nLab anomalous velocity

Idea

In crystalline materials whose Berry curvature $\Omega$ of the valence band is non-vanishing over the Brillouin torus of crystal momenta, the semiclassical equation of motion for movement of electron states in momentum space contains an extra “anomalous” term proportional to $\Omega$. This is a momentum-space analogue of what for electrons moving through position space in the presence of magnetic field flux density $F$ is the Lorentz force.

Accordingly, like the Lorentz force in the presence of an ambient magnetic field (the latter being the curvature of the electromagnetic gauge potential) is the cause of the Hall effect, so this anomalous velocity, induced instead by Berry curvature in momentum space, is the cause of the anomalous Hall effect.

References

The original articles:

• Ming-Che Chang, Qian Niu: Berry Phase, Hyperorbits, and the Hofstadter Spectrum, Phys. Rev. Lett. 75 (1995) 1348 [doi:10.1103/PhysRevLett.75.1348]

• Ming-Che Chang, Qian Niu: Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands, Phys. Rev. B 53 (1996) 7010 [doi:10.1103/PhysRevB.53.7010]

• Ganesh Sundaram, Qian Niu, §III.B in: Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Rev. B 59 (1999) 14915 [doi:10.1103/PhysRevB.59.14915]

From the point of view of relativistic field theory:

• Michael Stone, Vatsal Dwivedi, Tianci Zhou, §II.A in: Berry Phase, Lorentz Covariance, and Anomalous Velocity for Dirac and Weyl Particles, Phys. Rev. D 91 (2015) 025004 [arXiv:1406.0354, doi:10.1103/PhysRevD.91.025004]

Review:

• Di Xiao, Junren Shi, Qian Niu: Berry phase correction to electron density of states in solids, Phys. Rev. Lett. 95 (2005) 137204 [arXiv:cond-mat/0502340, doi:10.1103/PhysRevLett.95.137204]

• Nikolai A. Sinitsyn, around (59) in: Semiclassical theories of the anomalous Hall effect, Journal of Physics: Condensed Matter 20 2 (2008) [arXiv:0712.0183, doi:10.1088/0953-8984/20/02/023201]

• Di Xiao, Ming-Che Chang, Qian Niu, §III.A in: Berry Phase Effects on Electronic Properties, Rev. Mod. Phys. 82 (2010) 1959-2007 [arXiv:0907.2021, doi:10.1103/RevModPhys.82.1959]

• Y. D. Chong: Berry’s phase and the anomalous velocity of Bloch wavepackets, Phys. Rev. B 81 (2010) 052303 [arXiv:0912.0716, doi:10.1103/PhysRevB.81.052303]

• Mikhail A. Stephanov, Y. Yin, (12-13) in: Chiral Kinetic Theory, Phys. Rev. Lett. 109 (2012) 162001 [arXiv:1207.0747, doi:10.1103/PhysRevLett.109.162001]

• Michael Stone: Berry Curvature, Spin, and Anomalous Velocity, talk at ESI, Vienna (2014) [pdf, pdf]