# Chiral Alfvén Wave in Anomalous Hydrodynamics

###### Abstract

We study the hydrodynamic regime of chiral plasmas at high temperature. We find a new type of gapless collective excitation induced by chiral effects in an external magnetic field. This is a transverse wave, and it is present even in incompressible fluids, unlike the chiral magnetic and chiral vortical waves. The velocity is proportional to the coefficient of the gravitational anomaly. We briefly discuss the possible relevance of this “chiral Alfvén wave” in physical systems.

###### pacs:

11.15.-q, 47.75.+f, 52.35.Bj*Introduction.*—Hot and/or dense matter with chirality imbalance is considered to be realized in a
broad range of systems, including quark-gluon plasmas produced in heavy-ion collisions
Kharzeev:2007jp ; Fukushima:2008xe , the early Universe Joyce:1997uy ; Boyarsky:2011uy ,
compact stars and supernovae Charbonneau:2009ax ; Ohnishi:2014uea , and
Weyl semimetals Vishwanath ; BurkovBalents ; Xu-chern . The main reason
why such chiral matter has attracted much attention recently is that it exhibits
anomalous transport phenomena, related to quantum anomalies in field theory Adler ; BellJackiw .
Two prominent examples are the so-called chiral magnetic effect (CME)
Vilenkin:1980fu ; Nielsen:1983rb ; Alekseev:1998ds ; Fukushima:2008xe
and chiral vortical effect (CVE) Vilenkin:1979ui ; Erdmenger:2008rm ; Banerjee:2008th ; Son:2009tf ; Landsteiner:2011cp , which are the currents along the direction of a
magnetic field and vorticity in chiral matter.

On the theoretical side, chiral (or anomalous) hydrodynamics Son:2009tf and chiral kinetic theory Son:2012wh ; Stephanov:2012ki ; Chen:2012ca ; Manuel:2014dza , which describe anomalous transport phenomena in chiral plasmas, have been formulated. The chiral plasmas have also been found to exhibit a new type of density wave in an external magnetic field or with a vorticity, called the chiral magnetic wave (CMW) Newman:2005hd ; Kharzeev:2010gd and the chiral vortical wave (CVW) Jiang:2015cva , as well as the unstable collective mode in the presence of dynamical (color) electromagnetic fields, called the chiral plasma instability (CPI) Akamatsu:2013pjd ; Khaidukov:2013sja (see also Refs. Joyce:1997uy ; Boyarsky:2011uy ). For recent numerical and analytical applications of chiral (magneto)hydrodynamics, see Ref. Hongo:2013cqa and Refs. Giovannini:2013oga ; Boyarsky:2015faa , respectively.

In this paper, we show that there exists a new type of gapless collective excitation specific for charged chiral plasmas in an external magnetic field. Unlike the CMW and CVW, this is a transverse wave, and it is present even in incompressible fluids. This is somewhat similar to the Alfvén wave in normal charged plasmas, which propagates without compressing the medium, driven by magnetic tension forces Biskamp . As we will show, the velocity of this new mode is proportional to the coefficient of the mixed gauge-gravitational anomaly for chiral fermions. We call it the “chiral Alfvén wave” (CAW). Since the CAW is the first example of the transverse wave with a vector-type perturbation induced by chiral effects, it should provide a unique signature in physical observables. We briefly discuss its possible phenomenological implications. Throughout the paper, we set .

*Physical argument.*—Before going to a mathematical analysis based on chiral hydrodynamic equations,
we first provide a physical argument as to why a new type of wave
(which is different from the CMW and CVW) can exist for chiral fluids in
an external magnetic field. We consider a fluid of single right-handed chiral
fermions at high temperature and zero chemical potential .

As shown in Fig. 1, we take the external magnetic field in the positive direction and consider the perturbation of the local fluid velocity in the positive direction, with . For this local fluid velocity , the vorticity points in the positive direction. In the chiral fluid, the vorticity induces the local chiral vortical current, Vilenkin:1979ui ; Landsteiner:2011cp . This current then receives the Lorentz force, , in the negative direction. Notice that this is the opposite direction as the original fluid velocity , so the Lorentz force acts as a restoring force; it makes the perturbation of the fluid velocity oscillate, and this is the origin of a wave.

This argument holds even when the fluids are incompressible like the Alfvén wave in the normal charged fluids. This should be contrasted with sound waves in normal fluids and the CMW and CVW in chiral fluids, which can propagate only in compressible fluids. Apparently, the CVE specific for chiral fluids is essential for the presence of this wave—thus, the name chiral Alfvén wave.

In the following, we put this argument on a formal mathematical basis and derive its wave equation together with the explicit expression for the velocity of the CAW.

*Chiral hydrodynamics.*—Let us start with the generic hydrodynamics for plasmas of single right-handed
chiral fermions in external electromagnetic fields.
The hydrodynamic equations read Son:2009tf ^{1}^{1}1In this paper, we use the “mostly minus” metric signature
.

(1) | |||

(2) |

Here is the energy-momentum tensor, is the electric current, is the field strength, is the anomaly coefficient [see Eq. (7) below], , and , with being the local fluid velocity.

In the Landau-Lifshitz frame, and are given by Son:2009tf

(3) | |||

(4) |

with the energy density, the pressure, the charge density,
and
the vorticity.^{2}^{2}2Note that the definition of the vorticity here is different
from the one in Ref. Son:2009tf by a factor of 2.
The dissipative effects, such as the conductivity and viscosities, are
incorporated in and , which we ignore to simplify
our argument for a moment. (We discuss these corrections later.)
The transport coefficients and , corresponding to the CME
Vilenkin:1980fu ; Nielsen:1983rb ; Alekseev:1998ds ; Fukushima:2008xe
and CVE Vilenkin:1979ui ; Erdmenger:2008rm ; Banerjee:2008th ; Son:2009tf ; Landsteiner:2011cp ,
take the form Son:2009tf ; Neiman:2010zi

(5) | |||

(6) |

where is the temperature and is the chemical potential. The transport coefficients and are related to the chiral anomaly and mixed gauge-gravitational anomaly, respectively, as Son:2009tf ; Landsteiner:2011cp ; Gao:2012ix ; Golkar:2012kb ; Jensen:2012kj

(7) |

We now explicitly write down the hydrodynamic equations in an external magnetic field. We assume the bulk collective flow to be nonrelativistic, , despite individual constituents of the fluid being relativistic. To be specific, we consider the following counting scheme: , , and with three independent expansion parameters, and . We take the gauge field to be the same order as and , so that .

As usual, it is convenient to consider the longitudinal and transverse projections of Eq. (1) with respect to the fluid velocity Landau ,

(8) | |||

(9) |

Although hydrodynamic equations to the linear order in will be sufficient for the purpose of deriving the wave equation of the CAW, we first write down a more generic “Euler equation” taking into account chiral effects. To this end, we keep the terms to the order of . Then, Eqs. (8), (9) for the temporal and spatial components (), and (2) are given by

(10) | |||

(11) | |||

(12) | |||

(13) |

respectively. Here

(14) |

with . The terms on the right-hand sides of Eqs. (10) and (11) are of order and are ignored.

For plasmas with homogeneous and static , , and (which is the case where the variations of and are much smaller and slower than ), the hydrodynamic equations are further simplified to

(15) | |||

(16) |

Equation (16) is the incompressibility condition of fluids.

*Chiral Alfvén wave.*—We now assume homogeneous plasmas at high temperature , where
we can ignore the contribution of (and so ) note . Let us consider a small
perturbation of , similarly to the analysis in sound waves in hydrodynamics
and normal Alfvén waves in magnetohydrodynamics Biskamp .
The linearized chiral hydrodynamic equation (15) in a magnetic field
[to the order of by assuming
] is given by

(17) |

where . Using , the above equation can be rewritten as

(18) |

Taking the divergence of Eq. (18) and using Eq. (16), we have . To satisfy this equation, we set

(19) |

i.e., the direction of is taken to be perpendicular to .

Without loss of generality, we take the magnetic field in the direction, . From Eq. (18), we then obtain the wave equation

(20) |

where

(21) |

Substituting the plane-wave solution of the form , we get the dispersion relation

(22) |

so the wave propagates in the opposite direction as the magnetic field. (For the fluid with left-handed chiral fermions, the wave propagates with the velocity in the same direction as the magnetic field.) As the velocity of this collective mode is proportional to the coefficient related to the mixed gauge-gravitational anomaly for chiral fermions, this phenomenon is specific for chiral fluids, and it is not present in normal charged fluids. This is the CAW. The incompressibility condition (16) means that the CAW is a transverse wave, .

This clarifies the difference between the CAW and CMW, the latter of which is also a gapless collective excitation in chiral fluids with a magnetic field; the CMW is a longitudinal density wave and appears only in compressible fluids () Newman:2005hd ; Kharzeev:2010gd , while the CAW is present even in incompressible fluids (). Note also that the conventional Alfvén wave appears when the magnetic fields are dynamical Biskamp . In contrast, the CAW exists even when the magnetic fields are external.

For high-temperature chiral plasma, , and the velocity of the CAW, given by Eq. (21), is . On the other hand, the velocity of the CMW, given by with the susceptibility Kharzeev:2010gd , is also . Hence, and coincide up to (generally different) prefactors. Note that in our counting scheme, where .

When the dissipative effects are included, the terms and , with the kinematic viscosity and , are also added to the right-hand side of Eq. (20). Then, the dispersion relation of Eq. (22) becomes

(23) |

with . In this case, the CAW propagates with the velocity , but is damped by dissipation.

*Discussions.*—Let us discuss phenomenological implications of the CAW. One possible physical situation
is the chiral fluid in the early Universe. Above the electroweak phase transition temperature
where the symmetry is restored, the massless Abelian
gauge field that can propagate at long distances is the hypermagnetic field associated
with the hypercharge, rather than the ordinary magnetic field associated with
the charge. As the hypermagnetic field couples differently to right- and
left-handed electrons, the high-temperature plasma there is chiral Giovannini:1997eg .

One expects the emergence of the CAW in such chiral fluids with strong hypermagnetic fields, which should affect the polarization anisotropies of the cosmic microwave background radiation (CMBR). Since the CAW is the vector-type perturbation (i.e., the velocity ) while the CMW and CVW are scalar-type ones (i.e., the density ), the former should leave a peculiar signature that can be distinguished from the latter, e.g., parity-odd correlations of multipole amplitudes with different angular momenta in the CMBR. We defer this question to future work.

It would be interesting to study the possible roles of the CAW in the turbulence of chiral magnetohydrodynamics relevant to the evolution of the primordial magnetic field of the Universe. One can also consider the analogue of the CAW in rotating chiral fluids (without external electromagnetic fields), which is relevant to the physics of hot neutrino gas.

Finally, the CAW revealed in this paper may be understood in the language of chiral kinetic theory Son:2012wh ; Stephanov:2012ki ; Chen:2012ca ; Manuel:2014dza , in a way similar to Refs. Stephanov:2014dma ; Jiang:2015cva .

###### Acknowledgements.

We thank Y. Akamatsu, M. Hongo, X. G. Huang, and Y. Yin for useful comments. This work was supported by JSPS KAKENHI Grant No. 26887032.## References

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