[Paper Review] Berry curvature and 4-dimensional monopole in relativistic chiral kinetic equation

Jiunn-Wei Chen, Pu, Shi+2 more|arXiv (Cornell University)|Oct 31, 2012

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This paper derives a manifestly Lorentz-covariant relativistic chiral kinetic equation from Wigner functions of massless spin-1/2 fermions, revealing that the chiral anomaly arises from a 4-dimensional Euclidean Berry monopole in momentum space. The equation incorporates vorticity terms and reproduces the axial anomaly via a source term proportional to $E \cdot B$, unifying chiral magnetic and vortical effects under a geometric phase framework.

ABSTRACT

We derive a relativistic chiral kinetic equation with manifest Lorentz covariance from Wigner functions of spin-1/2 massless fermions in a constant background electromagnetic field. It contains vorticity terms and a 4-dimensional Euclidean Berry monopole which gives axial anomaly. By integrating out the zero-th component of the 4-momentum p, we reproduce the previous 3-dimensional results derived from the Hamiltonian approach, together with the newly derived vorticity terms. The phase space continuity equation has an anomalous source term proportional to the product of electric and magnetic fields ($F ilde{F} \sim E.B$). This provides a unified interpretation of the chiral magnetic and vortical effects, chiral anomaly, Berry curvature, and the Berry monopole in the framework of Wigner functions.

Motivation & Objective

• To derive a manifestly Lorentz-covariant chiral kinetic equation for relativistic fermions using Wigner functions.
• To identify the origin of the chiral anomaly in terms of a 4-dimensional Euclidean Berry monopole in momentum space.
• To unify the chiral magnetic effect (CME), chiral vortical effect (CVE), and axial anomaly within a single framework based on Wigner functions.
• To show that vorticity terms emerge naturally in the covariant formulation, which were previously missing in non-covariant approaches.
• To demonstrate that the phase space continuity equation is modified by an anomalous source proportional to $E \cdot B$, linked to the monopole flux.

Proposed method

• Derive the relativistic chiral kinetic equation from Wigner functions of spin-1/2 massless fermions in a constant electromagnetic background field.
• Use the Wigner function formalism to identify the Berry curvature in momentum space, leading to a 4-dimensional Euclidean monopole structure.
• Perform a Wick rotation to map Minkowski space poles to Euclidean space, enabling identification of the monopole flux via $\partial_\sigma (p_E^\sigma / p_E^4) = 2\pi^2 \delta^{(4)}(p_E)$.
• Integrate over the zero-th component of momentum $p_0$ to recover the 3-dimensional chiral kinetic equation, including vorticity terms not present in earlier Hamiltonian-based derivations.
• Apply the phase space continuity equation and derive the anomalous source term $\propto E \cdot B$ from the divergence of the current, linked to the monopole flux.
• Use analytic continuation $p_4 = i p_0$ and the identity $\pi \delta(x) = -\mathrm{Im}[1/(x + i\epsilon)]$ to relate Minkowski and Euclidean momentum integrals.

Experimental results

Research questions

• RQ1 How does the chiral anomaly emerge from a 4-dimensional Berry monopole in Euclidean momentum space?
• RQ2 What is the role of vorticity in the relativistic chiral kinetic equation, and how does it relate to the chiral vortical effect?
• RQ3 How does the Lorentz-covariant Wigner function approach reproduce and extend previous 3-dimensional chiral kinetic equations?
• RQ4 What is the geometric origin of the $E \cdot B$ source term in the phase space continuity equation?
• RQ5 How is the axial current non-conservation related to the flux of a 4-dimensional monopole in momentum space?

Key findings

• The chiral anomaly is interpreted as the flux of a 4-dimensional Euclidean Berry monopole in momentum space, with the monopole located at $p_E = 0$.
• The phase space continuity equation contains an anomalous source term proportional to $E \cdot B$, given by $\partial_\rho j_{R/L}^\rho = \mp \frac{Q^2}{4\pi^2} (E \cdot B)$.
• The 3-dimensional chiral kinetic equation is recovered by integrating over $p_0$, and it includes vorticity terms not derived in earlier Hamiltonian-based approaches.
• The Wigner function formalism naturally incorporates the Berry curvature, with $\delta(p^2) b^\sigma$ acting as the 4D monopole's field strength in Euclidean space.
• The 4D monopole structure reduces to the 3D Berry curvature $\bm{\Omega}$ upon on-shell projection $\int dp_0 \delta(p^2) b^\sigma = (0, \bm{\Omega}/2)$.
• The anomalous current non-conservation is rigorously derived from $\partial_\sigma^{p_E} (p_E^\sigma / p_E^4) = 2\pi^2 \delta^{(4)}(p_E)$, confirming the monopole flux as the source of the anomaly.