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We present a comprehensive experimental and theoretical study of thepyrochlore Tb$_2$Ge$_2$O$_7$, an exemplary realization of a material whoseproperties are dominated by competition between magnetic dipolar and electricquadrupolar correlations. The dipolar and quadrupolar correlations evolve overthree distinct regimes that we characterize via heat capacity, elastic andinelastic neutron scattering. In the first regime, above $T^*=1.1$ K,significant quadrupolar correlations lead to an intense inelastic mode thatcannot be accounted for within a scenario with solely magnetic dipole-dipolecorrelations. The onset of extended dipole correlations occurs in theintermediate regime, between $T^*=1.1$ K and $T_c = 0.25$ K, with the formationof a collective paramagnetic state characterized by extended ferromagneticcanted spin ice domains. Here, long-range order is impeded not only by theusual frustration operating in classical spin ice systems, but also by acompetition between dipolar and quadrupolar correlations. Finally, in thelowest temperature regime, below $T_c=0.25$ K, there is an abrupt andsignificant increase in the dipole ordered moment. The majority of the orderedmoment remains tied up in the ferromagnetic spin ice-like state, but anadditional $\mathbf{k}=(0,0,1)$ antiferromagnetic order parameter alsodevelops. Simultaneously, the spectral weight of the inelastic mode, which is aproxy for the quadrupolar correlations, is observed to drop, indicating thatdipole order ultimately wins out. Tb$_2$Ge$_2$O$_7$ is therefore a remarkableplatform to study intertwined dipolar and quadrupolar correlations in amagnetically frustrated system and provides important insights into the physicsof the whole family of terbium pyrochlores.

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