Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands

Chang, Ming Che; Niu, Qian

doi:10.1103/physrevb.53.7010

Cited by 563 publications

(501 citation statements)

References 48 publications

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“…One of the most interesting properties of gapped bilayer graphene is the existence of a finite Hall conductivity and orbital magnetism due to the flavor-dependent momentumspace vortices 14,17,30 in the broken symmetry states. Because the vorticity v is opposite for opposite valleys, the integrated Berry curvature gives rise to a Hall conductivity 18,19,31,32 with magnitude e 2 /h for each flavor, and a sign that changes with valley as well as with layer polarization. The Berry curvature reflects the handedness of Bloch electrons and captures intracell circulating currents which generate a finite orbital magnetic moment proportional to the angular momentum due to self-rotating Bloch wave packets.…”

Section: A Total Layer Density and Chern Number Classifications Of Cmentioning

confidence: 99%

“…The Berry curvature reflects the handedness of Bloch electrons and captures intracell circulating currents which generate a finite orbital magnetic moment proportional to the angular momentum due to self-rotating Bloch wave packets. 32 The Berry curvature of the system can be evaluated using 18…”

Section: A Total Layer Density and Chern Number Classifications Of Cmentioning

confidence: 99%

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Lattice theory of pseudospin ferromagnetism in bilayer graphene: Competing interaction-induced quantum Hall states

2011

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In mean-field-theory bilayer graphene's massive Dirac fermion model has a family of broken inversion symmetry ground states with charge gaps and flavor dependent spontaneous inter layer charge transfers. We use a lattice Hartree-Fock model to explore the lattice scale physics of graphene bilayers which has a strong influence on ordering energy scales and on the competition between distinct ordered states. We find that inversion symmetry is still broken in the lattice model and estimate that the transferred areal densities are ∼ 10 −5 electrons per carbon atom, that the associated energy gaps are ∼ 10 −2 eV, that the ordering condensation energies are ∼ 10 −7 eV per carbon atom, and that the differences in energy between competing ordered states are ∼ 10 −9 eV per carbon atom. We find that states with a quantized valley Hall effect are lowest in energy, but that the coupling of an external magnetic field to spontaneous orbital moments favors the broken time-reversal-symmetry states that have quantized anomalous Hall effects. Our theory predicts non monotonic behavior of the band gap at neutrality on the potential difference between layers, in qualitative agreement with recent experiments.

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Section: A Total Layer Density and Chern Number Classifications Of Cmentioning

confidence: 99%

Section: A Total Layer Density and Chern Number Classifications Of Cmentioning

confidence: 99%

Lattice theory of pseudospin ferromagnetism in bilayer graphene: Competing interaction-induced quantum Hall states

2011

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“…Interestingly, this intrinsic mechanism has recently been reinterpreted in terms of the Berry curvature of the occupied Bloch states. [7][8][9] Furthermore, recent first-principles studies based on the Berry phase formalism showed that the intrinsic AHE is important in various materials. 10,11 In particular, in itinerant ferromagnets such as Fe, the intrinsic AHC given by firstprinciples density functional calculations with the generalized gradient approximation (GGA) 10 has been found to agree rather well with the experimental AHC 12,13 .…”

Section: Introductionmentioning

confidence: 99%

Anomalous and spin Hall effects in hcp cobalt from GGA+Ucalculations

2012

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We have calculated the intrinsic anomalous and spin Hall conductivities, spin and orbital magnetic moments and also spin polarization of Hall currents in hexagonal cobalt within the density functional theory with the generalized gradient approximation (GGA) plus on-site Coulomb interaction (GGA+U ). The accurate all-electron full-potential linearized augmented plane wave method is used. We find that the anomalous Hall conductivity (AHC) (σ A xy ) and orbital magnetic moment (mo) of cobalt with the magnetization being along the c-axis (m//c) calculated in the GGA+U scheme with U = 1.6 eV and J = 0.9 eV agree rather well with the corresponding experimental values while, in contrast, the σ A xy and mo from the GGA calculations are significantly smaller than the measured ones. This suggests that moderate U = 1.6 eV and J = 0.9 eV are appropriate for Co metals. The calculated AHC and spin Hall conductivity (SHC) (σ S ) are highly anisotropic, and the ratio of the Hall conductivity for m//c to that for the magnetization being in-plane (m//ab) from the GGA+U calculations, is ∼16.0 for the AHC and ∼6.0 for the SHC. For m//c, the spin-up and spin-down Hall currents are found to flow in the same direction. The magnitude of the calculated Hall current spin polarization (P H ) is large and P H = −0.61. Remarkably, for m//ab, the spin-up and spin-down Hall currents are predicted to flow in the opposite directions. This indicates that the Hall current contains both spin-polarized charge current and pure spin current, resulting in the magnitude of the Hall current spin polarization (P H = −1.59) being larger than 1.0.

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“…As an electron wavepacket moves in kspace under the influence of an applied field, there are two contributions to its spatial velocity v. The first term, the group velocity, describes how the modified energy-momentum relation changes the velocity of the center of a wavepacket, which is an effect that would be present even for a point particle. The Karplus-Luttinger contribution, which can be derived quite systematically 6,23 , describes how a change in k induces a change in the real-space location because the Bloch states are changing: this change is…”

Section: Key Features Of the Anomalous Hall Effectmentioning

confidence: 99%

Pumping conductance, the intrinsic anomalous Hall effect, and statistics of topological invariants

et al. 2015

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The pumping conductance of a disordered two-dimensional Chern insulator scales with increasing size and fixed disorder strength to sharp plateau transitions at well-defined energies between ordinary and quantum Hall insulators. When the disorder strength is scaled to zero as system size increases, the "metallic" regime of fluctuating Chern numbers can extend over the whole band. A simple argument leads to a sort of weighted equipartition of Chern number over minibands in a finite system with periodic boundary conditions: even though there must be strong fluctuations between disorder realizations, the mean Chern number at a given energy is determined by the clean Berry curvature distribution, as in the intrinsic anomalous Hall effect formula for metals. This estimate is compared to numerical results using recently developed operator algebra methods, and indeed the dominant variation of average Chern number is explained by the intrinsic anomalous Hall formula. A mathematical appendix provides more precise definitions and a model for the full distribution of Chern numbers.

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Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands

Cited by 563 publications

References 48 publications

Lattice theory of pseudospin ferromagnetism in bilayer graphene: Competing interaction-induced quantum Hall states

Lattice theory of pseudospin ferromagnetism in bilayer graphene: Competing interaction-induced quantum Hall states

Anomalous and spin Hall effects in hcp cobalt from GGA+Ucalculations

Pumping conductance, the intrinsic anomalous Hall effect, and statistics of topological invariants

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