Dirac versus Weyl Fermions in Topological Insulators: Adler-Bell-Jackiw Anomaly in Transport Phenomena

Dirac metals (gapless semiconductors) are believed to turn into Weyl metals when perturbations, which break either time reversal symmetry or inversion symmetry, are employed. However, no experimental evidence has been reported for the existence of Weyl fermions in three dimensions. Applying magnetic fields near the topological phase transition from a topological insulator to a band insulator in ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ we observe not only the weak antilocalization phenomenon in magnetoconductivity near zero magnetic fields ($B<0.4\mathrm{T}$), but also its upturn above 0.4 T only for $E//B$. This “incompatible” coexistence between weak antilocalization and “negative” magnetoresistivity is attributed to the Adler-Bell-Jackiw anomaly (“topological” $E·B$ term) in the presence of weak antilocalization corrections.

Figure 1

(a) A topological phase transition occurs from a band insulator ($x<3\%$) to a topological insulator ($x>3\%$) around $x\approx 3\%$, where band touching emerges to form a Dirac cone at the $L$ point. Applying a magnetic field $B$ to this Dirac metal, the Dirac point described by the four-component Dirac spinor is split into two Weyl points described by the two-component Weyl spinors with opposite chirality (red, $\mathrm{\text{blue}}=\mathrm{\text{Weyl}}$, $\mathrm{\text{green}}=\mathrm{\text{Dirac}}$). The distance between these two Weyl points is proportional to $B$. The right figure depicts the structure of the Dirac points at $B=0$ and that of the Weyl points at $B\ne 0$ in the reciprocal space, where the Dirac points split along the direction of $B$. (b) Schematic diagram of the electrical transport measurements.

Figure 2

(a) Angle-dependent MR. “Negative” MR appears around $\theta =90°$, which originates from the Adler-Bell-Jackiw anomaly of Weyl fermions. (b) The scaling property of MR in the low-field region. The $x$ axis is the perpendicular field component of $B\cos \theta$. This dip was attributed to the three-dimensional weak antilocalization. (c) Longitudinal and transverse MR for a Bi single crystal. We could not find any negative signals in the longitudinal MR. (d) The transverse and longitudinal MR measured up to 50–60 T in a pulse magnet.

Figure 3

(a) Transverse magnetoconductance. The black circles represent the experimental data and the red lines express the theoretical fitting, where both weak antilocalization corrections and normal contributions from other bands are considered. (b) Longitudinal magnetoconductance. The overall factor ${\mathit{C}}_W{\mathit{B}}^2$ in front of weak antilocalization corrections is the key feature, which originates solely from the topological $E·B$ term in both the quantum Boltzmann equation and the equation of motion approaches. This is the fingerprint of the Weyl metallic state.

Figure 4

(a) Magnetoresistance of ${\rho }_{xx}(B_z)$ and ${\rho }_{xx}(B_y)$. (b) The ${\rho }_{yx}(B_z)$ and ${\rho }_{yx}(B_y)$ are compared. (c) Comparison of ${\rho }_{yx}(B_z)$ and ${\rho }_{yx}(B_x)$. The signal in the latter coefficient is nonzero because of the topological Hall effect, which results from the topological $E·B$ term. This topological Hall effect is regarded as another signature for the Weyl fermions. (d) Transport coefficient ${\rho }_{yx}(B_x)$ with the theoretical fitting based on the formula $a·B+b·B^3$.