Chiral Vortical Effects in Quantum Media

• Chiral Vortical Effects are macroscopic quantum phenomena in chiral media that generate directed currents along rotational axes due to the interplay of chemical potentials, vorticity, and quantum anomalies.
• Theoretical studies employ hydrodynamic, kinetic, and holographic methods to derive coefficients from gauge and gravitational anomalies, elucidating both temperature and density contributions.
• Applications span heavy-ion collisions, early-universe magnetogenesis, and condensed matter systems, with experimental signatures including hyperon polarization and induced magnetic fields.

The chiral vortical effect (CVE) is a macroscopic quantum phenomenon in chiral media—systems with particles of definite handedness—whereby a rotating, chirally asymmetric matter generates a nondissipative current along the axis of rotation. This axial or vector current arises from the interplay of quantum anomalies, vorticity, finite chemical potentials, and, in some cases, temperature effects. The CVE is a universal feature of chiral matter, with ramifications in particle physics (quark–gluon plasma), cosmology (magnetogenesis and baryogenesis), condensed matter (Weyl and Dirac semimetals), and astrophysics (compact star jets and hyperon polarization).

1. Theoretical Foundations and Anomaly Structure

The CVE originates from the anomalous nonconservation of chiral current in the presence of rotation—analogous to the chiral magnetic effect (CME) but with fluid vorticity $\omega^\mu$ rather than magnetic field $B^\mu$ as the source. For a relativistic fermion system with chemical potentials $\mu$ (vector) and $\mu_5$ (axial), at temperature $T$, the equilibrium CVE current can be written as:

$$j_5^\mu = C\, (\mu^2 + \mu_5^2)\, \omega^\mu + C_T\, T^2\, \omega^\mu$$

where $C$ is determined by the gauge anomaly, while $C_T$ can depend on microscopic details and often reflects contributions from the mixed gauge–gravitational anomaly (Kalaydzhyan, 2014, Mitkin et al., 2021, Huang et al., 2018). For a single Weyl fermion: $C = \frac{1}{2\pi^2}, \qquad C_T = \frac{1}{12}$

The general structure for the vector and axial currents in the presence of vorticity is:

Current	Coefficient (CVE)	Dependence
$j_A^\mu$	$C_T T^2 + C (\mu^2+\mu_5^2)$	Thermally-induced and chemical-potential-induced
$j_V^\mu$	$C_V \mu\mu_5$	Requires both vector and axial imbalance

Rotational response is rooted in the interplay of the quantum triangle anomaly (for the $\mu$-dependent part) and the gravitational anomaly (for the $T^2$ part) (Mitkin et al., 2021, Flachi et al., 2017).

2. Microscopic and Effective Descriptions

Relativistic Fluids and Hydrodynamics

Linear response/Kubo approaches and anomaly-matching (chemical shift) arguments yield the same result for the leading vortical conductivity (Mitkin et al., 2021). The axial current in an idealized chiral fluid in flat spacetime is:

$$\langle j_5^\mu(x) \rangle = n_5 u^\mu + \sigma_\omega \omega^\mu, \qquad \sigma_\omega = \frac{\mu^2}{4\pi^2} + \frac{T^2}{12}$$

where $u^\mu$ is the fluid four-velocity. In equilibrium and in the absence of acceleration or temperature gradients, the fluid helicity current and the microscopic axial current are separately conserved.

Anomalous Transport and Kinetic Theory

Chiral kinetic theory, extended to arbitrary particle spin, generalizes the CVE to massless particles of any spin $s$ (Huang et al., 2018):

$$\bm J_\chi = \hbar s \bm\Omega \Big[ \frac{\mu_+^2 + \mu_-^2}{4\pi^2} + \frac{T^2}{6} \Big]$$

For photons ($s=1$), the photonic CVE (or bosonic CVE) arises with a universal coefficient determined by the gravitational anomaly, $\xi_\omega^{(1)} = T^2/3$ (Prokhorov et al., 2020, Avkhadiev et al., 2017). Infrared regularization is crucial for consistent quantification in the massless (photon) regime.

Holographic and Strong-Coupling Realizations

Holographic field theory frameworks (AdS/CFT) provide non-perturbative access to CVE in strongly coupled systems. The 5D gravity dual incorporates both U(1)$_A$ and U(1)$_V$ gauge fields, massless scalar fields for momentum relaxation, and Chern–Simons terms encoding pure gauge and mixed gauge–gravitational anomalies (Morales-Tejera et al., 2020, Wu et al., 2016). The CVE current is extracted from the near-boundary expansion of the axial gauge field perturbation under a gravito-magnetic perturbation modeling rotation.

In equilibrium, the total vortical conductivity is:

$$\sigma_5 = 8\alpha(\mu^2+\mu_5^2) + 64\pi^2\lambda T^2$$

with $\alpha$, $\lambda$ fixed by anomaly coefficients in the bulk theory. For non-zero chemical potential but vanishing axial potential ($\mu_5=0$), this reduces accordingly (Morales-Tejera et al., 2020).

Nonconformal backgrounds (e.g., compactified D4-branes with smeared D0 charge) yield CVE with identical $\mu^2$-dependence, confirming the anomaly-governed topology of the effect independent of conformality (Wu et al., 2016).

3. Realizations, Phenomenology, and Extension to Non-equilibrium

Superfluids, Vortices, and Topological Defects

In confining phases represented by pionic superfluidity, the CVE emerges through anomaly loops with support from heavy baryons. The induced axial current is carried not by Goldstone bosons (pions) but by spin-aligned baryons within vortex cores. The standard CVE coefficient is thus controlled by the anomaly, $j_5^\mu = (\mu_I^2/2\pi^2)\omega^\mu$ for isospin chemical potential $\mu_I$ (1705.01650).

In rotating superfluids, the distinction between macroscopic and microscopic derivations becomes visible: chiral charge on vortex cores propagates at the speed of light and is not captured by a single local fluid velocity, leading to a factor-of-two enhancement in specific contexts (Kirilin et al., 2012).

In systems with Weyl nodes and topological skyrmions in k-space, mass (momentum) currents along vortices are set by the anomaly and skyrmion winding: $j_m = (1/2\pi^2)\mu\omega$ for two nodes (Volovik, 2017).

Early Universe Magnetogenesis and Baryogenesis

During the symmetric phase of the early Universe ($100~\mathrm{GeV} < T < 10~\mathrm{TeV}$), the CVE, together with the CME, plays a central role in simultaneously generating large-scale hypermagnetic fields and the matter–antimatter asymmetry (1908.10105, Abbaslu et al., 2020).

Key mechanism:

1. Vorticity fluctuations seed a hypermagnetic field via the CVE, even from zero initial field.
2. Once the seed is established, CME-driven exponential amplification takes over, saturating the field to $\sim10^{20}\,\mathrm{G}$.
3. The Abelian anomaly converts evolving hypermagnetic helicity into net lepton and baryon number.

This two-stage process is robust to rapid viscous damping of vorticity, as the initial CVE seeding occurs much faster than viscous decay.

Non-equilibrium Dynamics and Relaxation

In far-from-equilibrium strong coupling (holographic) settings, the CVE current does not instantaneously build up after a thermal quench. There is a calculable retardation time $\Delta t$ corresponding to the causal propagation from the horizon to the boundary (Morales-Tejera et al., 2020). The subsequent ring-down of the current is governed by the quasi-normal mode spectrum of the final equilibrium black brane, with the late-time decay set by the imaginary part of the lowest QNMs.

Nonperturbative Suppression and QGP

Nonperturbative gluonic interactions in the quark–gluon plasma (QGP) substantially suppress the CVE relative to its free-fermion value. Calculations in the Field Correlator Method yield a suppression factor $R(T,\mu) = I_0(T,\mu)/I_{\mathrm{free}}(T,\mu) < 1$, with $R\simeq0.3-0.5$ near RHIC/ALICE relevant parameters. This matches the phenomenological suppression required to explain the observed $\Lambda$ hyperon polarization in heavy-ion collisions (Abramchuk et al., 2023).

4. Generalizations: Spin, Statistics, and Curvature

Arbitrary Spin

The chiral vortical effect is not unique to fermions. For massless particles of arbitrary spin $s$, the CVE current is:

$$\bm J_{\chi} = \hbar\,s\,\bm\Omega\,\left[ \frac{\mu_+^2+\mu_-^2}{4\pi^2} + \frac{T^2}{6} \right]$$

(Huang et al., 2018)

For photons (spin-1), the equilibrium vortical conductivity is fully determined by the mixed gravitational anomaly: $\langle K^\mu \rangle = \frac{T^2}{3} \Omega^\mu$ (Prokhorov et al., 2020, Avkhadiev et al., 2017). Consistent IR regularization is essential for agreement between Kubo/thermal field theory and horizon-anomaly approaches.

Temperature and Statistics Dependence

The $T^2$ coefficient is generally not determined by anomaly alone, as originally conjectured. For fermions, $C_T>0$ (positive, e.g. $T^2/12$ per flavor), while for bosons $C_T$ can be negative or positive depending on the phase (e.g., $-T^2/(24)$ for pions at low $T$) (Kalaydzhyan, 2014). The sign and magnitude thus encode the nature of the light degrees of freedom.

Curvature Effects and the Gravitational Contribution

Finite curvature and mass corrections enter the CVE via the “chiral gap effect,” a mass-shift due to background scalar curvature $R$:

$$J_A^z = \left( \frac{T^2}{12} - \frac{m^2}{8\pi^2} - \frac{R}{96\pi^2} \right) \omega$$

The coefficient of the $R\omega$ term matches the mixed gauge–gravitational anomaly, as does the expression for the Chern–Simons current approximation (Flachi et al., 2017). This enables direct application to astrophysical systems (Kerr geometry), providing a microscopic seed for jet collimation.

5. Magnetohydrodynamics, Instabilities, and Fluid Helicity

MHD with Chiral Vortical Term

The CVE modifies the Ohm's law in anomalous MHD with a term proportional to $\omega^\mu$. In the presence of both magnetic and vortical effects (CME and CVE), the generalized induction equation is (1908.10105, Wang et al., 2023):

$$\partial_t \bm B = \nabla \times\left( \bm v \times \bm B + \xi_B \bm B + \xi_\omega \bm\omega \right) + \eta \nabla^2 \bm B - 2H \bm B$$

where $\xi_B$ (CME) and $\xi_\omega$ (CVE) are the anomalous transport coefficients.

Magnetovortical Instability

In a background with finite magnetic field and vorticity, the mutual evolution of magnetic and vortical fields can lead to the chiral magnetovortical instability (CMVI). When the vortical conductivity exceeds a threshold, certain Alfvén-like modes become unstable, resulting in a rapid transfer of chiral imbalance into cross-helicity and magnetic amplification (Wang et al., 2023).

Fluid Helicity Conservation

In ideal, barotropic hydrodynamics, the fluid helicity current $j_f^\mu = \frac{\mu^2}{4\pi^2} \omega^\mu$ is conserved in global equilibrium, ensuring the decoupling of microscopic chirality and macroscopic helicity even in the presence of weak external gravity (Mitkin et al., 2021).

6. Experimental and Observational Implications

Heavy-ion Collisions

The CVE is intimately linked to polarization observables in heavy-ion collisions. Off-central collisions impart large angular momentum to the QGP, generating vorticity and, via the CVE, measurable polarization of produced hyperons (e.g., $\Lambda$, $\Xi$) (Abramchuk et al., 2023, Sun et al., 2016). The suppression of the CVE by nonperturbative QCD dynamics is required for quantitative agreement with experimental data.

Early-Universe Baryogenesis and Magnetogenesis

Primordial vorticity during the electroweak epoch seeds hypermagnetic fields and, via the Abelian anomaly, the baryon asymmetry of the Universe (1908.10105, Abbaslu et al., 2020). Transient vorticity fluctuations can be highly efficient, and the final outcome is largely insensitive to viscous damping timescales.

Condensed Matter and Photonic Systems

Rotating Weyl/Dirac semimetals and photonic media can realize the CVE and its bosonic analog, respectively (Huang et al., 2018, Avkhadiev et al., 2017, Prokhorov et al., 2020). Table-top experiments could, in principle, probe the CVE via polarization-dependent phase shifts or net spin current in engineered rotating systems.

Astrophysics

In rotating neutron stars and black holes, curvature-induced corrections to the CVE can seed collimated axial flows, possibly relating to the formation of astrophysical jets along rotation axes (Flachi et al., 2017).

7. Open Issues, Model Dependencies, and Non-universal Aspects

• The $T^2$ coefficient in the vortical conductivity is only partly controlled by anomalies and, in specific models, can depend nonuniversally on the degrees of freedom and dynamical interactions (Kalaydzhyan, 2014, 1705.01650).
• In superfluids and topological phases, the transport of chiral charge can be dominated by zero modes or skyrmion defect cores, requiring consideration of microscopic regularization and nonhydrodynamic components (Kirilin et al., 2012, Volovik, 2017).
• Strong-coupling corrections and nonperturbative effects, notably in QGP, necessitate model-specific input (e.g., field-correlator methods) to account for real-world observables (Abramchuk et al., 2023).
• Out-of-equilibrium response, as constructed in holographic frameworks, reveals intrinsic relaxation timescales for the build-up of anomaly-induced currents (Morales-Tejera et al., 2020).

---

In summary, the chiral vortical effect is a universal anomalous transport phenomenon, fundamentally rooted in the chiral and gravitational anomalies of quantum field theory but with model-dependent manifestations. Its signatures span diverse domains, governed by a common set of symmetry and topological principles, but with quantitative details sensitive to microscopic structure, strong coupling, and dynamical out-of-equilibrium response.