Correlated phases in rhombohedral multilayer graphene

TL;DR

This work analyzes correlation-driven phases in rhombohedral $N$-layer graphene under two-valley Coulomb interactions using a low-energy $k\cdot p$ model. It derives analytic Lindhard susceptibilities for intra- and intervalley channels and compares the random phase approximation (RPA) with the parquet approximation (PA) to map instabilities in particle-hole and particle-particle sectors, establishing a scaling law for the critical temperature $T_c$ with layer number $N$ and identifying symmetry-driven phase selection. The PA uncovers a richer phase structure, including potential pair-density-wave (PDW) order in the PP channel, while RPA restricts to Stoner and intervalley-coherent (IVC) phases; the results show an upper bound for $T_c$ as $N\to\infty$ and a non-monotonic dependence of $T_c$ on chemical potential $\mu$. Symmetry plays a crucial role: $SU(4)$-symmetric interactions favor intervalley Stoner order in the density channel, whereas $SU(2)\times SU(2)$-symmetric interactions admit a broader set of instabilities and a layer-number-dependent crossover in the dominant PH channel.

Abstract

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.

Figures (17)

• Figure 1: (a) The unit cell of rhombohedral $N$-layer graphene that corresponds to Hamiltonian density given by Eq. (\ref{['eq:Nlayer-graphene-model']}) with the nearest-neighboring intralayer $\gamma_0$ and interlayer $\gamma_1$ orbital hoppings. (b) The dispersion relation of the effective low-energy model given by Eq. (\ref{['eq:dispresion']}) for four values of $N$. Here, $a$ is the lattice constant, the energy $\varepsilon$ is in the units of $g_N/a^N$.
• Figure 2: Diagrammatic representation of Eq. (\ref{['eq:def-gen-susceptibility-FT-v1']}).
• Figure 3: (a) Particle-hole, (b) transverse particle-hole and (c) particle-particle channels of the two-point correlation function $G_{l_1 l_2 l_3 l_4}$.
• Figure 4: Critical temperatures $T_c$ for the IVC and Stoner phases as a function of $N$ for $SU(4)$-symmetric interaction at $\mu=0$. The results for the Stoner phases given by Eqs. (\ref{['eq:rpa-t-st-m']}) and (\ref{['eq:rpa-t-st-d']}) are identical in both channels because of $V_0=V_1$.
• Figure 5: Phase diagram $(\mu, T)$ for the IVC phase in the $SU(4)$-symmetric case for $N=3$ in the random phase approximation.