Thermodynamic properties of quadrupolar states in the frustrated pyrochlore magnet Tb$_2$Ti$_2$O$_7$

The low-temperature thermodynamic properties of the frustrated pyrochlore Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ have been studied using the single crystal of $x=0.005$ sitting in a long range ordered phase in the $x$-$T$ phase diagram. We observed that the specific heat exhibits a minimum around 2 K and slightly increases on cooling, similar to a Schottky-like anomaly for canonical spin ices. A clear specific-heat peak observed at $T_{\rm c} = 0.53$ K is ascribable to the phase transition to a quadrupolar state, which contributes to a relatively large change in entropy, $S \simeq 2.7$ J K$^{-1}$mol$^{-1}$. However, it is still smaller than $R\ln2$ for the ground state doublet of the Tb ions. The entropy release persists to higher temperatures, suggesting strong fluctuations associated with spin ice correlations above $T_{\rm c}$. We discuss the field dependence of the entropy change for $H||[111]$ and $H||[001]$.