Chiral degeneracies and Fermi-surface Chern numbers in bcc Fe

The degeneracies in the spinor band structure of bcc Fe are studied from first principles. We find numerous isolated band touchings carrying chiral charges of magnitude one (Weyl points) or two (double-Weyl nodes), as well as nonchiral degeneracy loops (nodal rings). Some degeneracies are located on symmetry lines or planes in the Brillouin zone and others at generic low-symmetry points, realizing all possible scenarios consistent with the magnetic point group. We clarify the general theory relating the chiral band touchings to the Chern numbers of the Fermi sheets enclosing them and use this approach to determine the Chern numbers on the Fermi surface of bcc Fe. Although most Fermi sheets enclose Weyl nodes, in almost all cases the net enclosed charge vanishes for symmetry reasons, resulting in a vanishing Chern number. The exceptions are two inversion-symmetric electron pockets along the symmetry line $\mathrm{\Delta }$ parallel to the magnetization. Each of them surrounds a single Weyl point, leading to Chern numbers of $\pm 1$. These small topological pockets contribute a sizable amount to the nonquantized part of the intrinsic anomalous Hall conductivity, proportional to their reciprocal-space separation. Variation of the Fermi level (or other system parameters) may lead to a touching event between Fermi sheets, accompanied by a transfer of Chern number between them.

Figure 1

Examples of Fermi-sheet structures.

Figure 2

Band structure of bcc Fe with SOC included. Energies are measured from the Fermi level, and the bands are color coded by the expectation value of the spin component $S_z$ in units of $\hslash /2$, ranging from red (majority down-spin character) to blue (minority up-spin), as indicated by the color bar.

Figure 3

Symmetry points and lines in reciprocal space for bcc Fe with SOC included. The top volume is the half BZ between $k_z=0$ and $k_z=\pi /a$ (which becomes a full BZ when expanded using inversion through $\mathrm{\Gamma }$). The shaded area at the bottom is a 2D projection. Therein, $\overline{\mathrm{\Gamma }}$ is a projection of $\mathrm{\Gamma }\text{−}\mathrm{T}\text{−}\mathrm{H}\text{−}\mathrm{T}\text{−}\mathrm{\Gamma } (\mathrm{\Delta }$), $\overline{\mathrm{X}}$ is a projection of $\mathrm{N}\text{−}\mathrm{P}\text{−}\mathrm{N}\text{−}\mathrm{P}\text{−}\mathrm{N}$, and $\overline{\mathrm{M}}$ is a projection of $\mathrm{H}\text{−}\mathrm{T}\text{−}\mathrm{\Gamma }\text{−}\mathrm{T}\text{−}\mathrm{H}$. Note that $\mathrm{T}$ is not actually a symmetry point, but ${\mathrm{N}}^{\prime }$ is.

Figure 4

Degeneracies between bands six and seven in the half BZ of Fig. 3. (a) Black dots denote WPs of positive chirality (monopole sources of Berry curvature on band six). (b) Open circles denote WPs of negative chirality, and the single open square represents a negative double-Weyl node. Each chiral degeneracy is threaded by a vertical line to help locate it with respect to the projected high-symmetry lines (dashed gray lines). The solid gray lines on the $k_z=0$ plane represent nonchiral nodal rings.

Figure 5

Mirror symmetry labels ($+i$ in gray and $-i$ in white) of band six on the mirror plane at $k_z=0$. The degeneracy lines of band six with bands five and seven are drawn in thin (green) and thick (red) lines, respectively.

Figure 6

Slice Chern number $\tilde{C}$ [Eq. (7)] associated with the six lowest bands of bcc Fe. Since $\tilde{C}$ is symmetric about $k_z=0$ and $k_z=\pi /a$ [Eqs. (A4) and (A5)], only half of the range of $k_z$ is shown.

Figure 7

A WP between bands six and seven along the fourfold axis that projects onto $\overline{\text{M}}$ in Fig. 3. The top panel and the right inset show the band dispersions as one passes through the touching point along the symmetry axis and on the orthogonal plane, respectively. The left inset shows the dispersion along a vertical line that is shifted from the symmetry axis by $\delta k_x=0.05\times 2\pi /a$. The bands are color coded by the spin as in Fig. 2. The main bottom panel shows the evolution along the symmetry axis of the phase of the $C_4$ eigenvalues (symmetry labels) of the crossing bands.

Figure 8

A double-Weyl node between bands six and seven along the fourfold axis $\mathrm{\Delta }$ in Fig. 3. The top and bottom panels and the right inset have the same meaning as in Fig. 7. Near the crossing the two bands have majority-spin character, as indicated by the red color.

Figure 9

(a) Fermi contours of bcc Fe on the $\mathrm{\Gamma }\mathrm{N}\mathrm{H}$ plane at $k_z=0$, calculated without SOC. The majority-spin contours I–IV are in red; the minority-spin contours V–VIII in blue. Curved gray lines are accidental degeneracy lines (nodal rings) that cross the Fermi level at points of contact between like-spin Fermi sheets, where they change from solid $\left(E_{\mathrm{degen}}<E_F\right)$ to dashed $\left(E_{\mathrm{degen}}>E_F\right)$. The dashed-dotted line from $\mathrm{\Gamma }$ to $\mathrm{H}$ is a line of essential degeneracy. (b) Energy bands close to the Fermi level along the $\mathrm{\Gamma }\mathrm{H}$ line. In both panels, the spin-dependent band indices are indicated.

Figure 10

Fermi surface of bcc Fe with SOC included. The indices $a$ of the individual Fermi sheets $S_{na}$ on each band $n$ are indicated.

Figure 11

Fermi contours of bcc Fe on the $\mathrm{\Gamma }\mathrm{N}\mathrm{H}$ plane at $k_z=0$, with SOC included. (a) The contours are color coded by the band index $n$ and labeled $n_a$, where $a$ is the sheet index (it is omitted for bands with a single Fermi sheet). (b) The Fermi contours are color coded by the spin in the same way as the energy bands in Fig. 2, and the gray lines labeled $\alpha ,{\beta }_1,{\beta }_2,\gamma$, and $\delta$ are degeneracy lines (nodal rings) that cross the Fermi level at points of contact between Fermi sheets, where they change from solid $\left(E_{\mathrm{degen}}<E_F\right)$ to dashed $\left(E_{\mathrm{degen}}>E_F\right)$.

Figure 12

Same as Fig. 11, but for the $\mathrm{\Gamma }{\mathrm{N}}^{\prime }\mathrm{H}$ plane at $k_y=0$. The Fermi contours are labeled in the same way as in Fig. 11. Because the symmetry on this plane is reduced by the spin-orbit interaction, the displayed region has twice the area compared to Fig. 11 and there are no symmetry-protected nodal rings.

Figure 13

Details of the spinor band structure of Fig. 2. (a) Along the line $\mathrm{\Delta }$ close to the electron pocket ${10}_7$. (b) Along the line ${\mathrm{\Delta }}^{\prime }$ close to the pocket ${10}_2$. The spin-orbit-free energy bands of Fig. 9 are shown as dotted gray lines.

Figure 14

(a) Gluing together of two Fermi sheets along a symmetry-protected degeneracy line carrying a Berry flux of $\pi$ modulo $2\pi$, indicated as $\pm \pi$. (b) The symmetry has been broken weakly, gapping the degeneracy line and separating the Fermi sheets. A Berry flux of definite sign now exits one sheet and enters the other.

Figure 15

Evolution with $k_z$ of the Berry phase along the intersection loops between a Fermi sheet and a BZ slice, enforcing continuity from one slice to the next. (a) Fermi pocket ${10}_1$. (b) Topological pockets $({10}_6,{10}_7)$, treated as a composite Fermi sheet. (c) Fermi sheet $8_1$.