Tetrahedral Core in a Sea of Competing Magnetic Phases in Graphene

Abstract

We reveal the emergence of a robust tetrahedral magnetic ground state in monolayer graphene doped to the van Hove singularity (vHS). This noncoplanar, gapped spin configuration -- featuring four orthogonal moments -- has been previously identified as a candidate instability. Here, not only do we confirm its stability across all finite interactions using fully self-consistent, real-space-resolved calculations, but we also go beyond earlier work by charting the full surrounding phase diagram. In doing so, we unravel a cascade of symmetry-broken magnetic states -- pseudo-tetrahedral, planar, collinear, and modulated textures -- which we classify using spin structure factors and vector order parameters. These results stem from unrestricted Hartree-Fock simulations on large supercells with dense k-point sampling, enabling us to resolve interaction-driven magnetic and charge inhomogeneities. Our findings connect directly with recent ARPES and doping experiments near the vHS in graphene, and establish the tetrahedral state as the central correlated instability in this regime, offering predictive insight into emergent magnetism in correlated Dirac materials.

Figures (6)

• Figure 1: Magnetic ground-state phase diagram of graphene near quarter doping. (a) Schematic mean-field $(N_e, U)$ phase diagram based on computations at $k_B T = 10^{-7}t$, using a $6\times6$ supercell and $48\times48$$k$-point sampling. (b) For doping levels below the van Hove singularity (vHS), $N_e \le 0.75$, indicated by blue regions, all magnetic phases exhibit well-defined spin orders compatible with a $2\times2$ graphene supercell (see also End Matter Fig. \ref{['fig:2D_proj_6x6']}). Real-space sketches illustrate these phases: "Tetra" denotes the ideal tetrahedral configuration with four spins forming a perfect tetrahedron (found only at $N_e = 0.75$); canted tetrahedral states are its distorted variants and denoted by "Tetra$^*$". "Y" and "Y$^*$" refer to planar ferrimagnetic states with three distinct spin orientations, while "Ferri." indicates a two-moment ferrimagnetic phase. "Stripe" represents a collinear magnetic phase with uniform spin magnitudes. (c) Above the vHS (red regions), more complex magnetic orders arise, strongly dependent on both doping and Coulomb interaction $U$. These fall into two primary regimes characterized by dominant magnetic structure factors at the high-symmetry points $M_i$ and $X$ in the Brillouin zone. An intermediate mixed region, where both $S_M \ne 0$ and $S_X \ne 0$, is indicated with dashed lines. Phase boundaries are defined according to the order parameters described in Fig. \ref{['Fig_Order_parameters']}.
• Figure 2: Evolution of magnetic order parameters with electron density in the ground state at fixed interaction strength $U = 3t$. Upper table: Symbol correspondence with the four observables $A_m$, $T_m$, $S_M$, and $S_X$ described in the text. Background colors correspond to those used characterizing the regions of the phase diagram Fig. \ref{['Fig_Phase_Diag']}. Empty (filled) symbols represent data obtained on $6 \times 6$ ($24 \times 24$) supercells.
• Figure 3: Charge density in a $24\times24$ supercell at $U=3t$ and average density $N_e=0.76$. We observe non-periodic charge displacement, which creates zones of different density. The red stripe is at $N_e=0.75$ and displays exact tetrahedral order. Blue zones show perpendicular order (4 moments perpendicular to each other, in a plane) with $N_e$ around 0.7675. Yellow zones are at $N_e=0.76$ and show pseudo-tetrahedral orders.
• Figure 4: Maximum local magnetic moment as a function of the Coulomb interaction $U$ for different values of $N_k$, at $N_e=0.75$. Panels (a) and (b) are for a $6\times6$ supercell, while (c) results from a $24\times2$4 supercell. Lines without markers indicate results using a random initial state, while marked lines are for simulations that started from exact tetrahedral state. The inset in panel (a) shows the shift between $N_k=48\times48$ and $N_k=120\times120$ at low $U$.
• Figure 5: Raw data for the magnetic ground-state phase diagram, with the corresponding transition lines used for the schematic Fig. \ref{['Fig_Phase_Diag']}(a). Ground states were selected among 4 sets of results, each with different initial states (Random, Tetra, Y, and Ferri), using lowest total energy per site as the selection criterion.