Chiral fermion asymmetry in high-energy plasma simulations

ABSTRACT

The chiral magnetic effect (CME) is a quantum relativistic effect that describes the appearance of an additional electric current along a magnetic field. It is caused by an asymmetry between the number densities of left- and right-handed fermions, which can be maintained at high energies when the chirality flipping rate can be neglected, for example in the early Universe. The inclusion of the CME in the Maxwell equations leads to a modified set of magnetohydrodynamical (MHD) equations. The CME is studied here in numerical simulations with the Pencil Code. We discuss how the CME is implemented in the code and how the time step and the spatial resolution of a simulation need to be adjusted in presence of a chiral asymmetry. The CME plays a key role in the evolution of magnetic fields, since it results in a dynamo effect associated with an additional term in the induction equation. This term is formally similar to the α effect in classical mean-field MHD. However, the chiral dynamo can operate without turbulence and is associated with small spatial scales that can be, in the case of the early Universe, orders of magnitude below the Hubble radius. A chiral ${\alpha }_{\mu }$ effect has also been identified in mean-field theory. It occurs in the presence of turbulence, but is not related to kinetic helicity. Depending on the plasma parameters, chiral dynamo instabilities can amplify magnetic fields over many orders of magnitude. These instabilities can potentially affect the propagation of MHD waves. Our numerical simulations demonstrate strong modifications of the dispersion relation for MHD waves for large chiral asymmetry. We also study the coupling between the evolution of the chiral chemical potential and the ordinary chemical potential, which is proportional to the sum of the number densities of left- and right-handed fermions. An important consequence of this coupling is the emergence of chiral magnetic waves (CMWs). We confirm numerically that linear CMWs and MHD waves are not interacting. Our simulations suggest that the chemical potential has only a minor effect on the non-linear evolution of the chiral dynamo.

KEYWORDS:

• Relativistic magnetohydrodynamics (MHD)
• chiral magnetic effect
• turbulence
• MHD dynamos
• numerical simulations

Acknowledgments

We are grateful to Dmitri Kharzeev for numerous discussions on the effects of the chemical potential and chiral magnetic waves in chiral MHD. Further, we acknowledge the discussions with participants of the Nordita Scientific Program on Quantum Anomalies and Chiral Magnetic Phenomena, Stockholm (September – October 2018). The detailed comments on our manuscript by Matthias Rheinhardt and the anonymous referees are very much appreciated. I.R. acknowledges the hospitality of NORDITA, the Kavli Institute for Theoretical Physics in Santa Barbara and the École Polytechnique Fédérale de Lausanne. Simulations presented in this work have been performed with computing resources provided by the Swedish National Allocations Committee at the Center for Parallel Computers at the Royal Institute of Technology in Stockholm.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 https://github.com/pencil-code, DOI:10.5281/zenodo.2315093

2 The notation with the number 5 indicates that ${\mu }_5$ arises from quantum mechanics. Here, a Dirac field can be projected onto its left- and right-handed components using ${\gamma }^5\equiv i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$, where ${\gamma }^n$ with n=0,1,2,3 are the Dirac matrices.

3 The superscript “phys” indicates that the chemical potential is given in its usual physical dimension of energy; the symbol ${\mu }_5$ will later be used for a rescaled chiral chemical potential.

4 We note that in previous works (Brandenburg et al. 2017, Rogachevskii et al. 2017, Schober et al. 2018b), the symbol “μ” was used for the normalised chiral chemical potential. Due to the inclusion of the evolution of the ordinary chemical potential, a change of notation became necessary for the extended chiral MHD equations used in the present study.

5 If μ is incorporated, two additional evolution equations need to be solved.

Additional information

Funding

This project has received funding from the European Union's Horizon 2020 research and innovation program as ‘EPFL Fellow’ co-funded by the Marie Sklodowska-Curie Action grant No. 665667. We thank for support by the École polytechnique fédérale de Lausanne, Nordita, and the University of Colorado through the George Ellery Hale visiting faculty appointment. Support through the NSF Astrophysics and Astronomy Grant Program (grant 1615100), the Research Council of Norway (FRINATEK grant 231444), and the European Research Council (grant number 694896) are gratefully acknowledged.