Broken sublattice symmetry states in Bernal stacked multilayer graphene

Abstract

We analyze the ordered phases of Bernal stacked multilayer graphene in the presence of interaction induced band gaps due to sublattice symmetry breaking potentials, whose solutions can be analyzed in terms of light-mass and heavy-mass pseudospin doublets which have the same Chern numbers but opposite charge polarization directions. The application of a perpendicular external electric field reveals an effective Hund's rule for the ordering of the sublattice pseudospin doublets in a tetralayer, while a similar but more complex phase diagram develops with increasing layer number. I. FLAVOR ANTIFERRO STATES IN TETRALAYER GRAPHENEFrom the Hartree energy cost considerations, the metastable states with the lowest total energy are expected to be flavor antiferro when there is no external electric field perpendicular to the graphene layers. Within the energetically more favorable flavor antiferro states, we classify different Hall phases [1, 2]: layer antiferromagnetic (LAF) phase with the spin dependent but valley independent sublattice potential, quantum spin Hall (QSH) phase with both the spin and valley dependent sublattice potential, and quantum anomalous Hall (QAH) phase with the valley dependent but spin independent sublattice potential, as schematically shown in Fig. 1.

Exchange interaction, disorder, and stacking faults in rhombohedral graphene multilayers

“…Electron-electron interactions are included within a mean-field Hartree-Fock approximation [45,46,62] and, in particular, we follow the methodology applied to Bernal-stacked multilayer graphene in Refs. [60,61]. The total Hamiltonian isĤ tot =Ĥ +V MF wherê…”

“…Using the minimal model, the energy spectrum is isotropic around each K point and the eigenstates of the non-interacting Hamiltonian (1) at an arbitrary angle may be related to those at a specific angle by a stackingdependent unitary transformation [45,76]. We assume that the eigenstates of the interacting mean-field theory (2,3) also satisfy this rotational transformation [60], allowing for the k summations to be performed in only one specific direction with the exchange interaction (5) being determined via an integration with respect to the polar angle of wave vector k . This simplification dramatically reduces the numerical cost of the calculations allowing for a study of multilayers with a large number of layers.…”

“…That the odd antiferromagnetic state is the ground state agrees with Refs. [60,61] for the (2, 2) fault which is the same system as N = 4 Bernal-stacked multilayer. For benchmarking, with α g = 0.1 we find E g = 1.42 meV for the (2, 2) fault which compares with E g ≈ 1.44 meV for the second red circle in Fig.…”

“…In this paper, we generalize the Hartree Fock meanfield theory [45,46,[60][61][62] of the pseudospin LAF state to RMG with a large number of layers (up to N = 30) in order to determine the layer dependence of the interactioninduced band gap at charge neutrality. We find that the strength of the interacting state, as characterized by the separation of the bands exactly at the valley center, saturates with layer number, but that the actual band gap decreases with layer number; this agrees with density functional theory (DFT) which predicted the band gap for RMG with up to eight layers [64].…”

“…Owing to the antiferromagnetic configuration of four flavors, there is no net charge accumulation on summing over them and, thus, * ed.mccann@lancaster.ac.uk no cost in terms of Hartree energy. The evolution of similar gapped states with layer number has been discussed for both Bernal [60,61] and rhombohedral multilayers [17,[62][63][64].…”