Quadrupole order in the frustrated pyrochlore magnet Tb$_2$Ti$_2$O$_7$

We have studied the hidden long-range order (LRO) of the frustrated pyrochlore magnet Tb$_2$Ti$_2$O$_7$ by means of specific-heat experiments and Monte-Carlo (MC) simulations, which has been discussed as the LRO of quadrupole moments inherent to the non-Kramers ion of Tb$^{3+}$. We have found that the sharp specific-heat peak is collapsed into a broad hump by magnetic fields above 0.3 T for $H//[001]$. This result, qualitatively reproduced by MC simulations, suggests that a field-induced magnetic state overcomes the quadrupolar LRO state, as a similar case of a classical spin ice. The present results support the interpretation that Tb$_{2+x}$Ti$_{2-x}$O$_{7+y}$ is a unique material in the boundary between the quadrupolar ($x \geq x_c = -0.0025$) and spin-liquid ($x \leq x_c$) states, where the magnetic field along the [001] axis is a tuning parameter which induces the magnetic ordered state.