Correlated phases in rhombohedral N-layer graphene

We investigate the emergence of correlated electron phases in rhombohedral $N$-layer graphene due to two-valley Coulomb interactions within a low-energy $k \cdot p$ framework. Analytical expressions for Lindhard susceptibilities in intra- and intervalley channels are derived, and the critical temperatures for phase transitions are estimated using both the random phase approximation (RPA) and the parquet approximation (PA). Within RPA, only Stoner and intervalley coherent (IVC) phases are supported, while the PA reveals a richer phase structure including particle-particle (PP) channel instabilities. We establish a general scaling law for the critical temperature with respect to layer number $N$, highlighting an upper bound as $N \rightarrow \infty$, and demonstrate a non-monotonic decrease of the critical temperature with increasing chemical potential. The PA uncovers the role of interaction symmetry: $SU(4)$-symmetric interactions favor intervalley Stoner order in the density channel, whereas $SU(2) \times SU(2)$-symmetric interactions permit a broader set of phases. A crossover in the dominant instability occurs in the particle-hole channel at a critical layer number, suggesting the emergence of magnetic or IVC phases in thicker systems. We also identify conditions under which pair-density wave (PDW) order could form in the PP channel, though its physical realization may be constrained.