Magnetic photocurrents in multifold Weyl fermions

We examine the magneto-optical response of chiral multifold fermions. Specifically, we show that they are ideal candidates for observing the helical magnetic effect (HME) previously predicted for simple Weyl fermions. Unlike Weyl fermions, the HME is present in multifold fermions even in the simplest case where the low-energy dispersion is linear and spherically symmetric. In this ideal case, we derive an analytical expression for the HME and find it is proportional to the circular photogalvanic effect; for realistic parameters and accounting for the geometry of the setup, the HME photocurrent could be roughly the same order of magnitude as the circular photogalvanic effect observed in multifold fermions. Additional nonlinear and symmetry-breaking terms will ruin the quantization but not hurt the observation of the HME.

Figure 1

Comparison of transitions in an untilted and a tilted Weyl cone. In an untilted cone, the initial and final states are related by $CP$ and there is no HME. In the tilted cone, the initial and final states are not related by $CP$, and it is possible to partially blockade the cone. This allows the HME current to be nonzero.

Figure 2

Pauli blockade in a spin-1 (left) and spin-3/2 (right) multifold fermion. Solid black arrows indicate allowed transitions from occupied to empty bands, while dashed arrows indicate blockaded transitions. The shaded blue region indicates the Fermi sea. The Pauli blockaded multifold fermions exhibit the HME.

Figure 3

Normalized HME photocurrent vs normalized frequency for a spin-1 cone (top) and a spin-3/2 cone (bottom) for $T/\mu =0.01,0.05,0.1,0.2$, and 0.5. The current is normalized in units of $j_0=\frac{e^{3}I\tau }{3\pi {\hslash }^2{\epsilon }_{0}c}\frac{eB{v}_F^2}{\hslash {\mu }^2}$ where $v_F$ is the velocity of the fastest band. The frequency is normalized in units of chemical potential $\mu$. At $T=0$, the current is zero until a critical frequency where the upper band is unoccupied at the momentum required for the transition. In the spin-3/2 case, there is also a second critical frequency where the current drops to zero because both upper bands are unoccupied at the momentum required for the transition.

Figure 4

The dispersion relation (top) and normalized photocurrent vs normalized frequency (bottom) for the double spin-1/2 fermion split into a spin-1 fermion and trivial band, described by the Hamiltonian in Eq. (19). The photocurrent is normalized in units of $\frac{e^{3}I\tau }{3\pi {\hslash }^2{\epsilon }_{0}c}\frac{eB{v}_F^2}{\hslash E_0^2}$, while the frequency is normalized in units of $E_0=\mathrm{\Delta }/4$. The chemical potential is taken to be $\mu =0$. The polarization is along the magnetic field. The normalized temperature is $T/E_0=0.01,0.05,0.1,0.2$, and 0.5. At $T=0$, the current is zero for small frequencies. Above a critical frequency, the Pauli blockade is lifted only for transitions from trivial to chiral bands, and there is a current. Above a second critical frequency, the Pauli blockade is also lifted for transitions from chiral to trivial bands, and the current is approximately canceled, but is not exactly zero

Figure 5

The chiral Landau levels and first five achiral Landau levels for a spin-1 fermion (left) and a spin-3/2 fermion (right).