Intrinsic ac anomalous Hall effect of nonsymmorphic chiral superconductors with an application to ${\mathrm{UPt}}_3$

We identify an intrinsic mechanism of the anomalous Hall effect for nonsymmorphic chiral superconductors. This mechanism relies on both a nontrivial multiband chiral superconducting order parameter, which is a mixture of pairings of even and odd angular momentum channels, and a complex normal-state intersublattice hopping, both of which are consequences of the nonsymmorphic group symmetry of the underlying lattice. We apply this mechanism to the putative chiral superconducting phase of the heavy-fermion superconductor ${\mathrm{UPt}}_3$ and calculate the anomalous ac Hall conductivity in a simplified two-band model. From the ac Hall conductivity and optical data we estimate the polar Kerr rotation angle and compare it to the measured results for ${\mathrm{UPt}}_3$ [Schemm et al., Science 345, 190 (2014)].

Figure 1

Crystal structure of ${\mathrm{UPt}}_3$. Blue disks denote the positions of $\mathrm{U}$ atoms. There is a $\mathrm{Pt}$ atom (not shown) between each nearest-neighbor intralayer pair of $\mathrm{U}$ atoms. The vectors ${\mathbf{e}}_i$ and ${\mathbf{r}}_i$ connect two nearest-neighbor intralayer and interlayer $\mathrm{U}$ atoms, respectively. The coordinate system is chosen such that $\widehat{x}\parallel {\mathbf{e}}_1$.

Figure 2

Starfish Fermi surface (FS). (a) FS in the three-dimensional Brillouin zone of ${\mathrm{UPt}}_3$; (b) FS contours in the plane of $k_z=\pi$. The red (blue) line is the $E_{+}^{\left(n\right)}\left(\mathbf{k}\right)=0$ $[E_{-}^{\left(n\right)}\left(\mathbf{k}\right)=0]$ constant energy contour. Parameters used are $(t,t_z,t^{\prime },\alpha ,\mu )=(1,-4,1,2,12)$.

Figure 3

Bogoliubov quasiparticle energy line nodes of the BdG Hamiltonian $\widehat{\mathcal{H}}\left(\mathbf{k}\right)$ at $k_z=\pi$. The parameter ${\mathrm{\Delta }}_0=0.1t$. Other parameters used are the same as in Fig. 2.

Figure 4

Numerical results for $\mathrm{Im}{\sigma }_H^{\left(1\right)}\left(\omega \right)$. Left panel: small frequency regime $\omega /t\le 14$; right panel: large frequency regime $\omega /t\ge 10$. Note that the vertical axis scales of the two figures are different. The unit of ${\sigma }_H$ is $e^2/\hslash d$, with $d$ the $\widehat{c}$-axis lattice spacing of ${\mathrm{UPt}}_3$. Parameters used are $(t,t_z,t^{\prime },\alpha ,\mu ,{\mathrm{\Delta }}_0,\eta )=(1,-4,1,2,12,0.1,1.0)$.

Figure 5

Numerical results for $\mathrm{Re}{\sigma }_H^{\left(1\right)}\left(\omega \right)$. Left panel: small frequency regime; right panel: large frequency regime. Note that the scales of the vertical axes in the two figures are different. In the right figure, the red dashed line is a high-frequency asymptotic result. Parameters used are the same as in Fig. 4.

Figure 6

Diagrammatic representation of the nonvanishing contributions to ${\sigma }_H$, where wiggly lines are photons and double solid lines with arrows are Green's functions given by Eq. (16). The photon polarization is labeled by $i,j=x,y$. $a,b$ are sublattice labels, and $s$ is the spin label. If $s=\{\uparrow ,\downarrow \}$, then $\overline{s}=\{\downarrow ,\uparrow \}$. Note that, in each diagram, the spin labels on a right vertex are opposite to that on the corresponding left vertex. This is because, in each diagram, each Green's function contributes one superconducting order parameter that pairs electrons of opposite spin, while all normal-state Hamiltonian matrix elements, including SOC, only connect electrons of the same spin.

Figure 7

Comparison between the numerical results for $\mathrm{Im}{\sigma }_H^{\left(1\right)}$ (thick black line) and that for $\mathrm{Im}{\sigma }_H$ (dashed red line). Left panel: small $\omega /t\le 14$; right panel: large $\omega /t\ge 10$. Notice that the vertical axis scales of the two panels are different. Parameters used are the same as in Fig. 4.

Figure 8

Comparison between the numerical results for $\mathrm{Re}{\sigma }_H^{\left(1\right)}$ (thick black line) and that for $\mathrm{Re}{\sigma }_H$ (dashed red line). Left panel: small $\omega /t\le 14$; right panel: large $\omega /t\ge 10$. Note that the vertical axis scales of the two panels are different.

Figure 9

Normal-state energy dispersions along high-symmetry directions of the hexagonal Brillouin zone at $k_z=\pi$. The two energy band dispersions are $E_{\pm }^{\left(n\right)}\left(\mathbf{k}\right)={\xi }_{\mathbf{k}}\pm \sqrt{g_{\mathbf{k}}^2+{\left|{\epsilon }_{\mathbf{k}}\right|}^2}$, with $E_{+}^{\left(n\right)}$ plotted in full blue line and $E_{-}^{\left(n\right)}$ in the dashed red line. The two bands are degenerate along the symmetry axis $\mathrm{A}-\mathrm{L}$ because ${\epsilon }_{\mathbf{k}}=0$ at $k_z=\pi$ and the SOC vanishes along these directions as well.