Superconductivity Provides Access to the Chiral Magnetic Effect of an Unpaired Weyl Cone

The massless fermions of a Weyl semimetal come in two species of opposite chirality, in two cones of the band structure. As a consequence, the current $j$ induced in one Weyl cone by a magnetic field $B$ [the chiral magnetic effect (CME)] is canceled in equilibrium by an opposite current in the other cone. Here, we show that superconductivity offers a way to avoid this cancellation, by means of a flux bias that gaps out a Weyl cone jointly with its particle-hole conjugate. The remaining gapless Weyl cone and its particle-hole conjugate represent a single fermionic species, with renormalized charge $e^{*}$ and a single chirality $\pm$ set by the sign of the flux bias. As a consequence, the CME is no longer canceled in equilibrium but appears as a supercurrent response $\partial j/\partial B=\pm (e^{*}e/h^2)\mu$ along the magnetic field at chemical potential $\mu$.

Figure 1

Left panel: Slab of a Weyl superconductor subject to a magnetic field $B$ in the plane of the slab (thickness $W$ less than the London penetration depth). The equilibrium chiral magnetic effect manifests itself as a current response $\partial j/\partial B=\pm \kappa (e/h{)}^2\mu$ along the field lines, with $\kappa$ a charge renormalization factor and $\mu$ the equilibrium chemical potential. The right panel shows the flux-biased measurement circuit and the charge-conjugate pair of Weyl cones responsible for the effect, of a single chirality $\pm$ determined by the sign of the flux bias.

Figure 2

Effect of a flux bias on the band structure of a Weyl superconductor. The plots are calculated from the Hamiltonian (3) in the slab geometry of Fig. 1 (parameters: $m_0=0$, ${\mathrm{\Delta }}_0=0.2$, $\beta =0.5$, $\mu =-0.05$, $k_y=0$, $W=100$, $B_z=0$). The color scale indicates the charge expectation value, to distinguish electronlike and holelike cones. As the flux bias is increased from $\mathrm{\Lambda }=0$ in panel (a), to $\mathrm{\Lambda }=0.1$ and 0.4 in panels (b) and (c), one electron-hole pair of Weyl cones merges and is gapped by the pair potential. What remains in panel (c) is a single pair of charge-conjugate Weyl cones, connected by a surface Fermi arc. This is the phase that supports a chiral magnetic effect in equilibrium.

Figure 3

Chirality switch of a pair of charge-conjugate Weyl cones, induced by a sign change of the flux bias $\mathrm{\Lambda }=-0.45$, 0.15, and 0.45 in panels (a), (b), and (c), respectively. All other parameters are the same in each panel: $m_0=0$, ${\mathrm{\Delta }}_0=0.6$, $\beta =0.5$, $W=100$, $k_y=0$, $\mu =-0.05$, and $B_z=0.001a_0^{-2}h/e$. The charge color scale of the band structure is as in Fig. 2. Particles in the zeroth Landau level propagate through the bulk in the same direction both in the electronlike cone and in the holelike cone, as determined by the chirality $\chi =-\mathrm{sign}\mathrm{\Lambda }$ [49]. A net charge current appears in equilibrium because $\mu <0$, so there is an excess of electronlike states at $E>0$. [States at $E<0$ do not contribute to the equilibrium current (11).] The particle current is canceled by the Fermi arc that connects the charge-conjugate Weyl cones. The Fermi arc carries an approximately neutral current; hence, the charge current in the chiral Landau level is not much affected by the counterflow of particles in the Fermi arc.

Figure 4

Data points: numerical calculation of the equilibrium supercurrent in the flux-biased circuit of Fig. 1. The parameters are $m_0=0$, ${\mathrm{\Delta }}_0=0.6$, $\beta =0.5$, $\mathrm{\Lambda }=0.45$, $W=100$, and $k_{\mathrm{B}}T=0.01$; the green data points are for a fixed $\mu$ with a variation of $B_z$ and the blue points are for a fixed $B_z$ with a variation of $\mu$. The data are antisymmetrized, as indicated, to eliminate the background supercurrent from the flux bias. The solid curves are the analytical prediction (10), with $\kappa =0.775$ following directly from Eq. (9) (no fit parameters). The $B_z$-dependent data is also shown with a zoom in to very small magnetic fields, down to ${10}^{-7}{a}_0^{-2}h/e$, to demonstrate that the linear $B_z$ dependence continues when $l_m>W$.

Figure 5

Same as Fig. 4 but using the current-biased circuit shown in the inset. No antisymmetrization of the data is needed because the measured current is perpendicular to the current bias.