Various electronic crystal phases in rhombohedral graphene multilayers

TL;DR

Rhombhohedral graphene multilayers exhibit rich correlated electronic order when Coulomb interactions compete with nearly flat topological bands. The authors use self-consistent Hartree–Fock calculations built on an ab initio Slater–Koster tight-binding model to map the phase diagram as a function of carrier density $n$ and displacement field $U$, discovering a cascade of isospin transitions and an array of electronic crystal phases, including Wigner crystals and anomalous Hall crystals with nonzero Chern numbers. These topological crystal phases are nearly degenerate with Fermi-liquid states, and external pressure can tune the competition, shifting the WC–AHC boundary while preserving band geometry. The work connects thermodynamic signatures, especially inverse compressibility $K^{-1} = rac{ ext{d} abla ext{d} abla}{ ext{d}n}$, to experimental observations and highlights the role of hBN alignment, disorder, and correlations beyond mean-field as avenues for future study.

Abstract

We systematically investigate the emergence of electronic crystal phases in rhombohedral multilayer graphene using comprehensive self-consistent Hartree Fock calculations combined with \textit{ab initio} tight binding model. As the carrier density increases, we uncover an isospin cascade sequence of phase transitions that gives rise to a rich variety of ordered states, including electronic crystal phases with non-zero Chern numbers. We further show the nearly degeneracy of these topological electronic crystals hosting extended quantum anomalous Hall effect (EQAH) in the mean field regime and characterize pressure driven phase transitions. Finally, we discuss the thermodynamic signatures, particularly the behavior of the inverse compressibility, in light of recent experimental observations.

Figures (8)

• Figure 1: Single particle band structure (a) of rhombohedral penta layer graphene (R5G) near the $K$ point for a displacement field of $U = 0.035~\text{eV}$, and the corresponding density of states (b). The van Hove singularity in the first conduction band originates from the flat band bottom. The shaded area lables the region when the electron filling is $0.5\times10^{12}$ cm$^{-2}$ and $1.2\times10^{12}$ cm$^{-2}$, respectively.
• Figure 2: Cascade of isospin phase transitions in R5G. QM denotes the quarter metal phase, HM the half metal phase, TQM the three quarter metal phase, and FM the full metal phase. The Hartree Fock results indicate a first order phase transition as the carrier density $n$ increases.
• Figure 3: Hartree–Fock mean-field phase diagrams of rhombohedral graphene multilayers as functions of electron doping density $n$ and displacement field $U$: (a) tetralayer (R4G), (b) pentalayer (R5G), and (c) hexalayer (R6G). QM, HM, TQM, and FM denote the quarter-metal, half-metal, three-quarter-metal, and full-metal phases, respectively. PIP indicates a partially isospin-polarized state. Across the isospin-driven phase transitions, symmetry-broken electronic crystal phases—Wigner crystal (WC) and anomalous Hall crystal (AHC)—can also emerge. The annotated regions additionally label the number of isolated flat bands participating at the Fermi level: 1 in QM, 2 in HM, 3 in TQM, and 4 in FM.
• Figure 4: Competition among different electronic crystal phases in R5G as the electron density $n$ varies, with the displacement field fixed at $U = 0.035~\text{eV}$. (a) The energy differences between the square-lattice states ($C=0/1$) and the hexagonal-lattice states ($C=0/1$) are relatively small. In certain density ranges, the square-lattice phases compete closely with the hexagonal ones. Density profile $(\rho(r)-\left<\rho(r)\right>)\Omega_0$ when $n=1\times 10^{12}$ cm$^{-2}$ for (b) hexagonal AHC (c) square AHC. $\Omega_0$ is the superlattice size.
• Figure 5: Pressure induced phase transition in rhombohedral pentalayer graphene (R5G) (a) The same phase diagram as Fig. \ref{['fig3']} when $P=0.5$ GPa (b) when $P=1.0$ GPa. The same color legend has been adopted here. (c) Hartree Fock phase diagram as a function of pressure $P$ and doping density $n$ at fixed displacement field $U=0.035~\mathrm{eV}$. AHM represents the anomalous Hall metal (AHM) region. (d) Trace condition $T$ corresponding to the anomalous Hall crystal (AHC) phases identified in (c). The dark colored region marks the optimal trace condition, located near the WC--AHC phase transition boundary.