Exotic magnetic phase diagram and extremely robust antiferromagnetism in Ce$_2$RhIn$_8$

The antiferromagnetic heavy-fermion compound Ce$_2$RhIn$_8$ belongs to the same family and bears similarities with the well-studied prototypical material CeRhIn$_5$, which demonstrates a unique behavior under applied magnetic field. Here, we report specific-heat measurements on a high-quality single crystal of Ce$_2$RhIn$_8$ in magnetic fields up to 35 T applied along both principal crystallographic directions. When the magnetic field is applied along the $a$ axis of the tetragonal crystal structure, two additional field-induced antiferromagnetic phases are observed, in agreement with previous reports. One of them is confined in a small area of the field-temperature phase diagram. The other one, which develops above $\sim$2.5 T, is very robust against the field. Its transition temperature increases with field, reaches a maximum at $\sim$12 T, and then starts to decrease. However, it tends to saturate towards the highest field of our measurements. For a field applied along the $c$ axis, the N\'{e}el temperature, $T_N$, initially decreases with field, as expected for a typical antiferromagnet. Surprisingly, an additional phase emerges above $\sim$10 T. It is most likely to be of the same origin as its counterpart observed above 2.5 T for the other field orientation. This phase is also unusually robust: Its transition temperature increases all the way up to 35 T, where it exceeds the zero field $T_N =$~2.85 K. Finally, for both field directions, the phase diagrams contain rarely observed tricritical points of second-order phase transitions.