Existence of a thermodynamic spin-glass phase in the zero-concentration limit of anisotropic dipolar systems

The nature of ordering in dilute dipolar interacting systems dates back to the work of Debye and is one of the most basic, oldest and as-of-yet unsettled problems in magnetism. While spin-glass order is readily observed in several RKKY-interacting systems, dipolar spin glasses are the subject of controversy and ongoing scrutiny, e.g., in LiHo_xY_1-xF₄, a rare-earth randomly diluted uniaxial (Ising) dipolar system. In particular, it is unclear if the spin-glass phase in these paradigmatic materials persists in the limit of zero concentration or not. We study an effective model of LiHo_xY_1-xF₄ using large-scale Monte Carlo simulations that combine parallel tempering with a special cluster algorithm tailored to overcome the numerical difficulties that occur at extreme dilutions. We find a paramagnetic to spin-glass phase transition for all Ho⁺ ion concentrations down to the smallest concentration numerically accessible, 0.1%, and including Ho⁺ ion concentrations that coincide with those studied experimentally up to 16.7%. Our results suggest that randomly diluted dipolar Ising systems have a spin-glass phase in the limit of vanishing dipole concentration, with a critical temperature vanishing linearly with concentration. The agreement of our results with mean-field theory testifies to the irrelevance of fluctuations in interactions strengths, albeit being strong at small concentrations, to the nature of the low-temperature phase and the functional form of the critical temperature of dilute anisotropic dipolar systems. Deviations from linearity in experimental results at the lowest concentrations are discussed.