Zero-Field Long Range Order at $T \sim 40$ mK in the Proximate Quantum Spin Ice Ce$_2$Sn$_2$O$_7$ and Phase Diagram for Magnetic Fields Along [1,1,0]
The Ce$^{3+}$ pseudospin-1/2 degrees of freedom in the pyrochlore magnets Ce$_2$X$_2$O$_7$, with $X$ = Zr, Hf, or Sn, possess dipole-octupole character. The XYZ nearest-neighbor model Hamiltonians which have successfully described their properties make them attractive candidates for quantum spin ice ground states. We report new heat capacity measurements to very low temperatures on a high-quality single crystal of Ce$_2$Sn$_2$O$_7$ grown by hydrothermal techniques. Our zero-field measurements uncover a clear first order transition to long-ranged order at $T \sim 0.04$ K, well below the downturn in the broader Schottky-like anomaly that is a common feature in the zero-field heat capacity for cerium pyrochlores. This observation settles the debate as to whether Ce$_2$Sn$_2$O$_7$ possesses a QSI ground state in zero field - it does not. However, we also find compelling evidence suggesting that spin ice physics is nearby, and remains relevant to Ce$_2$Sn$_2$O$_7$. Application of a magnetic field along the $[1,1,0]$ direction leads to an evolution of this peak to a weak anomaly at higher temperature more characteristic of a continuous, mean field transition. At higher fields along $[1,1,0]$, the Schottky-like anomaly bifurcates similar to expectations for independent polarized $α$ and orthogonal $β$ chains in classical spin ice. These new experimental results demonstrate richness to the phase diagram for Ce-based pyrochlores.