Chinese Physics C Vol. 44, No. 7 (2020) 074106 Chiral magnetic effect for chiral fermion system* 1) 2) Ren-Da Dong(董仁达) Ren-Hong Fang(方仁洪) De-Fu Hou(侯德富) Duan She(佘端) Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS), Central China Normal University, Wuhan 430079, China Abstract: The chiral magnetic effect is concisely derived by employing the Wigner function approach in the chiral fermion system. Subsequently, the chiral magnetic effect is derived by solving the Landau levels of chiral fermions in detail. The second quantization and ensemble average leads to the equation of the chiral magnetic effect for righthand and lefthand fermion systems. The chiral magnetic effect arises uniquely from the contribution of the lowest Landau level. We carefully analyze the lowest Landau level and find that all righthand (chirality is +1) fermions move along the direction of the magnetic field, whereas all lefthand (chirality is −1) fermions move in the opposite direction of the magnetic field. Hence, the chiral magnetic effect can be explained clearly using a microscopic approach. Keywords: CME, Landau levels, chiral fermions DOI: 10.1088/1674-1137/44/7/074106 1 Introduction the thermodynamic potential Ω. The macroscopic elec- tric current jz along the z-axis can be obtained from the thermodynamic potential Ω. Another study on the CME Quark gluon plasma (QGP) is created in high energy addressing Landau levels is related to the second quantiz- heavy ion collisions, constituting extremely hot and dense ation of the Dirac field. In Ref. [21], the authors determ- matter. An enormous magnetic field can be generated by ined the Landau levels and corresponding Landau wave- high energy peripheral collisions [1-3]. One of the predic- functions for the massive Dirac equation in a uniform tions in QGP is that positively and negatively charged magnetic field, likewise with chemical potential µ and particles seperate along the direction of the magnetic chiral chemical potential µ5. Then, they second-quant- field, which is related to chiral magnetic effect (CME) [4- ized the Dirac field and expanded it by these solved 6]. Numerous efforts have been made to determine the Landau wavefunctions and creation/construction operat- CME in experiments [7-9]. However, due to background ors. The density operator ρˆ can then be determined from noise, no definite CME has been revealed to date. Numer- Hamiltonian Hˆ and particle number operator Nˆ of the ous theoretical methods likewise investigated the CME, system. Finally, they derived the macroscopic electric such as AdS/CFT [10, 11], hydrodynamics [12-14], fi- current jz along the z-axis through the trace of density op- nite temperature field theory [15-18], quantum kinetic erator ρˆ and electric current operator ˆjz , which is simply theory [19], lattice method [20], et al. the CME equation. In this article, we study the CME in detail by determ- From the study on CME for massive Dirac fermions ination of Landau levels. For the massive Dirac fermion through Landau levels, we conclude that the contribution system, several studies on CME addressed Landau levels. to CME arises uniquely from the lowest Landau level, In Ref. [15], Fukushima et al. proposed four methods to while the contributions from higher Landau levels cancel derive the CME. One of these methods made use of each other. However, because of the mass m of the Dirac Landau energy levels for the massive Dirac equation with fermion, the physical picture of the CME for the massive chemical potential µ and chiral chemical potential µ5 in a Dirac fermion system is not as clear as in the massless homogeneous magnetic background B = Bez to construct fermion case, as the physical meaning of the chiral chem- Received 25 January 2020, Published online 12 May 2020 * R.-H. F. is Supported by the National Natural Science Foundation of China (11847220); D.-F. H. is in part Supported by the National Natural Science Foundation of China (11735007, 11890711) 1) E-mail: [email protected] 2) E-mail: [email protected] Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main- tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub- lishing Ltd 074106-1 Chinese Physics C Vol. 44, No. 7 (2020) 074106 ical potential µ5 for the massive fermion case is not en- we show a succinct derivation of CME employing the tirely understood. To address this issue, we list the low- Wigner function approach, which we can use to obtain est Landau level as follows (we set the homogeneous the CME as a quantum effect of the first order in the h¯ ex- magnetic background B = Bez along the z-axis and as- pansion. Subsequently, we turn to determine the Landau sume eB > 0, which is also appropriate for following sec- levels for the chiral fermion system. Because chiral fer- tions), mions are massless, the equations of righthand and 0 1 lefthand parts of the chiral fermion field decouple with B '0 C B C B 0 C 1 + each other, which allows us to deal with righthand and ; = B C i(yky zkz); λ = ; 0λ(ky kz; x) c0λ B ' C e ( 1) (1) lefthand fermion fields independently. Taking the @B F0λ 0 AC L 0 righthand fermion field as an example, we first solve the q q energy eigenvalue equation of the righthand fermion field =λ 2+ 2 = λ 2+ 2+ = with energy E m kz , where F0λ ( m kz kz) m, in an external uniform magnetic field and obtain a series and '0 is the zeroth harmonic oscillator wavefunction of Landau levels. Then, we perform the second quantiza- along the x-axis. To simplify the following discussions, tion for righthand fermion field, which can be expanded we set ky = 0. The z-component of the spin operator for by complete wavefunctions of Landau levels. Finally, the z = 1 σ ;σ z = the single particle is S 2 diag(q 3 3), implying S 0λ CME can be derived through the ensemble average, ex- 1 plicitly indicating that the CME uniquely arises from the (+ ) λ. When λ = +1, E = m2 + k2 > 0, then + in Eq. 2 0 z 0 lowest Landau level. By analyzing the physical picture (1) describes a particle with momentum k and spin pro- q z for the lowest Landau level, we conclude that all z = + 1 λ = − = − 2 + 2 < jection S 2. When 1, E m kz 0, then righthand (chirality is +1) fermions move along the posit- 0− in Eq. (1) describes an antiparticle with momentum ive z-direction, and all lefthand (chirality is -1) fermions − z = − 1 kz and spin projection S 2. Thus, in the homogen- move along the negative z-direction. This is the main res- eous magnetic background B = Bez, we obtain a picture ult of this study. This result can qualitatively explain why for the lowest Landau level (with ky = 0): All particles a macroscopic electric current occurs along the direction spin along the (+z)-axis, while all antiparticles spin along of the magnetic field in a chiral fermion system, called the (−z)-axis; however, the z-component momentum of the CME. We emphasize that the CME equation is de- particles and anti-particles can be along both the (+z)-ax- rived by determining Landau levels, without the approx- is or the(−z)-axis. A net electric current is difficult to ob- imation of a weak magnetic field. tain along the magnetic field direction from the point of The rest of this article is organized as follows. In Sec. view of the lowest Landau level for the massive fermion 2, we present a succinct derivation for the CME using case. Wigner function approach. In Sec. 3, we determine the In this article, we focus on a massless fermion (also Landau levels for the righthand fermion field. In Secs. 4 referred to as the “chiral fermion ”) system, where we and 5, we perform the second quantization of the show that it is easy to obtain a net electric current along righthand fermion system and obtain CME through the the magnetic field direction, seen from the picture of the ensemble average. In Sec. 6, we discuss the physical pic- lowest Landau level. The chiral fermion field can be di- ture of the lowest Landau level. Finally, we summarize vided into two independent parts, namely the righthand this study in Sec. 7. Some derivation details are presen- and lefthand parts. First, we set up the notation. The elec- ted in the appendixes. tric charge of a fermion/antifermion is e. The chemical Throughout this article, we adopt natural units, where potential for righthand/lefthand fermions is µR=L , which h¯ = c = kB = 1. The convention for the metric tensor is can be employed to express the chiral and ordinary chem- gµν = diag(+1;−1;−1;−1). The totally antisymmetric Levi- µνρσ 0123 ical potentials as µ5 = (µR − µL)=2 and µ = (µR + µL)=2, re- Civita tensor is ϵ with ϵ = +1, which is in agree- spectively. The chemical potential µ describes the imbal- ment with Peskin [25], but not with Bjorken and Drell ance of fermions and anti-fermions, while the chiral [26]. The Greek indices, µ,ν;ρ,σ, run over 0;1;2;3, or chemical potential µ5 describes the imbalance of t; x;y;z, whereas Roman indices, i; j;k, run over 1;2;3 or righthand and lefthand chirality.