Chiral Anomaly in Quantum Systems

• Chiral anomaly is a quantum violation of axial current conservation in systems with massless fermions, driven by topological field configurations.
• It underpins key phenomena such as the chiral magnetic effect and negative magnetoresistance in Dirac and Weyl semimetals, impacting transport properties.
• The anomaly's topological robustness and clear experimental signatures connect quantum field theory, condensed matter physics, and gravitational systems.

The chiral anomaly is a profound quantum mechanical effect first identified in high-energy physics but now recognized as a central feature underlying a diverse array of phenomena in gauge field theories, condensed matter systems, fluids, and even gravitational backgrounds. It refers to the explicit quantum violation of the classical conservation law for the axial (chiral) current of massless fermions in the presence of gauge or other external fields. Operationally, this manifests as the non-conservation of left- and right-handed fermion charge, described by the anomalous divergence equation for the axial current. The coefficient of the anomaly is fixed by topological properties of the quantum fields—specifically, gauge field configurations and band-structure monopole charges—and is protected from renormalization under many circumstances, although important exceptions arise in certain interacting or Lorentz-violating backgrounds.

1. Fundamental Definition and Field-Theoretic Origin

The archetype of the chiral anomaly arises in quantum field theory for massless Dirac fermions coupled to gauge fields. The classical action is invariant under global axial rotations, yielding a conserved axial current $J_5^\mu = \bar\psi \gamma^\mu \gamma^5 \psi$. However, quantum effects (specifically, regularization of the triangle diagram) produce an explicit divergence: $$\partial_\mu J_5^\mu = \frac{e^2}{16\pi^2} \varepsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma} = \frac{e^2}{2\pi^2} \mathbf{E}\cdot\mathbf{B}$$ where $F_{\mu\nu}$ is the electromagnetic field tensor and the numerical coefficient encodes the anomaly. This quantum breakdown is nonperturbative, universal across massless fermion flavors, and intimately tied to topological indices (Atiyah-Singer theorem for the Dirac operator) (Liao, 2016).

The physical interpretation is spectral flow: in a background $E\cdot B$, left-handed fermion states are continuously transformed into right-handed ones, and vice versa, violating separate conservation of these charges. In QCD, the anomaly is responsible for processes such as instanton-induced chirality change.

2. Macroscopic Manifestations: Chiral Magnetic Effect and Transport

When systems with chiral fermions are placed in external fields, the chiral anomaly leads to distinctive transport effects. The most studied is the Chiral Magnetic Effect (CME), wherein a vector (electric) current is generated along the direction of an applied magnetic field, proportional to a chiral chemical potential $\mu_5$: $J^\mu = C_A \mu_5 B^\mu$ with $C_A$ the anomaly coefficient (Liao, 2016). This current is non-dissipative and arises from the requirement of the Second Law in anomalous hydrodynamics (Son & Surowka formulation).

Hydrodynamic models incorporating anomalies, such as Anomalous-Viscous Fluid Dynamics (AVFD), combine background viscous hydrodynamics (e.g., VISHNU) with coupled anomalous fluid equations for RH and LH currents (Liao, 2016). These frameworks govern the time evolution of charge separation, relaxational corrections, and the interplay with viscous transport.

In condensed matter, the anomaly sets the foundation for phenomena in Dirac and Weyl semimetals, including negative longitudinal magnetoresistance and planar Hall effects, observable experimentally through magnetotransport and charge separation signals (Nandy et al., 2017).

3. Topological Structure, Index Theorems, and Robustness

While the basic anomaly equation is fixed by short-distance physics, recent results have revealed a deeper topological factorization. For generalized Hermitian Dirac operators—including higher-order, inhomogeneous, or momentum-dependent forms—the global anomaly is given by (Xavier et al., 5 Jun 2025): $$A_\text{global} = N_3 \times \frac{1}{8\pi^2} \int \mathrm{tr}(F \wedge F)$$ where $N_3$ is a phase-space (momentum-space) topological invariant (Volovik’s invariant), responsible for topological stability of Fermi points/surfaces and non-dissipative transport (e.g., chiral separation effect).

The universality of the anomaly coefficient can break down in Lorentz-violating or interaction-dependent backgrounds. In spin-1 (higher-spin) fermionic systems, non-quantized anomaly coefficients result from nontopological contributions arising from momentum-dependent background fields mixed with the gauge sector (Mukherjee et al., 20 Mar 2025). In interacting 1D systems (Luttinger liquids), the anomaly acquires susceptibility-dependent factors that reflect the interplay of short-distance regularization and interactions (Rylands et al., 2021, Wang et al., 17 Jan 2024).

4. Extensions and Generalizations: Fermi Surface, Non-Fermionic, and Gravitational Anomalies

The traditional view tied the anomaly to point-like Weyl nodes, but recent work demonstrates that Berry curvature flux through the Fermi surface suffices—even in systems lacking paired Weyl points. In noncentrosymmetric and spin-orbit-coupled metals, the chiral anomaly is a Fermi-surface property, with transport signatures (negative longitudinal magnetoresistance, planar Hall effect) dictated by the net Berry flux piercing the Fermi surface (Cheon et al., 2022, Das et al., 2023).

Remarkably, non-fermionic systems such as magnon bands in 2D honeycomb magnets exhibit (1+1)D chiral anomalies when subjected to edge, nonuniform field, or strain, leading to controlled imbalance of chiral magnon populations and negative strain-resistivity (Liu et al., 2023). The underlying structure remains a spectral flow between chirality sectors, generalized beyond standard Fermi statistics.

Anomaly-induced transport extends to fluids: classical, stationary Beltrami flows in Euler fluids possess the anomaly (with coefficient 2 per Dirac fermion), driving equilibrium currents of both chiral and electric type and demonstrating that this effect is fundamentally rooted in central extensions of phase-space algebras (Wiegmann et al., 2022).

Gravitational anomalies arise for chiral fermions in curved spacetime. In the presence of dynamical, circularly-polarized gravitational waves, the nonconservation of axial charge is tied to the gravitational Pontryagin density $R\wedge R$, leading to net chiral flux production proportional to the Stokes-V parameter of the radiative field (Rio, 2021).

5. Experimental Probes and Observables

The macroscopic consequences of the chiral anomaly are observable in a variety of experiments:

• Heavy ion collisions: Charge separation along magnetic fields measured via azimuthal correlations; beam-energy scans and isobaric collision strategies are used to isolate anomalous contributions from background flow (Liao, 2016).
• Weyl/Dirac semimetals: Magnetoresistance and Hall measurements, including squeezing tests to exclude current jetting in multifold fermion systems (Balduini et al., 30 Apr 2024); planar Hall angular dependence confirms anomaly-driven origin (Nandy et al., 2017).
• Quantum dots and edges: Real-space chiral anomaly realized by fractional charge transfer in helical edge quantum spin Hall setups, read out via single-electron transistors (Fleckenstein et al., 2016).
• Phonon dynamics: Chiral anomaly resonances in A$_1$ phonons of Weyl semimetals, visible in infrared reflectivity and Raman scattering as magnetic-field-induced modes (Rinkel et al., 2016).
• Magnon and spin transport: Nonlocal magnon chiral anomaly and negative strain-resistivity in 2D magnets; spin Nernst and magnetic spin Hall effects in SOC metals (Das et al., 2023, Cheon et al., 2022).

6. Lattice, Finite-System, and Non-Hermitian Considerations

On a finite lattice, the chiral anomaly survives as an operator-level nonconservation, implemented by gauging (typically) Z$_2$ fermion parity rather than U(1), as larger symmetry groups encounter quasi-anomalies or full projection of matter states due to volume-dependent consistency conditions (Radicevic, 2018). The anomaly is caused by symmetry-broken, deeply bound states far below the Fermi level, and is perfectly quantized at the Fermi surface when local chiral symmetry is preserved (Wang et al., 10 Sep 2025).

In non-Hermitian systems—effective descriptions of dissipation—the anomaly structure is altered, with modifications to the functional form and normalization coefficients, and novel terms arising from the mixing of gauge fields and their adjoints. This has direct consequences for anomalous Hall currents, Chern–Simons couplings, and the quantization of topological magnetoelectric effects (Sayyad et al., 2021).

7. Controversies, Robustness, and Theoretical Frameworks

Robustness of the anomaly coefficient—its unrenormalization under interactions—is tightly linked to the topological nature of the underlying symmetry (mixed anomalies between U(1) and translation in solids) (Wang et al., 17 Jan 2024). However, adiabatic vs. non-adiabatic formulations (Berry monopole at all timescales vs. exact path-integral regularization) can yield artifacts or spurious anomalies, as highlighted by Fujikawa’s analysis (Fujikawa, 2017). Consistent vs. covariant anomaly formulations require careful consideration, especially in the context of tight-binding or lattice regularization.

In summary, the chiral anomaly unifies quantum-field-theoretic, topological, and transport perspectives across high-energy, condensed-matter, and classical systems. Its implications range from quantum number nonconservation to interaction-independent topological responses, from signal signatures in collision or magnetotransport experiments to conceptual advances in the theory of anomalies. The precise nature and quantization of the anomaly coefficient are determined by the interplay of symmetry, topology, and regularization, making the phenomenon central to the current frontier of research in physics.