Fermion-fermion interaction driven phase transitions in rhombohedral trilayer graphene

The effects of short-range fermion-fermion interactions on the low-energy properties of rhombohedral trilayer graphene are comprehensively investigated using the momentum-shell renormalization group method. We take into account all one-loop corrections and establish the energy-dependent coupled evolutions of independent fermionic couplings that carry the physical information stemming from the interplay of various fermion-fermion interactions. With detailed numerical analysis, we observe that the ferocious competition among all fermion-fermion interactions drives fermionic couplings to four distinct fixed points, dubbed ${\text{FP}}₁, {\text{FP}}₂, {\text{FP}}₃$, and ${\text{FP}}₄$, in the interaction-parameter space. These fixed points primarily dictate the fate of the system in the low-energy regime and are always associated with some instabilities characterized by specific symmetry breakings, leading to certain phase transitions. To determine the favorable states arising from the potential phase transitions, we introduce a number of fermion-bilinear source terms to characterize the underlying candidate states. By comparing their related susceptibilities, we find that the dominant states correspond to spin-singlet superconductivity, spin-triplet pair-density-waves, and spin-triplet superconductivity for fixed points ${\text{FP}}_1,3, {\text{FP}}₂$, and ${\text{FP}}₄$, respectively. These results provide valuable insights into the low-energy properties of rhombohedral trilayer graphene and analogous materials.