Intertwined Magnetic Dipolar and Electric Quadrupolar Correlations in the Pyrochlore Tb$_2$Ge$_2$O$_7$.

We present a comprehensive experimental and theoretical study of the pyrochlore Tb$_2$Ge$_2$O$_7$, an exemplary realization of a material whose properties are dominated by competition between magnetic dipolar and electric quadrupolar correlations. Tb$_2$Ge$_2$O$_7$ possesses a low-lying crystal field level that disrupts the clean separation of energy scales commonly found in other rare earth pyrochlores, imbuing this material with complex phase behavior. The dipolar and quadrupolar correlations evolve over three distinct regimes that we characterize via heat capacity, elastic and inelastic neutron scattering. In the first regime, above $T^*=1.1$ K, significant quadrupolar correlations lead to an intense inelastic mode that cannot be accounted for within a scenario with solely dipole correlations. The onset of extended dipole correlations occurs in the intermediate regime, between $T^*=1.1$ K and $T_c = 0.25$ K, which constitutes a collective paramagnetic regime characterized by extended short-range ordered spin ice domains. Here, long-range order is impeded not only by the usual frustration, but also by a competition between dipolar and quadrupolar correlations. Finally, in the lowest temperature regime, below $T_c=0.25$ K, there is an abrupt and significant increase in the dipole ordered moment. The majority of the moment remains tied up in the ferromagnetic spin ice-like state, but an additional $\mathbf{k}=(0,0,1)$ antiferromagnetic order parameter also develops. Simultaneously, the spectral weight of the inelastic mode, which is a proxy for the quadrupolar correlations, is observed to drop, indicating that dipole order ultimately wins out. Tb$_2$Ge$_2$O$_7$ is therefore a remarkable platform to study intertwined dipolar and quadrupolar correlations in a magnetically frustrated system and provides important insights into the physics of the whole family of terbium pyrochlores.