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

1. Problem Statement

The Adler-Bell-Jackiw (ABJ) chiral anomaly describes the non-conservation of chiral current in quantum field theory, typically expressed as $\partial_\mu J^\mu_5 \propto \mathbf{E} \cdot \mathbf{B}$. While this phenomenon is well-established in continuum theories with Lorentz symmetry, its manifestation in lattice systems (condensed matter) presents challenges:

• Symmetry Breaking: Lattice Weyl fermions lack continuous Lorentz and rotational ($SO(3)$) symmetries at the microscopic level due to anisotropic dispersion and lattice discretization.
• Experimental Ambiguity: Previous signatures, such as negative magnetoresistance (NMR), rely on semiclassical approximations and long-range disorder. Recent experiments show complex, anisotropic behaviors that deviate from semiclassical predictions. Short-range disorder can alter magnetoresistance dependence, making longitudinal conductivity ($\sigma_L$) alone an unreliable probe for the chiral anomaly.
• Core Question: Does the fundamental form of the anomaly equation remain invariant in lattice systems without Lorentz symmetry, and can a robust, universal experimental signature be identified that is independent of material-specific parameters?

2. Methodology

The authors employ a combination of analytical derivation and full numerical simulation using a lattice Weyl fermion model.

• Model: A tight-binding Hamiltonian for Weyl fermions on a lattice with two Weyl points of opposite chirality. The model explicitly breaks $SO(3)$ rotational symmetry ($J_\perp \neq J_z$) and Lorentz invariance.
• Theoretical Framework:
  • Gauge Choice: The system is analyzed under a strong magnetic field $\mathbf{B}$ using a rotated Landau gauge to align the vector potential with the field direction.
  • Quantum Limit: The analysis focuses on the regime where the Fermi level crosses only the lowest (chiral) Landau level.
  • Disorder: Short-range disorder is modeled via a delta-correlated potential, treated using the Self-Consistent Born Approximation (SCBA) within the Kubo formalism.
• Numerical Approach:
  • Exact diagonalization of the Hamiltonian in the presence of the magnetic field.
  • Calculation of longitudinal ($\sigma_L$) and Hall ($\sigma_H$) magnetoconductivities.
  • Computation of the Density of States (DOS) and specific heat density ($c_V$).

3. Key Contributions and Results

A. Invariance of the Anomaly Equation

Despite the absence of Lorentz and rotational symmetries in the microscopic lattice model, the authors prove that the chiral anomaly equation retains its universal form:
$$\partial_\mu J^\mu_5 = \frac{e^2}{2\pi^2\hbar^2} \mathbf{E} \cdot \mathbf{B}$$
This holds true in the quantum limit, provided the lattice constants are equal in all directions. The derivation shows that the anomaly arises from UV physics (topological charge conservation) and is robust against strong lattice effects and anisotropic dispersion.

B. Non-Universality of Conductivity

The study reveals that the raw longitudinal ($\sigma_L$) and Hall ($\sigma_H$) conductivities are non-universal:

• They depend heavily on microscopic parameters (e.g., $J_\perp/J_z$, $m$), the direction of the magnetic field, and nonlinear dispersion effects.
• Nonlinear dispersion induces a "vacuum shift" (via the parameter $\nu_B$), modifying the magnetic field dependence of the DOS and, consequently, the conductivity.
• The conductivities factorize into a product of a universal geometric part (dependent on $\mathbf{E}$ and $\mathbf{B}$ angles) and a non-universal part proportional to the Density of States (DOS) at the Fermi level ($\rho(\mu)$).

C. Proposal of a Universal Observable ($\kappa$)

To eliminate the material-specific DOS dependence, the authors propose a rotationally invariant observable $\kappa$:
$$\kappa = \sigma \left( \frac{c_V}{T} \right)^2$$
Where:

• $\sigma = \sqrt{\sigma_L^2 + \sigma_H^2}$ is the Euclidean norm of the conductivities.
• $c_V$ is the specific heat density. At low temperatures, $c_V/T \propto \rho(\mu)$, effectively canceling the DOS factor in the conductivity.

Key Properties of $\kappa$:

1. Universal Scaling: $\kappa$ exhibits a universal $B^2$ dependence.
2. Angular Dependence: $\kappa \propto |\cos \Theta|$, where $\Theta$ is the angle between $\mathbf{E}$ and $\mathbf{B}$.
3. Emergent Symmetry: Although the microscopic model and low-energy effective theory lack $SO(3)$ symmetry, $\kappa$ displays an emergent $SO(3)$ symmetry. It remains invariant regardless of the orientation of the fields or the specific lattice parameters, provided $\Theta$ is fixed.

4. Significance

• Robust Signature: The paper provides the first theoretically rigorous, material-independent signature of the chiral anomaly in lattice systems. By combining transport ($\sigma$) and thermodynamic ($c_V$) measurements, experimentalists can distinguish the anomaly from other magnetotransport effects caused by disorder or anisotropy.
• Theoretical Insight: It demonstrates that the topological nature of the chiral anomaly is robust enough to survive the explicit breaking of Lorentz and rotational symmetries in condensed matter systems.
• Experimental Guidance: The proposed observable $\kappa$ offers a clear target for experimental verification in Weyl semimetals (e.g., TaAs, NbP), resolving ambiguities in previous magnetoresistance studies where angular anisotropy complicated the interpretation of the anomaly.

In summary, the work establishes that while raw conductivity is sensitive to lattice details, a specific combination of conductivity and specific heat yields a universal quantity that faithfully reflects the underlying chiral anomaly, exhibiting emergent rotational symmetry in a non-rotational system.