Discrete quantum geometry and intrinsic spin Hall effect

We show that the quantum geometry of the Fermi surface can be numerically described by a three-dimensional discrete quantum manifold. This approach not only avoids singularities in the Fermi sea, but it also enables the precise computation of the intrinsic Hall conductivity resolved in spin, as well as any other local properties of the Fermi surface. The method assures numerical accuracy when the Fermi level is arbitrarily close to singularities, and it remains robust when Kramers degeneracy is protected by symmetry. The approach is demonstrated by calculating the anomalous Hall and spin Hall conductivities of a two-band lattice model of a Weyl semimetal and a full-band ab initio model of zinc-blende GaAs.

Figure 1

Illustration of different approaches to describe intrinsic anomalous Hall effect as a geometric property of the Fermi surface. (a) The slice-based approach. The Fermi surface (${\epsilon }_F=0$) corresponds to a cubic tight-binding model ${\epsilon }_{\mathbf{k}}=-2t(cos{k}_x+cos{k}_y+cos{k}_z)$, where $t$ is the hopping between nearest neighbors. (b) The new vertex-based method by establishing a three-dimensional discrete manifold of the Fermi surface. (c) The local Berry phase associated with each vertex given by the adjacent triangles. (d) The flux of Berry curvature is over-counted by three times for ${▵}_{123}$. (e) Manifold matching between vertices and edges on opposing Brillouin-zone boundaries. The red and orange polygons correspond to the ones shown in (b) near the boundaries.

Figure 2

Intrinsic (spin) Hall effect in a lattice model of a Weyl semimetal. (a) The band structure given by the model with $k_y=0, k_0=\frac{\pi }{2}, t_x=t$, and $m=2t$. (b) Fermi surface obtained as a 3D discrete quantum manifold (${\epsilon }_F=0.45t$). The vertices are colored by the local Berry curvature. (c) The Fermi surface at ${\epsilon }_F=1.05t$, where the shape represents the symmetry of the lattice. (d) The intrinsic anomalous Hall conductivity computed using Eqs. (9) and (10). Here, ${\sigma }_{\mu \nu }$ is normalized to the Weyl-point conductivity ${\sigma }_0=e^2/2h$. (e) and (f) The intrinsic spin-Hall conductivity tensor evaluated using Eq. (11). The results are first converted to the corresponding electrical conductivity and then normalized to ${\sigma }_0$.

Figure 3

(a) The band structure of GaAs along high-symmetry points. Both the heavy hole (red) and the light hole (blue) bands are twofold degenerate. [(b)–(e)] The 3D discrete quantum manifold of the Fermi surface at $\epsilon ={\epsilon }_V-0.02\text{eV}$. (b) and (c) correspond to the heavy hole band, whereas (d) and (e) correspond to the light hole band. [(f)–(i)] The electron spin $s_z$ illustrated on the Fermi surfaces corresponding to (b)–(e), respectively.

Figure 4

(a) The anomalous Hall conductivity (dark) and the intrinsic spin Hall conductivity (blue) near the valence band top of zinc-blende GaAs. The red line denotes the Hall conductivity without orienting the spins of degenerate states. (b) Two coincident triangles due to Kramers degeneracy. The illustration is shifted for demonstration. (c) Swapping one Kramers pair in the triangle. (d-e) The discrete heavy-hole Fermi surface at ${\epsilon }_F={\epsilon }_V-0.02\text{eV}$ without orienting the degenerate spins. The vertex color illustrates the local Berry phases ${\varphi }_p$. (f) Comparison between the Fermi-sea and the Fermi-surface results. The inset shows the Fermi-surface at ${\epsilon }_F=0.5t$. (g) The convergence of numerical results after mesh refinements. The Insets compare the zeroth and the sixth refinement for the heavy-hole band of GaAs.