Density-Jump Transitions in the Debye-H\"uckel Theory of Spin Ice and Electrolytes

Debye-H\"{u}ckel theory, originally developed to describe dilute electrolyte solutions, has proved particularly successful as a description of magnetic monopoles in spin ice systems such as Dy$_{2}$Ti$_{2}$O$_{7}$. For this model, Ryzhkin et al. [JETP Lett. 95, 302-306 (2012)] predicted a phase transition in which the monopole density abruptly changes by several orders of magnitude but to date this transition has not been observed experimentally. Here we confirm that this transition is a robust prediction of Debye-H\"{u}ckel theory, that does not rely on approximations made in the previous work. However, we also find that the transition occurs in a regime where the theory breaks downs as a description of a Coulomb fluid and may be plausibly interpreted as an indicator of monopole crystallisation. By extending Ryzhkin's model, we associate the density jump of Debye-H\"uckel theory with the monopole crystallisation observed in staggered-potential models of `magnetic moment fragmentation', as well as with crystallisation in conserved monopole-density models. The possibility of observing a true density-jump transition in real spin ice and electrolyte systems is discussed.