Particle-hole symmetry and bifurcating ground-state manifold in the quantum Hall ferromagnetic states of multilayer graphene

Abstract

The orbital structure of the quantum Hall ferromagnetic states in the zero-energy Landau level in chiral multilayer graphene (AB, ABC, ABCA, etc. stackings) is determined by the exchange interaction with all levels, including deep-lying states in the Dirac sea. This exchange field favors orbitally coherent states with a U(1) orbital symmetry if the filling factor ν is not a multiple of the number of layers. If electrons fill the orbital sector of a fixed spin/valley component to one-half, e.g., at ν = ±3, ±1 in the bilayer and at ν = ±2, ±6 in the ABCA four-layer, there is a transition to an Z2×U(1) manifold. For weak interaction, the structure in the zero-energy Landau band compensates for the different exchange interaction on the sublattices in the Landau orbitals; on the other side, the ground state comes in two copies that distribute charge on the sublattices differently. We expect a sequence of similar bifurcations in multilayers of Bernal stacking.PACS numbers: 73.43. Cd,73.22.Pr In their pristine form, monolayer graphene, 1 bilayer graphene 2 and trilayer graphene 3 are zero-gap semiconductors with a Landau level at zero energy, i.e., where the valence and the conduction bands touch in the absence of a magnetic field. This level significantly affects the integer quantum Hall effect 4 (IQHE). Quantum Hall ferromagnetism (QHF), 5 combined with smaller terms such as the Zeeman energy, fully resolves this level, 6,7 Particularly interesting are the multilayers, because their zero-energy Landau band (ZLB) has orbital degeneracies above the ubiquitous spin and valley quasidegeneracies.Recently, Shizuya has pointed out that the exchange field created by the sea of filled Landau levels (LL's) is essential for correctly identifying the QHF ground states in bilayer graphene 8 and ABC trilayer. 9 This exchange field favors orbitally coherent states, in contrast to the Hund's rule picture 10,11 that predicts the order the zero-energy orbitals are filled. We make use of this observation, but no longer treat the Dirac sea as inert. We identify a number of QHF states, and find that the interplay of the sublattice structure and exchange leads to bifurcations of the ground state manifold if the number of layers s is even and the filling factor is such as to half-fill the orbital sector of fixed spin and valley at zero energy.Hamiltonians.The following class of Hamiltonians concisely describes the low-energy physics of chiral (rhombohedral) stacking series, i.e., the AB, ABC, ABCA... multlilayers (s > 1 is the number of layers) if the layers are energetically equivalent,where π = p x + ip y with p = −i ∇ − eA, v = √ 3aγ 0 /2 ≈ 10 6 m/s is derived from the γ 0 intralayer nearest-neighbor hopping amplitude, and γ 1 ≈ 0.4 eV is the interlayer hopping between dimer sites (i.e., sites exactly above/below one another). For bilayer graphene,Ĥ (2)ξ is obtained by a SchriefferWolf transformation 12 from the Slonczewski-WeissMcClure tight-binding model of graphite, 13 expanded to first order in the momentum difference q − ξK from...

Symmetry characterization of the collective modes of the phase diagram of the ν=0 quantum Hall state in graphene: Mean-field phase diagram and spontaneously broken symmetries

“…Using symmetry considerations, it can be proven that g x = g y ≡ g ⊥ [21,37], so there are only two independent coupling constants g ⊥ , g z . The potential V DS (x, x ′ ) represents the mean-field interaction of the ZLL with the (inert) Dirac sea compound by all the occupied states with n ≤ −2 [20,26,45]. As shown in Appendix A 2, this potential is diagonal within the ZLL; its explicit expression is given by Eq.…”

“…One possible way to take it into account is through a static screening of the Coulomb interaction in the large-N approximation [21,[35][36][37][38]. Other approaches consider dynamical screening within the same approximation [15][16][17] or allow for LL mixing in the TDHFA formalism [26]. As we are mainly interested in the low-energy modes describing the phase transitions, static screening is expected to provide a good approximation in this limit; on the other hand, screening by LL mixing is not expected to describe correctly the dispersion relation near k = 0 [26], which is precisely the most interesting region for our purposes.…”

“…A nice discussion about the coupling of the ±n LLs for strong Coulomb interactions is presented in Ref. [26].…”

“…The corresponding effective Hamiltonian of the system is Ĥ = Ĥ0 + ĤC + Ĥsr . After neglecting trigonal warping effects and other small corrections [6,23,26,34,41], the single-particle Hamiltonian Ĥ0 reads:…”

Electron-hole asymmetric integer and fractional quantum Hall effect in bilayer graphene

“…For example, the n = 0 state, which is a canted antiferromagnet at large perpendicular magnetic field, can be tuned into a ferromagnetic state by a large parallel magnetic field, or a layer-polarized state under large perpendicular electric field (5)(6)(7)(8)(9)(10)(11). The order in which orbital states are filled remains an open question, with suggestions of full polarization or orbitally coherent states, depending on system parameters (12)(13)(14)(15)(16). The interplay between externally applied fields and intrinsic electron-electron interactions, both of which break the degeneracies of bilayer graphene, produces a rich phase diagram.…”