Orbital Lamb shift and mixing of the pseudo-zero-mode Landau levels in A B C -stacked trilayer graphene

We devote this work to the study of the mean-field phase diagram of the ν = 0 quantum Hall state in bilayer graphene and the computation of the corresponding neutral collective modes, extending the results of recent works in the literature. Specifically, we provide a detailed classification of the complete orbital-valley-spin structure of the collective modes and show that phase transitions are characterized by singlet modes in orbital pseudospin, which are independent of the Coulomb strength and suffer strong many-body corrections from short-range interactions at low momentum. We describe the symmetry breaking mechanism for phase transitions in terms of the valley-spin structure of the Goldstone modes. For the remaining phase boundaries, we prove that the associated exact SO(5) symmetry existing at zero Zeeman energy and interlayer voltage survives as a weaker mean-field symmetry of the Hartree-Fock equations. We extend the previous results for bilayer graphene to the monolayer scenario. Finally, we show that taking into account Landau level mixing through screening does not modify the physical picture explained above.

Section: B Projection Onto the Zero Landau Levelmentioning

confidence: 99%

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

Nova

Zapata

2017

“…Each zero-mode level, subjected to quantum fluctuations of the valence band, gets shifted differently within the LLL, just like the Lamb shift 19 in the hydrogen atom. This orbital Lamb shift was first noted 20 for bilayer graphene and is also realized in an analogous fashion 21 in rhombohedral (ABC-stacked) trilayer graphene, which is a "chiral" trilayer generalization of bilayer graphene. This orbital shift is considerably larger in scale than intrinsic spin or valley breaking, and one has to take it into account in clarifying the fine structure of the LLL in multilayers.…”

Section: 18mentioning

confidence: 67%

“…It turned out that ABA trilayers are distinct in zero-mode characteristics from ABC trilayers examined earlier. 21 We have, in particular, seen that both valley and eh symmetries are markedly broken in the LLL of ABA trilayers. These asymmetries appear in the one-body spectra {ǫ h n } already to first order in nonleading hopping parameters (such as γ 2 , γ 5 and ∆ ′ ), and are enhanced via the Lamb-shift contributions {ǫ v n } and the Coulomb interaction acting within the LLL.…”

Section: Summary and Discussionmentioning

confidence: 93%

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Interaction-enhanced electron-hole and valley asymmetries in the lowest Landau level ofABA-stacked trilayer graphene

Shizuya

2014

Self Cite

In a magnetic field graphene trilayers support a special multiplet of 12 zero(-energy)-mode Landau levels with a threefold degeneracy in Landau orbitals. A close look is made into such zero-mode levels in ABA-stacked trilayers, with the Coulomb interaction taken into account. It turns out that the zero-mode Landau levels of ABA trilayers are greatly afflicted with electron-hole and valley asymmetries, which come from general hopping parameters and which are enhanced by the Coulomb interaction and the associated vacuum effect, the orbital Lamb shift, that lifts the zeromode degeneracy. These asymmetries substantially affect the way the zero-mode levels evolve, with filling, via Coulomb interactions; and its consequences are discussed in the light of experiments.

“…We set θ m>M,ξσ = 0 and utilize the identity ∞ n=0 |F n ′ n (k)| 2 = 1 for arbitrary n ′ , for reducing infinite summations of exchange integrals to finite sums, plus a featureless divergent constant we omit. 8,9,16 The exchange field is still due to all filled levels; we merely treat the low-lying ones as inert. The charge distribution w ± for the optimized parameters are shown in Fig.…”

Section: For Bilayer Grapheneĥ (2)mentioning

confidence: 99%

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

Tőke

2013

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...