Eclipsing Kitaev: off-diagonal exchange governs the correlated high-field phases of $β$-Li$_2$IrO$_3$
We report a high-field thermodynamic study of the hyperhoneycomb Kitaev material $β$-Li$_2$IrO$_3$, using magnetotropic susceptibility to resolve its low-temperature field-angle phase diagram across the principal crystallographic planes in magnetic fields up to $60$ T. Rather than evolving directly from the low-field incommensurate state into a polarized regime, the system exhibits a strongly direction-dependent sequence of correlated phases. Most notably, for fields in the $ac$-plane, we identify an additional high-field phase that is absent in the other principal planes and exists only within a restricted region of field-angle space. This phase structure is naturally explained by the competition between magnetic field and bond-directional exchange interactions. Using a symmetry-based description supported by microscopic calculations within the $J$-$K$-$Γ$ model, we show that off-diagonal $Γ$ exchange couples the ferromagnetic and staggered magnetic orders and thereby stabilizes the observed correlated high-field phases. The measured angular dependence of the critical fields is quantitatively captured by this theory, identifying $Γ$ exchange as the key interaction controlling the high-field response. These results clarify why the promise of a field-induced spin liquid -- the notion that suppressing magnetic order might reveal the underlying Kitaev physics -- remains unfulfilled in candidate materials: even when the Kitaev interaction is large, off-diagonal exchange stabilizes symmetry-constrained correlated phases that instead preempt the polarized state.