Nodal semimetals in $d\ge 3$ to sharp pseudo-Landau levels by dimensional reduction

Nonuniform strain applied to graphene's honeycomb lattice can induce pseudo-Landau levels in the single-particle spectrum. Various generalizations have been put forward, including a particular family of hopping models in $d$ space dimensions. Here we show that the key ingredient for sharp pseudo-Landau levels in higher dimensions is dimensional reduction. We consider particles moving on a $d$-dimensional hyperdiamond lattice which displays a semimetallic band structure, with a $(d-2)$-dimensional nodal manifold. By applying a suitable strain pattern, the single-particle spectrum evolves into a sequence of relativistic Landau levels. We develop and solve the corresponding field theory: Each nodal point effectively generates a Landau-level problem which is strictly two dimensional to leading order in the applied strain. While the effective pseudovector potential varies across the nodal manifold, the Landau-level spacing does not. Our theory paves the way for strain engineering of single-particle states via dimensional reduction and beyond global minimal coupling.

Figure 1

Strain-induced Landau levels in $d=3$: (a) Diamond lattice with (b) semimetallic density of states. (c) Diamond lattice under tetraxial strain, resulting in (d) sharp relativistic Landau levels which emerge via dimensional reduction.

Figure 2

Brillouin zone (shaded) of the diamond lattice, together with the momentum-space network of nodal lines which cross at the high-symmetry $X$ points. Also shown are the directions of the pseudomagnetic field (arrows) emerging under tetraxial strain, together with the corresponding semiclassical trajectories for low-energy electrons (hatched ellipses). The pseudomagnetic field is singular at the $X$ points.

Figure 3

Illustration of dimensional reduction in $d=3$. (a) Dirac velocities $v_x$ and $v_y$, together with the amplitude of the pseudomagnetic field $|{\mathit{B}}_{\mathit{K}}|$, plotted along a nodal line connecting two $X$ points. (b) Semiclassical trajectories at selected points along the nodal line in the momentum-space plane perpendicular to the nodal line.