Interplay between magnetic exchange, multipolar interactions and virtual crystal field fluctuations in non-Kramers pyrochlore magnets

It has been shown that multipolar degrees of freedom can play a vital role in causing exotic phases in pyrochlore magnets, such as octupoles in Kramers Nd³⁺ or Ce³⁺ ions and quadrupoles in non-Kramers Pr³⁺ and Tb³⁺ ions. Terbium-based pyrochlores are peculiar for hosting virtual crystal field excitation (VCFE) due to the small energy separation between the two low-lying crystal electric field doublets. Here we consider a two-doublets system with magnetic bilinear exchange and electric quadrupole-quadrupole interaction. Via a mean field approach, we show that the proposed model exhibits complex dipolar and quadrupolar phases. Different dipolar order parameters coexist due to VCFE, which also leads to a “parasitic” ferroquadrupolar order accompanying the dominant antiferroquadrupolar order. We find for a set of model parameters locating the system near the dipolar/quadrupolar phase boundary that, upon cooling, such system may undergo a two-step thermal transition into the ultimate low-temperature dipolar phase with an intermediated quadrupolar ordered state. We also propose a range of acceptable parameters for the Tb₂Ge₂O₇ pyrochlore that allows us to reproduce some of the main inelastic neutron scattering features.