Abstract
In planar tilted Dirac cone systems, the tilt parameter can be made space-dependent by either a perpendicular displacement field, or by chemical substitution in certain systems. We show that the symmetric partial derivative of the tilt parameter generates non-Abelian synthetic gauge fields in these systems. The small velocity limit of these gauge forces corresponds to Rashba and Dresselhaus spin-orbit couplings. At the classical level, the same symmetric spatial derivatives of tilt contribute to conservative, Lorentz-type and friction-like forces. The velocity dependent forces are odd with respect to tilt and therefore have opposite signs in the two valleys when the system is inversion symmetric. Furthermore, toggling the chemical potential between the valence and conduction bands reverses the sign of the all these classical forces, which indicates these forces couple to the electric charge of the carriers. As such, these forces are natural extensions of the electric and magnetic forces in the particular geometry of the tilted Dirac cone systems.
Highlights
In solid-state physics, the lattice breaks the Lorentz symmetry of the vacuum. In certain lattices, such as the honeycomb lattice of graphene, the Lorentz symmetry emerges in a lower energy scale and with a velocity scale vF, which is much smaller than the seed of light,1 with the following effective Hamiltonian: HD = vF γ 0γ i pi + mvF2 γ 0, (1)
Dirac solids at a deeper level can be attributed to an emergent effective Minkowski spacetime, from which the Lorentz symmetry immediately and quite naturally follows
We further obtain the effect of curvature on the semi-classical motion and show that the tilt parameter gives rise to new forces which will be required in appropriate extensions of the Boltzmann equation in such spacetimes
Summary
In solid-state physics, the lattice breaks the Lorentz symmetry of the vacuum. But in certain lattices, such as the honeycomb lattice of graphene, the Lorentz symmetry emerges in a lower energy scale and with a velocity scale vF , which is much smaller than the seed of light, with the following effective Hamiltonian: HD = vF γ 0γ i pi + mvF2 γ 0,. (ii) The second line of thought is to attribute the tilt in the dispersion relation to a new spacetime structure This approach is pioneered by Volovik in three-dimensional Weyl semimetals and is followed by others. Ojanen and coworkers propose that spatially varying timereversal (TR) and inversion (I) breaking sources in Weyl semimetals are equivalent to a curved spacetime for chiral fermions [85] Such structures give rise to synthetic gauge fields. Unlike strain induced changes in the metric of the spacetime, the changes introduced by TR or I breaking agents in ( 2D materials) is not a small effect These ideas are further extended to meta-materials based on Weyl semimetals by Ojanen and coworkers [86] to design the structure of the spacetime. We further obtain the effect of curvature on the semi-classical motion (geodesics) and show that the tilt parameter gives rise to new forces which will be required in appropriate extensions of the Boltzmann equation in such spacetimes
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