Anomaly-Induced Transport Phenomena

• Anomaly-induced transport phenomena are macroscopic effects where quantum anomalies generate non-dissipative currents in systems like quark–gluon plasma and Weyl semimetals.
• They include key effects such as the chiral magnetic effect and chiral vortical effect, with coefficients uniquely fixed by anomaly coefficients and topological invariants.
• Holographic models and experimental observations from heavy-ion collisions to condensed matter systems validate these transport mechanisms and their underlying topological nature.

Anomaly-induced transport phenomena are macroscopic manifestations of quantum anomalies in relativistic fluids and quantum materials. These phenomena encompass a range of nondissipative currents, including the chiral magnetic effect, chiral vortical effect, and related magneto- and thermoelectric responses, whose existence and magnitude are fixed by the fundamental anomaly coefficients of the underlying theory. Central to recent theoretical and experimental developments, these effects bridge quantum field theory, hydrodynamics, holography, and condensed matter physics. The presence of explicit symmetry breaking and interplay between different anomaly sources further enriches the landscape of anomaly-induced transport, expanding its influence to non-anomalous current sectors and strongly coupled systems such as the quark–gluon plasma and Weyl semimetals.

1. Quantum Anomalies and Macroscopic Transport

Quantum anomalies—specifically gauge (triangle) and mixed gauge–gravitational anomalies—arise from the non-conservation of chiral or axial currents in the quantum effective action in the presence of external gauge fields or curved spacetime. In four dimensions, the canonical triangle anomaly relates the non-conservation of the axial current to the topological density $F_{\mu\nu}\tilde F^{\mu\nu}$, while the mixed anomaly couples gauge fields to the gravitational curvature tensor.

These microscopic anomalies manifest macroscopically as nondissipative transport currents in the hydrodynamics of chiral fluids. The green function/topological invariants approach (Zubkov et al., 2018) unifies these responses as first-order derivatives of the effective action with respect to background gauge fields, with all such currents quantized by momentum-space topological invariants (e.g., Chern numbers or node separation in Weyl systems).

For a system of Weyl fermions, the anomaly-induced contributions to the charge and energy currents (in equilibrium and to first order in gradients) take the form: $$J^i_{\text{anom}} = \sigma_B B^i + \sigma_V \omega^i$$ with chiral magnetic and vortical conductivities that depend solely on chemical potential, temperature, and the anomaly coefficients: $$\sigma_B = \sum_i \chi_i \frac{q_i^2}{4\pi^2} \mu,\quad \sigma_V = \sum_i \chi_i \left[\frac{q_i^3}{4\pi^2} \mu^2 + \frac{q_i}{12} T^2 \right]$$ (Loganayagam et al., 2012). The $T^2$ term in $\sigma_V$ directly signals the presence of the mixed gauge–gravitational anomaly (Landsteiner et al., 2011).

2. Principal Anomaly-Induced Transport Effects

2.1 Chiral Magnetic Effect (CME)

The CME is the generation of an electric current parallel to an applied magnetic field in the presence of a chiral chemical potential or imbalance. The CME current for a single Dirac fermion is (Kharzeev, 2013): $$\vec{j}_\text{CME} = \frac{e^2}{2\pi^2} \mu_5 \vec{B}$$ This effect is topologically protected and non-dissipative: its coefficient cannot be renormalized by interactions (Adler–Bardeen non-renormalization) and is robust even at strong coupling, as confirmed by holographic calculations (Pena-Benitez, 2013).

2.2 Chiral Vortical Effect (CVE)

The CVE is the induction of an electric or axial current along the vorticity vector in a rotating relativistic fluid. Its coefficient contains both a chemical-potential–dependent piece (from the triangle anomaly) and a temperature-squared piece (from the mixed gauge–gravitational anomaly): $$\vec{j}_\text{CVE} = \frac{\mu^2}{4\pi^2} \vec{\omega} + \frac{T^2}{12} \vec{\omega}$$ (Hongo et al., 2019, Landsteiner et al., 2011). The gravitational $T^2$ term can be traced holographically to contributions localized at the event horizon of charged black-brane backgrounds (Landsteiner, 2016).

2.3 Anomalous Hall and Chiral Separation Effects

Anomalous Hall currents in Weyl materials are fixed by band structure: in a time-reversal–breaking Weyl metal with node separation $b_z$, the intrinsic Hall conductivity is

$\sigma_{xy} = \frac{e^2}{2\pi^2} b_z$

(Burkov, 2015). The chiral separation effect (CSE) generates an axial current in response to a vector chemical potential and magnetic field, with

$\vec{j}_\text{CSE} = \frac{\mu}{2\pi^2} \vec{B}$

(Zubkov et al., 2018).

2.4 Magneto- and Thermoelectric Responses

Anomaly-induced transport manifests in anomalous magnetoconductivities, Nernst and thermal Hall effects, and Ettingshausen/Righi-Leduc effects in Weyl semimetals (Zeng et al., 2020, Das et al., 2019). These phenomena feature unique scaling with field strength (e.g., $\sim B^2$ for planar Hall/Nernst), angle, and temperature, often violating conventional Mott and Wiedemann-Franz relations.

3. Holographic Realizations and Broken Symmetries

Holographic duality provides a nonperturbative framework for computing anomaly-induced transport coefficients at strong coupling. The five-dimensional bulk action typically includes Einstein-Maxwell dynamics, pure gauge and mixed gauge–gravitational Chern-Simons terms, and, for explicit symmetry breaking, a charged scalar field (Tamang et al., 11 Nov 2025).

When global symmetries (e.g., axial $U(1)_A$ and an extra $U(1)_W$) are explicitly broken via a scalar source, the anomaly-induced transport is redistributed between anomalous and non-anomalous current sectors:

• In the unbroken (symmetric) limit, non-anomalous (e.g., $J_W$) conductivities vanish, and all anomaly-induced transport resides in the anomalous sector.
• In the strong-breaking limit, only a specific linear combination of broken currents survives, and anomaly-induced conductivities for non-anomalous currents reach half the value of the corresponding anomalous conductivities in the symmetric case.
• At finite breaking, conductivities interpolate monotonically, reflecting a smooth redistribution of anomaly-induced transport (Tamang et al., 11 Nov 2025).

The analytic expressions for the full matrix of chiral magnetic ($\sigma^B$) and chiral vortical ($\sigma^V$) conductivities depend explicitly on chemical potentials, temperature, anomaly coefficients, and the symmetry breaking mass $M$, as detailed in the following table for selected cases:

Conductivity	$M=0$ (Symmetric)	$M\to\infty$ (Strong Breaking)
$\sigma_{va}^B$	$\frac{\mu_v}{2\pi^2}$	$\frac{\mu_v}{4\pi^2}$
$\sigma_{wv}^B$	$0$	$\frac{\mu_v}{4\pi^2}$
$\sigma_{vv}^B$	$\frac{\mu_a}{2\pi^2}$	$\frac{\mu_a+\mu_w}{4\pi^2}$

Here, as $M$ increases, anomaly-induced transport "leaks" from the originally anomalous to the non-anomalous sector, with all conductivities exhibiting distinct $M$ dependence.

4. Methodologies: Kubo Formulae, Hydrodynamics, and Topology

Transport coefficients are typically obtained via Kubo formulae relating zero-frequency, zero-momentum limits of retarded correlators to the conductivities: $$\sigma^B_{sb} = \lim_{p_k \to 0} \frac{i}{2p_k} \epsilon_{ijk} \langle J^i_s J^j_b \rangle_{\omega=0}$$

$$\sigma^V_s = \lim_{p_k \to 0} \frac{i}{2p_k} \epsilon_{ijk} \langle J^i_s T^{0j} \rangle_{\omega=0}$$

The boundary U(1) and energy currents are defined holographically via differentiating the renormalized on-shell action with respect to boundary values of gauge fields and the metric (Tamang et al., 11 Nov 2025). In equilibrium hydrodynamics, the anomaly-induced terms appear at first order in derivative expansion and are dictated by entropy constraints (Kharzeev, 2013, Landsteiner, 2016). The full classification of possible anomaly-induced effects includes higher-order gradient corrections, collective excitations such as the chiral magnetic and chiral Hall waves, and nonlinear couplings between sectors (Bu et al., 2018, Bu et al., 2016).

The topological/fundamental origin of anomaly-induced transport is formalized through Wigner-transform and Green function approaches, establishing quantization and non-renormalization of the coefficients (Zubkov et al., 2018).

5. Physical Implications and Experimental Relevance

Anomaly-induced transport is realized across a diverse range of systems:

• Quark-gluon plasma (QGP): Charge separation and collective flows in heavy-ion collisions reflect the CME, CVE, and their higher-order generalizations, serving as probes for QCD anomaly dynamics (Buzzegoli et al., 2022, Kharzeev, 2013).
• Weyl and Dirac semimetals: Negative longitudinal magnetoresistance, anomalous Hall conductivity, planar Hall/Nernst, and nonlinear responses are unambiguous signatures of chiral anomaly transport, confirmed experimentally (Burkov, 2015, Das et al., 2019, Zeng et al., 2020, Gopalakrishnan et al., 27 May 2025).
• Strongly correlated or symmetry-broken systems: Holographic models show that explicit symmetry breaking redistributes anomaly-induced current into otherwise non-anomalous sectors—this effect can be significant in system with approximate symmetries, motivating targeted experiments in e.g. nodal metals or strongly coupled ultracold gases (Tamang et al., 11 Nov 2025).

Key experimental signatures include specific scaling relations (field, angle, temperature), non-monotonic dependence on topological charge, violation of conventional thermoelectric relations, as well as redistributed response upon symmetry breaking.

6. Interplay with Strong Coupling and Disorder

Anomaly-induced transport coefficients are protected against renormalization and are independent of the strength of interactions, as demonstrated both at weak coupling and in holographic (strong coupling) approaches (Pena-Benitez, 2013, Megias, 2018). Even in the presence of strong momentum relaxation (disorder, massive gravity), the chiral magnetic and vortical conductivities remain set by anomaly coefficients, whereas ordinary electric conductivity is suppressed or vanishes (Megias, 2018).

The universality of these results is seen in the detailed classification of responses, the persistence of topological quantization, and a host of nontrivial checks between field theory, holography, and experimental observations.

7. Theoretical Extensions and Open Directions

The current understanding of anomaly-induced transport encompasses linear and nonlinear phenomena, boundary-localized responses in conformal field theories (1804.01648), topological protection mechanisms, and a hierarchy of higher-order effects. Explicit symmetry breaking introduces a channel for anomaly-induced currents to affect non-anomalous sectors; this suggests broader implications for engineering quantum transport in synthetic and real materials.

Future directions include systematic exploration of these effects in time-dependent or out-of-equilibrium settings, coupling to dynamical gauge fields (chiral magnetohydrodynamics), incorporation of spatial inhomogeneity and strong disorder, and extension to multi-anomaly scenarios or crystalline topological systems.

Theoretical advances in holographic model building, nonperturbative field theory, and topological methods continue to refine and expand the anomaly–transport correspondence and its experimentally accessible manifestations.