Universal observable as a signal of chiral anomaly in lattice Weyl fermions

Abstract

The Adler-Bell-Jackiw chiral anomaly is shown to retain its Lorentz-invariant form, $\partial_μJ^μ_5 \propto \mathbf{E} \cdot \mathbf{B}$, in lattice Weyl systems beyond moderate magnetic fields, where neither Lorentz nor rotational symmetry is present. We show that the longitudinal and Hall magnetoconductivities factorize into a product of a universal part, governed by the chiral anomaly, and a non-universal part that depends on the density of states at the Fermi level. A rotationally invariant observable $\varkappa = σ(c_V/T)^2$ is introduced as a robust signature of the anomaly, where $σ$ denotes the Euclidean norm of the longitudinal and Hall conductivities and $c_V$ is the specific heat density. This quantity follows a universal $B^2$ dependence and scales as $|\cosΘ|$, with $Θ$ being the angle between $\mathbf{E}$ and $\mathbf{B}$. Through analytical derivation and full numerical simulation, we establish that $\varkappa$ remains universal independent of system parameters and of the orientation of the magnetic or electric field for fixed $Θ$. The emergent SO(3) symmetry in $\varkappa$ persists despite the absence of isotropy in both the microscopic model and the low-energy effective theory.

Figures (3)

• Figure 1: a,b:The eigenvalue $E_n(k_y',k_z)$ of different band. ($n=0$(a) is the chiral band, $n=1$(b) is the first band above chiral band). We fix $J_z=1,J_{\perp}=5,m=0.5,\Theta=0.1\pi,\Phi=0.25\pi,\nu_B=0.005$, and fix the cut of operator $\hat{a}$ to be 400 in the occupation representation. c:Schematic illustration of the chiral anomaly mechanism in the lattice. d:Band dispersion along the magnetic field direction.
• Figure 2: (a):We fix $\mu/J_z=0.3,m=0.5,\gamma/J_z=0.3,J_x/J_z=10$.The relationship between longitudinal conductivity and magnetic field strength when both electric and magnetic fields are aligned in the z-direction. The real line is obtained through analytical approximation. The qualitative difference between the blue and black lines reflects the correction to conductivity induced by vacuum shift due to nonlinear effects. (b): The relationship between longitudinal conductivity and the direction of the magnetic field $\hat{\mathbf{B}}=(\sin{\theta}\cos{\phi},\sin{\theta}\sin{\phi},\cos{\theta})$, while maintaining $\angle(\hat{\bf B},\hat{\bf E})=\pi/6$. (c,d): The $\sigma_L$ and $\sigma_H$ as functions of $\Theta$ under different parameters. We keep the direction of the magnetic field in the y-z plane . (e):The Dos as a function of $\theta$. (f): The temperature dependence of low-temperature heat capacity $C_{N,V}(T)$. We fix $\nu_B=0.02$ in Fig.b,f and $\mu/J_z=0.05,m=0.5,\gamma/J_z=0.3$ in (b) - (f).
• Figure 3: (a):The figure demonstrates that the same result for $\varkappa$ is obtained regardless of whether a linear approximation is made near the Weyl points or not. (b,c,d): We fix $\mu/J_z=0.05,m=0.5,\gamma/J_z=0.3$, and Keep the direction of the electric field in the z direction and the magnetic field in the y-z plane . These figure shows that $\varkappa,\tilde{\sigma}_H,\tilde{\sigma}_L$ falls into the same function of $\Theta$ for different $J_{\perp}/J_z$. The inset displays a statistical histogram of the relative errors between numerical calculations and analytical approximate expressions.