Anomalous Chiral Anomaly in Spin-1 Fermionic Systems

Abstract

Chiral anomaly is a key feature of Lorentz-invariant quantum field theories: in presence of parallel external electric and magnetic fields, the number of massless Weyl fermions of a given chirality is not conserved. In condensed matter, emergent chiral fermions in Weyl semimetals exhibit the same anomaly, directly tied to the topological charge of the Weyl node, ensuring a quantized anomaly coefficient. However, many condensed matter systems break Lorentz symmetry while retaining topological nodes, raising the question of how chiral anomaly manifests in such settings. In this work, we investigate this question in spin-1 fermionic systems and show that the conventional anomaly equation is modified by an additional nontopological contribution, leading to a nonquantized anomaly coefficient. This surprising result arises because spin-1 fermions can be decomposed into 2-flavor Weyl fermions coupled to a Lorentz-breaking, momentum-dependent non-Abelian background potential. The interplay between this potential and external electromagnetic fields generates the extra term in the anomaly equation. Our framework naturally generalizes to other Lorentz-breaking systems beyond the spin-1 case.

Figures (4)

• Figure 1: (a) Two Lorentz-violating scenarios: the $S1$–$S1$ model, consisting of two spin-1 nodes with opposite topological charges $C$, where Lorentz symmetry is broken at both nodes; and the $S1$–$2W$ model, consisting of a spin-1 node and a double-Weyl node with opposite charges, where only the spin-1 node breaks Lorentz symmetry. (b) Structure of a spin-1 node: Three spin-1 bands, together with a higher-energy flat band, emerge from two-flavor Weyl fermions via band mixing and a Lorentz-violating perturbation [Eq. (\ref{['non-Abelian3']})]. A tunable parameter $g$ generalizes this setup: the two flat bands at $g=1$ (dashed) acquire dispersion at large momentum when $g\ne 1$ (solid), but the topological charge and the linearly dispersing bands are unaffected by $g$ [Eq. (\ref{['non-Abelian4']})]. (c) Final formulation: The generalized models studied for chiral anomaly, corresponding to the $S1$–$S1$ and $S1$–$2W$ cases shown in (a) [Eqs. (\ref{['eff hamiltonian']}) and (\ref{['effs12w']}), respectively].
• Figure 2: Anomaly--induced longitudinal magnetoconductivity $\Delta\sigma_L\propto n'^2$ as a function of the non--topological coupling $g$. The $S1-S1$ and $S1-2W$ cases both show dependence on $g$ arising from broken Lorentz symmetry, unlike the Lorentz symmetric $2W-2W$ case, shown in dashed for comparison.
• Figure 3: A few low-energy Landau levels for (a) conventional Weyl and (b) 2-flavor Weyl fermions. The figure illustrates the fact that the number of chiral Landau levels crossing the Fermi level at zero energy is one in (a) and two in (b), which are equal to their respective flavor numbers. In each case, we show only one chiral node. The node with the opposite chirality (not shown) is identical except for the chiral mode(s) having equal but opposite slope.
• Figure 4: (a) Landau levels (LLs) for spin-1 fermions at $g=1$. The low-energy levels can be clubbed into three kinds: one chiral mode (shown in red), similar to the Weyl case, crossing zero energy; two other bands (shown in blue) meeting the Fermi level asymptotically at large $k_z$, sharing the same sign of slope as the chiral mode, though their slope is energy-dependent; and an infinite number of bands (we show only a few here in cyan) with slope of opposite sign crossing the zero energy. (b-d) show the evolution of the low-energy Landau levels as $g$ is changed from $1$ to $0$. The Weyl-like chiral mode in red remains unaffected, but the two other kinds of levels deform on changing $g$ (for clarity, we have shown only one level in cyan).