square shoulder. The liquid-glass transition lines exhibit both melting by cooling and melting by compression as well as associated diffusion anomalies, similar to the ones observed in water. Differently from all previously investigated models, we find for small shoulder widths a glass-glass line that is disconnected from the liquid phase. Upon increasing the shoulder width such a glass-glass line merges with the liquid-glass transition lines, featuring two distinct end point singularities that give rise to logarithmic decays in the dynamics. We analytically explain these findings by considering the interplay of different repulsive length scales.