Anomalous Chiral Anomaly in Spin-1 Fermionic Systems

1. Problem Statement

The chiral anomaly is a fundamental phenomenon in quantum field theory where the conservation of chiral charge is violated in the presence of parallel electric ($\vec{E}$) and magnetic ($\vec{B}$) fields. In Lorentz-invariant systems (like high-energy physics or standard Weyl semimetals), the anomaly equation is:
$$\partial_\mu J^\mu_5 - n \frac{e^2}{2\pi^2} \vec{E} \cdot \vec{B} = 0$$
where $n$ is the number of Weyl fermion flavors, which corresponds to the topological charge of the Weyl node. This coefficient $n$ is quantized (an integer).

The Core Question: Many condensed matter systems break Lorentz symmetry (e.g., due to lattice effects or specific band structures) while retaining topological nodes. Previous studies on multi-Weyl semimetals suggested the anomaly remains quantized. However, the behavior of the chiral anomaly in spin-1 fermionic systems (which feature three bands touching at a point: two linearly dispersing and one flat) remains unexplored. The authors investigate whether the breakdown of Lorentz symmetry in these systems alters the quantization of the anomaly coefficient.

2. Methodology

The authors employ a combination of effective field theory construction and the Fujikawa path integral method to derive the anomaly equation.

• Model Construction:
  • They consider a spin-1 fermionic system described by the Hamiltonian $H = \vec{S} \cdot \vec{k}$, where $\vec{S}$ are $SO(3)$ generators. This system has a topological charge of $\pm 2$ but includes a non-topological flat band.
  • They analyze two configurations:
    1. $S_1-S_1$ System: Two spin-1 nodes with opposite topological charges.
    2. $S_1-2W$ System: One spin-1 node and one double-Weyl node (asymmetric).
• Hamiltonian Decomposition:
  • The key insight is decomposing the spin-1 Hamiltonian into a Lorentz-invariant part (equivalent to a 2-flavor Weyl system) and a Lorentz-symmetry-breaking term.
  • They introduce a tunable parameter $g$ to generalize the Hamiltonian: $H_{S1}^{\lambda, g} = H_{2W} + h_{LB}^{\lambda, g}$.
  • Crucially, they demonstrate that the Lorentz-breaking term $h_{LB}$ can be interpreted as a momentum-dependent non-Abelian background potential ($A_\mu^a$) coupled to the 2-flavor Weyl fermions.
• Derivation of Anomaly:
  • Using the Fujikawa method, they calculate the change in the path integral measure under chiral transformations.
  • They compute the Jacobian of the transformation, which yields the anomaly equation. The calculation involves evaluating the trace of the operator $\Gamma_5 e^{-(\not{D})^2/M^2}$ in the limit $M \to \infty$.
  • They account for the non-Hermitian nature of the Dirac operator in the asymmetric ($S_1-2W$) case by using Laplacian eigenstates.

3. Key Contributions

1. Identification of Non-Topological Contribution: The paper establishes that in spin-1 systems, the chiral anomaly equation acquires an additional non-topological term arising from the momentum dependence of the emergent non-Abelian potential.
2. Loss of Quantization: Unlike standard Weyl or multi-Weyl semimetals where the anomaly coefficient is strictly quantized by the topological charge, the coefficient in spin-1 systems becomes non-quantized and dependent on the parameter $g$ (which controls the dispersion of the flat band).
3. Distinction from Multi-Weyl Semimetals: The authors clarify why multi-Weyl semimetals retain quantization (their effective non-Abelian potentials are momentum-independent, leading to vanishing commutators) whereas spin-1 systems do not (momentum-dependent potentials lead to non-zero commutators).
4. Asymmetry Effects: They quantify how nodal asymmetry (in the $S_1-2W$ case) further modifies the anomaly coefficient.

4. Key Results

The derived anomaly equations for the two models are:

• For the $S_1-S_1$ System (Symmetric):
  $$\partial_\mu J^\mu_5 - (2 - 3g) \frac{e^2}{2\pi^2} \vec{E} \cdot \vec{B} = 0$$
  Here, the anomaly coefficient is $n' = 2 - 3g$.

  • When $g=1$ (original model), $n'=1$. This is a "fortuitous" cancellation between the topological charge ($n=2$) and the non-topological contribution.
  • When $g=2/3$, $n'=0$, implying the chiral current is conserved despite the presence of topological nodes.
  • For any other $g$, the coefficient is non-integer and non-quantized.

• For the $S_1-2W$ System (Asymmetric):
  $$\partial_\mu J^\mu_5 - \left(2 - \frac{3g}{2}\right) \frac{e^2}{2\pi^2} \vec{E} \cdot \vec{B} = 0$$
  Here, $n' = 2 - 1.5g$. At $g=1$, $n'=0.5$, indicating a fractional anomaly contribution due to the asymmetric coupling of the non-Abelian potential.

• Observable Consequence (Longitudinal Magnetoconductivity):
  The longitudinal magnetoconductivity (LMC), a signature of the chiral anomaly, is proportional to the square of the anomaly coefficient: $\Delta \sigma_L \propto (n')^2$.

  • In spin-1 systems, $\Delta \sigma_L$ is tunable via the parameter $g$.
  • Unlike standard Weyl systems where LMC diverges at charge neutrality (due to vanishing density of states), the flat band in spin-1 systems provides a finite density of states, resulting in finite LMC even at charge neutrality.

5. Significance

• Theoretical Impact: This work challenges the conventional wisdom that topological charge alone dictates the chiral anomaly coefficient in condensed matter. It proves that Lorentz symmetry breaking, when coupled with momentum-dependent non-Abelian fields, can decouple the anomaly from topological quantization.
• Experimental Relevance: The results provide a clear experimental signature for spin-1 semimetals (such as CoSi-family compounds). The non-quantized, tunable longitudinal magnetoconductivity serves as a direct probe of the underlying Lorentz-breaking physics and the specific band structure parameter $g$.
• Broader Context: The framework generalizes to other Lorentz-breaking systems. The authors draw parallels to non-Abelian anomalies in Quantum Chromodynamics (QCD) and the electroweak theory, suggesting a deep connection between condensed matter anomalies and high-energy gauge theories.

In summary, the paper demonstrates that in spin-1 fermionic systems, the chiral anomaly is not solely a topological invariant but is significantly modified by dynamical, non-topological factors arising from Lorentz symmetry breaking, leading to a rich, tunable phenomenology distinct from standard Weyl semimetals.