PHYSICAL REVIEW D 102, 014045 (2020)

Mode decomposed chiral magnetic effect and rotating

† ‡ Kenji Fukushima,1,* Takuya Shimazaki ,1, and Lingxiao Wang 2,1, 1Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 2Physics Department, Tsinghua University and Collaborative Innovation Center of Quantum Matter, Beijing 100084, China

(Received 21 April 2020; accepted 19 July 2020; published 29 July 2020)

We present a novel perspective to characterize the chiral magnetic and related effects in terms of angular decomposed modes. We find that the vector current and the chirality density are connected through a surprisingly simple relation for all the modes and any mass, which defines the mode decomposed chiral magnetic effect in such a way free from the chiral chemical potential. The mode decomposed formulation is useful also to investigate properties of rotating fermions. For demonstration, we give an intuitive account for a nonzero density emerging from a combination of rotation and as well as an approach to the chiral vortical effect at finite density.

DOI: 10.1103/PhysRevD.102.014045

I. INTRODUCTION In other words, the coefficient in front of B in the CME, i.e., the chiral magnetic conductivity [9], must be odd and Chiral fermions exhibit fascinating transport properties, time-reversal even. Interestingly, the CME is anomaly the origin of which is traced back to the quantum anomaly protected and the chiral magnetic conductivity at zero associated with chiral symmetry. Gauge theories including frequency is unaffected by higher-order corrections. There quantum chromodynamics (QCD) may accommodate topo- are a number of theoretical and experimental efforts to logically winding configurations and a chirality imbalance detect CME signatures; parity-odd fluctuations (called the is induced [1,2]. Thanks to the index theorem, one can γ-correlator) in the heavy-ion collisions [10–15] and the quantify the chirality change in response to the winding negative magnetoresistance in Weyl/Dirac semimetals number even without solving QCD problems. There are [16–19] are representative examples. some indirect evidences for the such as the 0 0 A finite chirality imbalance plays a pivotal role in the decay rate of π → 2γ, anomalously heavy η meson [3,4], CME, which supposedly results from topological excita- etc. It is still an ambitious problem to establish more direct tions in the QCD case. For theoretical convenience, a finite experimental probes to the chiral anomaly [5]. chirality is imposed conventionally by μ5 coupled to the The chiral magnetic effect (CME) is a well-investigated chirality charge. Sometimes, however, μ5 has caused example of macroscopic realization of the chiral anomaly, controversies—the chemical potential is generally well which was originally proposed in the context of the defined for a conserved charge but the chirality charge is relativistic heavy-ion collision experiment [6] (see also not conserved due to the chiral anomaly. Since it is not a Ref. [7] for an earlier work) and an elegant formula with the conserved charge, μ5 must vanish in equilibrium. In other chiral chemical potential μ5 was found in Ref. [8]. The words, a finite μ5 as a theoretical device makes sense only CME generates a vector current j in parallel to an external magnetic field B in the presence of a finite chirality out of equilibrium. It would be therefore desirable to setup the CME without μ5. In fact, a background electromagnetic imbalance. In ordinary matter, such a vector current E ∝ B μ proportional to B is prohibited by parity and time-reversal field like is an alternative of 5 used for experiments symmetry, so the CME requires the chirality imbalance. in Weyl/Dirac semimetals. For more details on the absence of the CME in equilibrium, see Ref. [20]. Let us turn to other intriguing phenomena in a chiral * medium similar to the CME, namely, the chiral separation [email protected] † – [email protected] effect (CSE) [21 23] and the chiral vortical effect (CVE) ‡ [email protected] [24–30]. In later discussions, we will shed light on the CSE and the CVE as well as the CME using the mode Published by the American Physical Society under the terms of decomposed wave functions. In the CSE, an axial-vector the Creative Commons Attribution 4.0 International license. j Further distribution of this work must maintain attribution to current 5 is generated in parallel to an external magnetic the author(s) and the published article’s title, journal citation, field B in the presence of the chemical potential μ. The and DOI. Funded by SCOAP3. CVE is quite analogous to the CSE; an axial-vector current

2470-0010=2020=102(1)=014045(13) 014045-1 Published by the American Physical Society FUKUSHIMA, SHIMAZAKI, and WANG PHYS. REV. D 102, 014045 (2020) is induced by a vorticity ω instead of B [31–33]. Contrary factor in a form of the chemical potential shift [53,54]. to the CME, the CSE and the CVE are not anomaly From this analogy between the rotational and finite-density protected and their coefficients could be affected by effects, we can say that states with a certain angular infrared scales such as the mass, the temperature, and so momentum are reminiscent of those with a certain particle on [29,34–37]. Such a clear distinction between the CME number in the “canonical” ensemble. The Legendre trans- and the CSE/CVE will be unraveled in our mode decom- formation gives the grand canonical ensemble with the posed analysis. Our calculations will lead to a surprisingly chemical potential, and a similar machinery would work to simple relation between the vector current and the chirality translate the angular momentum modes into the “grand density, which had been unseen until we made a mode canonical” ensemble with a certain ω. One might think that decomposed formulation. we just recapitulate known calculations in the cylindrical Interestingly, both strong magnetic field B and large coordinates, but our formulation can reveal properties of vorticity ω are highly relevant to the heavy-ion collisions rotating fermions especially under B in an intuitive manner. (see Refs. [15,38] for recent reviews). Noncentral collisions As mentioned above, we will give a plain explanation of would produce a quark-gluon plasma (or hot hadronic induced density discussed in Ref. [45], through which matter at lower collision energies) under strong B and large subtleties about the angular momentum associated with the ω [15,39–41]. Now, chiral transport phenomena in general magnetic field will be manifested. have grown up to be one major subject in not only heavy- Finally, let us briefly mention a possibility that our ion collision but general physics [42,43]. So far, B-induced calculation may be directly applied to electron vortex and ω-induced phenomena have been considered individu- beams in optics [55–62]. Unlike the topological vortex ally, but an interplay between B and ω would be becoming in a superfluid, what is called the vortex is a more and more attractive [44–52]. Among various theo- fermion beam whose wave functions have a helical phase retical attempts, we shall pay our special attention to structure. For discussions from the view point of Berry Ref. [45] in which a finite density ∝ ω · B was discovered. phases, see Refs. [55,63–65]. It is also possible that light The authors in Ref. [45] proposed redistribution of the waves could have orbital angular momenta as speculated vector charge induced by ω · B, which may carry a vital role first by Poynting and this has been actually realized in magnetohydrodynamics where B is too large to be experimentally [66,67]. regarded as the first order in the gradient expansion [27]. The azimuthal dependence of electron vortex beams is We will reproduce this result as a benchmark test of our eilφ where l ∈ Z is a quantum number identified with the formulation and discuss an intuitive picture to deepen our orbital angular momentum. Although a relativistic exten- understanding on the induced density. sion of the electron vortex beam is a theoretically chal- In this work, we shall intensively investigate fermion lenging subject [68], nonrelativistic vortex beams have systems with B and analyze the mode decomposition. We been observed experimentally and their applications have will express the density, the chirality density, the vector become an exciting research field [63,69–76]. In this work, current, and the axial-vector current in terms of the angular we express various physical quantities using the angular momentum modes. Our most prominent finding is an momentum decomposed modes, and we can regard mode- elegant mode-by-mode equation between the vector current by-mode expressions as contributions from wave functions j and the chirality density ρ5, as we already mentioned of vortex beams. Then, the mode decomposed CME as we above. Notably, the equation holds not only for the lowest advocate in this work should be directly tested with the landau levels (LLLs) but also higher Landau levels for any vortex beams if the helicity of the vortex beams could be mass, while a similar equation between the axial-vector projected out. We can say, therefore, that we put forward an current j5 and the density ρ is valid for the LLLs only. The experimental possibility to realize chirally induced effects existence of such an elegant relation reminds us of the fact in tabletop setups in the future. It should be also empha- that the CME is anomaly protected. We should also sized that the fermion vortex beams have a lot of future emphasize that the anomaly nature of the CME appears prospects. The intrinsic orbital angular momentum of the from the LLL contribution, but the relation we found is for vortex beams could take an arbitrarily large value in all Landau levels. We are proposing this mode decomposed principle. For example, it has been reported that electron CME as an extended interpretation of the CME without μ5. vortex beams up to l ∼ 1000 (i.e., the orbital angular We note that we can straightforwardly recover the familiar momentum ∼1000ℏ) have been experimentally observed formula of the CME by taking the mode sum with proper [76]. Generally speaking, it is difficult to design experi- weights with μ5. ments with large vorticity ω under reasonable control, but We can apply our mode decomposed formulation for such states with large l would provide us with an alternative rotating fermion systems. It is known that there is a and controllable probe to investigate rotating fermions. reciprocal relationship between B and ω, but their micro- Although our theoretical background lies in QCD scopic descriptions are totally different. Once all angular physics, we will employ the convention of electrons and modes are available, we can introduce ω in the weight positrons in this work. If necessary, it would be easy to

014045-2 MODE DECOMPOSED CHIRAL MAGNETIC EFFECT AND … PHYS. REV. D 102, 014045 (2020) generalize our formulas for the purpose of QCD studies in work, we shall utilize the cylindrical coordinates, ðr; φ;zÞ, terms of quarks by adjusting electric charges. Throughout under a constant magnetic field, B ¼ Bez, so that the Landau this paper, we employ the physical units of ℏ ¼ c ¼ 1. wave functions are exponentially localized on the transverse ðr; φÞ plane. II. SOLUTIONS OF THE DIRAC EQUATION IN Throughout this work, we employ the Dirac representa- THE CYLINDRICAL COORDINATES tion for the γ matrices, i.e., We present explicit forms of the solutions of the Dirac I 0 0 σi equation in the cylindrical coordinates with the quantum γ0 ¼ ; γi ¼ ; ð1Þ number l corresponding to the orbital angular momentum. 0 −I −σi 0 Those expressions also explain our notations and conven- tions used throughout this paper. where I is the 2 × 2 unit matrix and σi’s are the Pauli The vacuum structures in quantum field theories should matrices. In this convention, γ5 takes a nondiagonal form of not be modified solely by rotation once the boundary effects 0 I are properly taken into account not to violate the causality γ5 ¼ : ð2Þ [25,44]. In contrast, as demonstrated in Refs. [48,49,77,78], I 0 the vacuum structures would change if the fermionic wave functions are localized by coupling to external environments The solutions of the free Dirac equation with the electric and thus become insensitive to the boundary effects. In this charge convention, e<0, read in the Dirac representation,

0 1 ðεð↑Þ þ mÞΦ ðχ2; φÞ B n;l;k n;l C −iεð↑Þ tþikz B C e n;l;k B 0 C uð↑Þ ðr; φ;z;tÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B C ð Þ n;l;k B 2 C 3 εð↑Þ þ m kΦn;lðχ ; φÞ n;l;k @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 i jejBð2n þjljþl þ 2ÞΦn;lþ1ðχ ; φÞ for the state with the positive helicity, the positive energy, respectively. The z component of the three momentum is and (the z component of) the total angular momentum, denoted by k. A mixture of l and l þ 1 in the spinor 1 J Jz ¼ l þ 2. In the above expression, n and l form the components stems from twofold realization of the same z l þ 1 ðl þ 1Þ − 1 Landau level index and l and s represent the z-component as 2 and 2. In the same way, the negative eigenvalues of the orbital and the spin angular momenta, helicity and positive energy states read

0 0 1 ð↓Þ −iεð↓Þ tþikz ðε þ mÞΦ ðχ2; φÞ ð↓Þ e n;l;k B n;l;k n;lþ1 C un;l;kðr; φ;z;tÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ð4Þ ð↓Þ 2 −i jejBð2n þjl þ 1jþl þ 1ÞΦn;lðχ ; φÞ εn;l;k þ m 2 −kΦn;lþ1ðχ ; φÞ

Z Z 1 2π 2π for the same total angular momentum, Jz ¼ l þ 2. In the 2 2 2 2 2 dxdyjΦn;lðχ ;φÞj ¼ dχ jΦn;lðχ ;φÞj ¼ : ð6Þ expressions (3) and (4), we introduced a dimensionless jeBj jeBj 2 1 2 2 variable χ ¼ 2 eBr and defined a function Φn;lðχ ; φÞ as This Landau wave function, representing a localized spatial profile, has an exponential damping factor for large r2.For sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð↑Þ ð↓Þ these states of un;l;k and un;l;k, the one particle eigenener- 2 n! −1χ2 jlj jlj 2 ilφ Φ ðχ ; φÞ¼ e 2 χ L ðχ Þe : ð Þ n;l ðn þjljÞ! n 5 gies are given by 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < ð↑Þ 2 2 εn;l;k ¼ jejBð2n þjljþl þ 2Þþk þ m ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ It would be useful to make it clear how we chose the : εð↓Þ ¼ jejBð2n þjl þ 1jþl þ 1Þþk2 þ m2: normalization of the above function, i.e., our convention is n;l;k

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R 2 We note that the normalization (6) corresponds to the dxdye−χ ¼ 2π=jeBj. In later discussions, in fact, we following convention for the Dirac spinor normalization: will consider the eB → 0 limit, which needs extra care Z jeBj about the treatment of 2π=jeBj which is cut off by the dxdy uð↑;↓Þ†uð↑;↓Þ ¼ 2εð↑;↓Þ: ð Þ S eB → 0 2π n;l;k n;l;k n;l;k 8 transverse area ⊥ for . The antiparticle solutions of the free Dirac equation in J ¼ l þ 1 This is reduced to the standard normalization in the the magnetic field with z 2 read limit of eB → 0, which can be understood from

0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 −i jejBð2n þjlj − lÞΦ ðχ2; φÞ B n;−l−1 C iε¯ð↑Þ t−ikz B 2 C ð↑Þ e n;l;k B −kΦn;−lðχ ; φÞ C vn;l;kðr; φ;z;tÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B C ð9Þ ð↑Þ B 0 C ε¯n;l;k þ m @ A ð↑Þ 2 ðε¯n;l;k þ mÞΦn;−lðχ ; φÞ for the positive helicity antiparticle state. The antiparticle states with the negative helicity read

0 1 2 −kΦn;−l−1ðχ ; φÞ B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C iε¯ð↓Þ t−ikz B 2 C ð↓Þ e n;l;k B −i jejBð2n þjl þ 1j − l þ 1ÞΦn;−lðχ ; φÞ C vn;l;kðr; φ;z;tÞ¼qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B C; ð10Þ ð↓Þ B −ðε¯ð↓Þ þ mÞΦ ðχ2; φÞ C ε¯n;l;k þ m @ n;l;k n;−l−1 A 0

which carries the same total angular momentum, positrons), in the same way, the LLLs are found to be ð↑Þ J ¼ l þ 1 n ¼ 0 l ≥ 0 J > 0 z 2. The antiparticle eigenenergies are written as states with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiand (i.e., z ), for which ð↑Þ ε¯ ¼ k2 þ m2. The range of l is ð−∞; ∞Þ,but 8 n¼0;l≥0;k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −S jeBj=ð2πÞ 0 for positrons are naturally concluded, respectively. For a schematic illus- tration, see Fig. 1.

III. MODE DECOMPOSED DENSITY DISTRIBUTIONS We will consider mode-by-mode contributions to the expectation values of the scalar (density), the pseudoscalar (chirality density), the vector, and the axial-vector oper- FIG. 1. Schematic illustration of the LLLs for negatively ators. We will later make use of the results in this section to charged particles (electrons) and positively charged antiparticles propose the mode decomposed version of the CME. Also, (positrons). The preferred spin and orbital alignments are con- we apply our results for a canonical formulation of rotating sistent with classical motion of charged particles. fermions.

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A. Density and chirality density ðεð↓Þ þ mÞ2 þ k2 ð↓Þ n;l;k 2 † ρ ¼ jΦn;lþ1j The fermion number density is given by ρ ¼hψˆ ψˆ i, n;l;k 2εð↓Þ ðεð↓Þ þ mÞ where ψˆ represents a Dirac field operator. It is a standard n;l;k n;l;k jejBð2n þjl þ 1jþl þ 1Þ procedure to expand ψˆ in terms of the complete basis of 2 þ jΦn;lj : ð15Þ ð↑;↓Þ ð↑;↓Þ ð↑;↓Þ 2εð↓Þ ðεð↓Þ þ mÞ un;l;k with particle annihilation operators aˆ n;l;k and vn;l;k n;l;k n;l;k ˆ ð↑;↓Þ† with antiparticle creation operators bn;l;k in the second bˆ ð↑;↓Þ†j0i quantization method. All annihilation and creation oper- Similarly, for the antiparticle contributions, states n;l;k ators are normal ordered. Subtleties of the normal-ordered lead to the expectation values as given by operators were discussed in Refs. [79,80]. In this work, however, we consider one-particle states individually, so we ðε¯ð↑Þ þ mÞ2 þ k2 ρ¯ð↑Þ ¼ − n;l;k jΦ j2 have dropped infinite classical numbers from the anticom- n;l;k ð↑Þ ð↑Þ n;−l 2ε¯ ðε¯ þ mÞ mutation relation. Then, we can express ρ as a linear n;l;k n;l;k superposition of different ðn; l; kÞ contributions, which we jejBð2n þjlj − lÞ − jΦ j2 ð Þ can symbolically represent in the following form: ð↑Þ ð↑Þ n;−l−1 16 2ε¯n;l;kðε¯n;l;k þ mÞ

XZ ð↑Þ ð↓Þ ð↑Þ ð↓Þ and ρ ¼ ðρn;l;k þ ρn;l;k þ ρ¯n;l;k þ ρ¯ n;l;kÞ; ð12Þ n;l;k ðε¯ð↓Þ þ mÞ2 þ k2 ð↓Þ n;l;k 2 ρ¯ ¼ − jΦn;−l−1j n;l;k 2ε¯ð↓Þ ðε¯ð↓Þ þ mÞ where the first two (and the last two) represent the n;l;k n;l;k particle (and the antiparticle) contributions. The sum jejBð2n þjl þ 1j − l þ 1Þ 2 − jΦn;−lj : ð17Þ integral is a shorthand representation of the phase space ð↓Þ ð↓Þ 2ε¯n;l;kðε¯n;l;k þ mÞ sum over ðn; l; kÞ with the proper weight having the mass dimension three. More explicitly, the phase space integral is The overall minus sign of ρ¯ comes from the anticommu- replaced as tation relation of bˆ. For our later quantitative discussions, it is important to note that the spatially integrated Z d2k 1 X jeBj X quantities (denoted by r here) are, corresponding to ⊥ ⇔ 0 → ; ð Þ 2 13 Eq. (6), quantized as ð2πÞ S⊥ 2π n;l n;l Z 2π rð↑;↓Þ ¼ dxdy ρð↑;↓Þ ¼ ; P n;l;k n;l;k jeBj S 0 Z where ⊥ is the transverse area and denotes a weighted 2π sum that can reproduce the phase space integral in the zero r¯ð↑;↓Þ ¼ dxdy ρ¯ð↑;↓Þ ¼ − ð Þ n;l;k n;l;k jeBj 18 magnetic limit and the weight goes to S⊥jeBj=ð2πÞ for sufficiently large S⊥jeBj=ð2πÞ. For a precise definition, see ðn; l; kÞ Ref. [46]; we need to cope with a finite sized boundary for any . ρð↓Þ condition and this is beyond the current scope. We make density plots for n;l;k in Figs. 2 and 3 to ð↑;↓Þ ð↑;↓Þ visualize spatial distributions of the s-wave and d-wave These expectation values of ρn;l;k and ρ¯n;l;k depend on ð↑Þ† the state. If we take an expectation value with aˆ n;l;kj0i,we 0.6 can immediately find the density constituent, using the 3 3 0.5 explicit solutions of the Dirac equation, as 2 2 1 1 0.4 yk 0 yk 0 0.3 ðεð↑Þ þ mÞ2 þ k2 ρð↑Þ ¼ n;l;k jΦ j2 -1 -1 0.2 n;l;k ð↑Þ ð↑Þ n;l -2 -2 2εn;l;kðεn;l;k þ mÞ 0.1 -3 -3 jejBð2n þjljþl þ 2Þ 0 þ jΦ j2: ð Þ -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 ð↑Þ ð↑Þ n;lþ1 14 xk xk 2εn;l;kðεn;l;k þ mÞ ð↓Þ FIG. 2. Density plot of ρn;l;k from one LLL with n ¼ 0, l ¼ −1 1 (i.e., Jz ¼ − 2) with all dimensionful quantities rescaled with k. In the same way, we consider an expectation value The mass is chosen as m=k ¼ 1. The left panel is for the magnetic ð↓Þ† 2 2 corresponding to a state, aˆ n;l;kj0i, which turns out to be strength jejB=k ¼ 1 and the right one for jejB=k ¼ 3.

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0.6 nothing but the momentum direction and ð↑; ↓Þ corre- 3 3 0.5 sponds to the spin direction. Supposing that states with a 2 2 particular k ≠ 0 are prepared, LLLs are states with the 0.4 1 1 largest chirality, i.e.,

yk 0 yk 0 0.3 Z -1 -1 0.2 ð↓Þ ð↓Þ 2π k -2 -2 r ¼ dxdy ρ ¼ − ; 0.1 5;LLL 5;LLL jeBj ε -3 -3 Z k 0 2π k -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 r¯ð↑Þ ¼ dxdy ρ¯ð↑Þ ¼ ð Þ xk xk 5;LLL 5;LLL 22 jeBj εk ð↓Þ FIG. 3. Density plot of ρ from another LLL with n ¼ 0, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n;l;k 2 2 5 after the spatial integration, where εk ¼ k þ m . In the l ¼ −3 (i.e., Jz ¼ − 2) with all dimensionful quantities rescaled with k. The mass is chosen as m=k ¼ 1. The left panel is for the massless limit of m → 0, we see that k=εk reduces to the jejB=k2 ¼ 1 jejB=k2 ¼ 3 sign function of k. magnetic strength and the right one for . ð↑;↓Þ ð↑;↓Þ In this paper, we would not plot ρ5n;l;k nor ρ¯ 5n;l;k, for they look indistinguishably similar to Figs. 2 and 3 on the LLL states (there are essentially the same plots shown in qualitative level. Ref. [59]). In our assignment l ¼ −1 for the ↓ spin 1 corresponds to the s-wave state with Jz ¼ − 2, so that the density is peaked around the origin in Fig. 2. As the B. Vector and axial vector currents magnetic field increases from the left panel to the right We can further proceed to the vector and the axial vector z † 0 z z † 0 z panel, the wave function becomes more localized near the currents, i.e., j ¼hψˆ γ γ ψˆ i and j5 ¼hψˆ γ γ γ5ψˆ i mode origin, as is clear in both Figs. 2 and 3. Then, such a by mode. Interestingly, for the vector components, we find suppression factor by 1=ðeBÞ in Eq. (18) is naturally such simple expressions as understood from reduction of distributed areas. Figure 3 l ¼ −3 J ¼ − 5 zð↑Þ k zð↓Þ k is a plot for the or z 2 mode. In this case, the j ¼ jΦ j2;j¼ jΦ j2 ð Þ n;l;k ð↑Þ n;l n;l;k ð↓Þ n;lþ1 23 centrifugal force makes the wave function peaks farther εn;l;k εn;l;k from the origin and there is a hollow around the origin. Next, we shall consider the chirality density, i.e., aˆ ð↑;↓Þ†j0i † for n;l;k states and ρ5 ¼hψˆ γ5ψˆ i, in the same way, which can be symbolically decomposed again as k k ¯zð↑Þ 2 ¯zð↓Þ 2 j ¼− jΦn;−lj ; j ¼− jΦn;−l−1j ð24Þ XZ n;l;k ð↑Þ n;l;k ð↓Þ ð↑Þ ð↓Þ ð↑Þ ð↓Þ ε¯n;l;k ε¯n;l;k ρ5 ¼ ðρ5n;l;k þ ρ5n;l;k þ ρ¯5n;l;k þ ρ¯5n;l;kÞ: ð19Þ n;l;k ˆ ð↑;↓Þ† for bn;l;k j0i states. Here, it is a quite interesting and Here, simple calculations immediately lead to the following profound observation that the following relations should expressions: hold:

ð↑Þ k 2 ð↓Þ k 2 zð↑Þ ð↑Þ zð↓Þ ð↓Þ ρ ¼ jΦn;lj ; ρ ¼ − jΦn;lþ1j ð20Þ j ¼ ρ ;j¼ −ρ ; 5n;l;k εð↑Þ 5n;l;k εð↓Þ n;l;k 5n;l;k n;l;k 5n;l;k n;l;k n;l;k ¯zð↑Þ ð↑Þ ¯zð↓Þ ð↓Þ jn;l;k ¼ −ρ¯5n;l;k; jn;l;k ¼ ρ¯5n;l;k ð25Þ ð↑;↓Þ† for aˆ j0i states and n;l;k ð↑;↓Þ† ˆ ð↑;↓Þ† for any aˆ n;l;k j0i and bn;l;k j0i states. We would emphasize ð↑Þ k 2 ð↓Þ k 2 that the above simple proportionality holds even beyond ρ¯ ¼ jΦn;−lj ; ρ¯ ¼ − jΦn;−l−1j ð21Þ 5n;l;k ε¯ð↑Þ 5n;l;k ε¯ð↓Þ the LLLs. n;l;k n;l;k The axial vector part is a little more complicated. After several line calculations, we arrive at the following bˆ ð↑;↓Þ†j0i for n;l;k states. These are surprisingly simple expres- expressions: sions as compared to counterparts of the fermion number density. Unlike the fermion density, we see that the net ðεð↑Þ þ mÞ2 þ k2 k jzð↑Þ ¼ n;l;k jΦ j2 chirality is vanishing after taking the mode sum over . This 5n;l;k ð↑Þ ð↑Þ n;l is because the combination of ð↑; ↓Þ and k uniquely fixes 2εn;l;kðεn;l;k þ mÞ whether the chirality is positive or negative. Usually for jejBð2n þjljþl þ 2Þ 2 massless fermions, the chirality is determined by the spin − jΦn;lþ1j ð26Þ 2εð↑Þ ðεð↑Þ þ mÞ and the momentum directions; in the present setup, k is n;l;k n;l;k

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ð↑Þ† for aˆ n;l;kj0i states and particles and there is no counterpart relation for the ð↑Þ particle states. The same can be said about the antiparticle ðεð↓Þ þ mÞ2 þ k2 LLL states. jzð↓Þ ¼ − n;l;k jΦ j2 5n;l;k ð↓Þ ð↓Þ n;lþ1 2εn;l;kðεn;l;k þ mÞ IV. APPLICATION TO THE CHIRAL MAGNETIC jejBð2n þjl þ 1jþl þ 1Þ 2 þ jΦn;lj ð27Þ AND RELATED EFFECTS 2εð↓Þ ðεð↓Þ þ mÞ n;l;k n;l;k The CME is characterized by a finite vector current induced by a nonzero chirality under external magnetic aˆ ð↓Þ†j0i for n;l;k states. In the same way, fields. The most compact representation of the CME relies μ j ¼ ð↑Þ 2 2 on the chiral chemical potential, 5 and the formula is ðε¯ þ mÞ þ k 2 j¯zð↑Þ ¼ n;l;k jΦ j2 μ5=ð2π ÞeB (where our j is not an electric current but a 5n;l;k ð↑Þ ð↑Þ n;−l 2ε¯n;l;kðε¯n;l;k þ mÞ vector current). Theoretically speaking, it would be desir- μ jejBð2n þjlj − lÞ able to define the CME without resorting to 5 that is a − jΦ j2 ð Þ ð↑Þ ð↑Þ n;−l−1 28 troublesome object. A common alternative is a parity-odd 2ε¯n;l;kðε¯n;l;k þ mÞ background ∼E · B and, here, we are proposing a novel picture of the CME by means of the mode decomposition. ˆ ð↑Þ† for bn;l;kj0i states and A. Recovery of the chiral magnetic effect ðε¯ð↓Þ þ mÞ2 þ k2 j¯zð↓Þ ¼ − n;l;k jΦ j2 We first discuss how to derive the well-known formula 5n;l;k ð↓Þ ð↓Þ n;−l−1 2ε¯n;l;kðε¯n;l;k þ mÞ of the ordinary CME with μ5 within the present framework. In the presence of μ5, the wave functions we presented jejBð2n þjl þ 1j − l þ 1Þ 2 þ jΦn;−lj ð29Þ before are no longer eigenstates and the eigenenergies 2ε¯ð↓Þ ðε¯ð↓Þ þ mÞ n;l;k n;l;k should be reconsidered. Generally speaking, μ5 dependence is involved (but known; see Ref. [8]), and yet, the ð↑; ↓Þ ˆ ð↓Þ† for bn;l;kj0i states. One might think that the above expres- parts give chirality contributions with equal weights due to sions look similar to previous Eqs. (14) and (15), but the degeneracy with incremented n and conversion by k → −k. relative sign between two terms is different. Therefore, an Then, the current contributions cancel out due to an extra elegant mode-by-mode relation like Eq. (25) cannot gen- minus sign in Eq. (25), so that only the LLL contribution erally exist for the axial vector current. survives, as is the case in the standard CME computation. Once we perform the spatial integration, we can sort out Now, we further simplify the calculation by taking the two terms into one, which yields m → 0 limit. We see that the LLL states have definite Z chirality for m ¼ 0 and we do not have to rediagonalizeR the zð↑Þ zð↑Þ Hamiltonian. Taking the spatial average, ð1=S⊥Þ dxdy, j5n;l;k ¼ dxdy j5n;l;k   we can then express the total current as jejBð2n þjljþl þ 2Þ ¼ 1 − rð↑Þ ð Þ ð↑Þ ð↑Þ n;l;k 30 ε ðε þ mÞ z zð↓Þ ¯zð↑Þ ð↓Þ ð↑Þ n;l;k n;l;k j ¼ j þ j ¼ −ρ5; − ρ¯5; LLL LLL LLL LLLZ 1 2π jeBj S jeBj dk and ¼ − ⊥ Z S⊥ jeBj 2π 2π 2π zð↓Þ zð↓Þ ½− ðkÞθðμ ðkÞ − jkjÞ j5n;l;k ¼ dxdy j5n;l;k × sgn 5 sgn

  þ sgnðkÞθð−μ5 sgnðkÞ − jkjÞ jejBð2n þjl þ 1jþl þ 1Þ ð↓Þ ¼ − 1 − rn;l;k: ð31Þ jeBjμ εð↓Þ ðεð↓Þ þ mÞ ¼ 5 ; ð Þ n;l;k n;l;k 2π2 33 Any further simplification is, however, impossible unless we limit ourselves to the LLLs. Indeed, for the LLLs only, a where the factor in the first parentheses comes from the simple relation realizes as follows: spatial average on the wave functions, the second factor by jeBj=ð2πÞ is from the phase space weight in Eq. (13), and jzð↓Þ ¼ −ρð↓Þ ; j¯zð↑Þ ¼ −ρ¯ð↑Þ ð Þ J 5 LLL LLL 5 LLL LLL 32 the third factor is from the z sum (degeneracy). We again note that we explicitly took the m ¼ 0 limit just for for any m, for the second terms dropoff. We note that only computational simplicity, but the CME itself is insensitive the ð↓Þ states have the LLLs for negatively charged to m.

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B. Mode decomposed chiral magnetic effect to justify the LLL approximation. In this sense, Eq. (34) Next, we shall discuss the mode decomposed realization keeps more physical contents including higher LL of the CME, which is one of the central results in this work. correspondence. Before that, here, let us point out that Eq. (25) explains why z we could find no simple relation between j and ρ5 until we C. Contrast to the chiral separation effect make the helicity decomposition. That is, according to Now, it would be instructive to consider the CSE here. z zð↑Þ zð↓Þ ð↑Þ ð↓Þ Eq. (25), jn;l;k ¼ jn;l;k þ jn;l;k ¼ ρ5n;l;k − ρ5n;l;k, while the Unlike the CME, the CSE is mass dependent, which was ð↑Þ ð↓Þ recognized since one of the earliest works [22]. In the CSE, chirality is ρ5n;l;k ¼ ρ þ ρ , and they cannot be 5n;l;k 5n;l;k an axial vector current arises in finite-density matter in ð↑; ↓Þ equated. Thus, the projection is essential to establish response to an external magnetic field. The CSE formula such a simple relation. takes a simple form only in the massless limit, and in fact, ρ Historically, the CME was first addressed in terms of 5 our Eqs. (30) and (31) can get simplified for the m ¼ 0 case in Ref. [6] (denoted differently in the original literature) in only as the context of the heavy-ion collisions, and then ρ5 gave way to μ5 since Ref. [8], and now we are turning back to the k2 k2 jzð↑Þ ¼ rð↑Þ ; jzð↓Þ ¼ − rð↓Þ : ð Þ original description with ρ5. The point is that data from the 5n;l;k n;l;k 5n;l;k n;l;k 37 εð↑Þ2 εð↓Þ2 heavy-ion collisions are convoluted with all time evolution n;l;k n;l;k and spatial fluctuations and the helicity decomposed measurement is unfeasible. With electron vortex beams, In the same way as the CME derivation, with incremented n ð↑; ↓Þ e.g., however, it may be possible to prepare wave functions , the energy dispersion relations for become μ with either ð↑Þ or ð↓Þ projected out. If we do so, we do no degenerated at finite , and only the current contributions from the LLLs survive. From almost the same procedure as longer have to introduce μ5 but we can simply refer to Eq. (25) as the CME; more specifically to distinguish from Eq. (33), using Eq. (32), we can recover the CSE as the conventional CME, we may well call our Eq. (25) the follows: mode decomposed chiral magnetic effect. z zð↓Þ ¯zð↑Þ ð↓Þ ð↑Þ j5 ¼ j5; þ j5; ¼ −ρ − ρ¯ Let us reiterate the key expressions LLL Z LLL LLL LLL jeBj dk jeBjμ zð↑Þ ð↑Þ zð↓Þ ð↓Þ ¼ − θðμ − jkjÞ ¼ − : ð Þ 2 38 jn;l;k ¼ ρ5n;l;k;jn;l;k ¼ −ρ5n;l;k ð34Þ 2π 2π 2π for the particle states. It is worth emphasizing that this In this case, unlike Eq. (33), there is no contribution from the relation is not only for the LLL states but for any ðn; l; kÞ antiparticle at zero temperature. The overall minus sign is and any m. This feature makes a sharp contrast to the intuitively understandable; the axial vector current is iden- ordinary CME in which only the LLL contribution sur- tifiablewith the spin, and the particle LLL states have the spin vives, as we already discussed. We would stress that this is antiparallel to the magnetic field as sketched in Fig. 1. a great advantage experimentally since Eq. (34) holds for arbitrary (even vanishing!) magnetic field and fermion V. APPLICATION TO ROTATING FERMIONS mass. In other words, in the ordinary CME, the role of the magnetic field is to separate ð↑; ↓Þ states. Therefore, the The mode decomposed formulation turns out to be magnetic field would be anyway needed for such a purpose insightful for us to acquire intuitive understanding for of the helicity decomposition in real setups. rotational systems as well. We shall demonstrate concrete One might think that Eq. (34) invokes an analogy to the calculations and develop a perspective different from the (1 þ 1)-dimensional system, i.e., the Dirac matrices alge- ordinary method. To this end, first, we make a quick braically satisfy overview of the conventional field-theoretical treatment of rotating fermions. In the rotating frame, the Hamiltonian is μ μν modified into a cranking form, which is analogous to finite- γ γ5 ¼ −ε γν ð35Þ density effects as pointed out in Ref. [53]. Bearing this in (1 þ 1)-dimensional theories, where our convention is analogy in mind, we will introduce a working concept of 01 10 grand canonical and canonical formulation of rotation. ε ¼ −ε ¼ 1. Then, the above identity is immediately translated into A. Rotating fermions jz ¼ −ρ ;jz ¼ −ρ ð Þ LLL 5 LLL 5 LLL LLL 36 In the rotating frame, we should incorporate a nontrivial metric as well as vierbeins for spinors. The covariant for the effectively (1 þ 1)-dimensional LLLs in the case derivative in the Dirac equation then involves the spin with negatively charged particles. These relations, however, connection. After some calculations, we can find that the make sense only when the magnetic field is strong enough free Dirac equation is given simply by

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0 ˆ ½γ ðiD0 − ω · JÞ − γ · iD − mψ ¼ 0; ð39Þ This analogy to the canonical ensemble is, however, not perfect and there are several crucial differences. The most where the covariant derivative is Dμ ¼ ∂μ þ ieAμ with our important difference comes from the fact that we cannot convention e<0 (for electrons), and ðγ0; γÞ are defined in take the thermodynamic limit for rotating systems; other- Dirac representation. Here, Jˆ is the total angular momen- wise, the causality bound would be violated. The next tum operator defined as Jˆ ¼ ˆl þ sˆ ¼ x × i∇ þ Σ using the important difference is that the one particle energy shift in i Eq. (40) is not a constant but changes with Jz, while the spin matrix, Σ ¼ γ × γ. In the above, ω denotes the 4 density shift is just a constant by ε → ε − μ. Since the angular velocity vector, which is taken along the z fermion occupation should be one per phase space, direction, i.e., ω ¼ ωez, without loss of generality. fermions occupy low energy states up to ε < μ. In contrast, ð↑;↓Þ in the rotational case, such occupied states with ε < B. Grand canonical vs canonical formulations n;l;k ωJ of rotation z are not necessarily limited to low energy states but high angular momentum states may be possibly accumulated The modification in Eq. (39) clearly indicates that the ð↑;↓Þ near the boundary edges of the finite size system [44]. one-particle energy, εn;l;k is shifted by a rotation-induced Therefore, unlike the finite-density case, the canonical “chemical potential” as approach to rotating fermionic systems may exhibit quali- tative differences from what is expected from the conven- ð↑;↓Þ ð↑;↓Þ 1 tional grand canonical description at ω ≠ 0. Still, a ε →ε −μ l; μ l ¼ ω·J ¼ω lþ : ð40Þ n;l;k n;l;k eff eff 2 superposition of all canonical states with appropriate (ω dependent) weight factors would recover the conventional It is quite natural that a finite ω > 0 energetically favors results obtained in the rotating frame at ω ≠ 0. states with larger Jz > 0. In many-body systems, the total E energy total would be thus shifted as C. Rotation-induced density E → E − ωJ ; ð Þ As an example to show how to utilize our expressions for total total total 41 rotating fermions, let us consider the following problem. J Without rotation and chemical potential, the particle and where total represents the total angular momentum. Such an energy shift is reminiscent of the situation at finite the antiparticle contributions cancel out in Eq. (12). The chemical potential, i.e., interesting question is what would happen for ω ≠ 0 and eB ≠ 0? Then, for the particles and the antiparticles, an E → E − μN : ð Þ total total total 42 effective chemical potential is induced and the distribution ð↑;↓Þ ð↑;↓Þ functions should be θðωJz − εn;l;k Þ and θðωJz − ε¯n;l;k Þ, N Here, total is the total particle number. We see prominent respectively, at zero temperature. similarity between Eqs. (41) and (42). This finite-density For the calculation of the density ρ, as discussed in analogy of the rotational effect has been already pointed out Ref. [45], only the LLL contribution remains finite up to the in Ref. [53], which has inspired a phase diagram with the linear order in ω. For the LLLs, then, Jz < 0 for ω > 0 is finite-density axis replaced with the angular velocity [54]. chosen out for the negatively charged particles, and Jz > 0 μ > 0 For finite-density systems, we know that in the is for the antiparticles. It is clear from the above step- grand canonical ensemble plays a role of pressure to induce function distribution functions that only the antiparticle hN i > 0 a finite expectation value of density, i.e., total . LLL states make nonzero contributions. Therefore, the Alternatively, we can adopt the canonical ensemble picture density is attributed solely to the antiparticle LLL states N > 0 and realize a finite-density system by forcing total with ωJz > 0, and this is along the direction of the directly. In the thermodynamic limit (i.e., the limit of antiparticle LLL motion as illustrated in Fig. 1.To infinite volume and particle number), the grand canonical summarize, for m ¼ 0, we can express the spatially and the canonical descriptions should be completely averaged density as follows: equivalent. For the rotating fermionic system, the above correspon- Z 1 2π jeBj S⊥jeBXj=ð2πÞ dk dence of μ → ω and N → J implies a canonical total total ρ ¼ − θðωJz − jkjÞ J S⊥ jeBj 2π 2π alternative with which total is directly imposed instead of Jz>0 using ω ≠ 0. In fact, in finite size systems like nuclei ω S⊥jeBXj=ð2πÞ 1 treated in nuclear structure theory, it is a common and well- ¼ − l þ πS 2 established method to deal with rotation of deformed nuclei ⊥ J by performing the angular momentum projection. Indeed, z ωjeBj the angular momentum projection is required to restore ¼ − þðorbital partÞ: ð43Þ rotational symmetry spontaneously broken by deformation. 4π2

014045-9 FUKUSHIMA, SHIMAZAKI, and WANG PHYS. REV. D 102, 014045 (2020) Z  X 2 Therefore, the spin contribution to an induced density reads 1 2π jeBj dk k ð↑Þ jz ¼ θðωJ þ μ − ε Þ 5 ð↑Þ2 z n;l;k S⊥ jeBj 2π 2π ε ωjeBj n;l n;l;k ρ ¼ − ; ð Þ spin 2 44 k2 k2 4π − θðωJ þ μ − εð↓Þ Þþ θðωJ − μ − ε¯ð↑Þ Þ ð↓Þ2 z n;l;k ð↑Þ2 z n;l;k εn;l;k ε¯n;l;k which correctly coincides with Ref. [45]. The orbital part  2 may look proportional to S⊥ which diverges in the k ð↓Þ − θðωJz − μ − ε¯ Þ : ð45Þ thermodynamic limit. This pathological behavior arises ε¯ð↓Þ2 n;l;k from the magnetic contribution to the angular momentum, n;l;k and we need careful treatments of the canonical and the One might think that only the LLLs remain with incre- kinetic orbital angular momenta [81]. They are different by mented n, but in this case Jz enters the step functions and jeBjr2=2 [82] and this extra contribution (coming from the the cancellation is incomplete. Namely, the LLLs give Poynting vector) cancels the orbital part. We will present more detailed and explicit discussions on this subtle but  Z 1 −S⊥jXeBj=ð2πÞ dk interesting point in another publication. jz ¼ − θðωJ þ μ − jkjÞ 5;LLL z S⊥ 2π Jz<0 D. Chiral vortical effect Z  S⊥jeBXj=ð2πÞ The CVE is characterized by an axial vector current dk þ θðωJz − μ − jkjÞ induced in matter at ω ≠ 0. The current can exist at finite 2π Jz>0 value of either temperature or density, and we will consider jeBj the finite-density situation only in this paper. Then, the ¼ðω − 2μÞ þðorbital partÞ; ð46Þ 2 2 4π2 CVE formula should be j5 ¼ μ =ð2π Þω in the case of m ¼ 0 and jeBj¼0. The coexistence of finite magnetic field may change the formula, and it would be an intriguing and the first term is completely consistent with a com- question to generalize the CVE formula to the finite bination of the LLL relation (36) and the induced magnetic case. The extra magnetic contribution to the density (44). The second term is nothing but the CSE formula (38). orbital angular momentum is, however, a subtle problem ð↑Þ For higher LL contributions, we note that εn;l;k ¼ and we will discuss it in another publication. ð↓Þ ð↓Þ ð↑Þ In the massless limit, the axial vector current (after the εnþ1;l−1;k and ε¯n;l;k ¼ ε¯nþ1;lþ1;k. By incrementing n and spatial average) is given by shifting l accordingly, we get

Z   1 X dk k2 k2 jz ¼ ½θðωJ þ ω þ μ − εð↓Þ Þ − θðωJ þ μ − εð↓Þ Þ þ ½θðωJ − μ − ε¯ð↑Þ Þ − θðωJ − ω − μ − ε¯ð↑Þ Þ 5 S 2π ð↓Þ2 z n;l;k z n;l;k ð↑Þ2 z n;l;k z n;l;k ⊥ n>0;l εn;l;k ε¯n;l;k Z 1 X dk k2 ¼ ½θðωJ þ ω þ μ − εð↓Þ Þ − θðωJ þ μ − εð↓Þ Þþθð−ωJ − μ − εð↓Þ Þ − θð−ωJ − ω − μ − εð↓Þ Þ S 2π ð↓Þ2 z n;l;k z n;l;k z n;l;k z n;l;k ⊥ n>0;l εn;l;k Z 1 X dk k2ω ≃ ½δðμ − εð↓Þ Þþδð−μ − εð↓Þ Þ: ð Þ S 2π ð↓Þ2 n;l;k n;l;k 47 ⊥ n>0;l εn;l;k

Here, in the last step, we made expansion in terms of completely correct and this factor 1=3 problem has been ω ≪ 1, which is adopted in the standard derivation of the already resolved well. CVE formula. For μ > 0, the second term from the The remaining 2=3 contributions are accounted for by the antiparticle is vanishing. The term involving Jz is of higher magnetization current. The magnetization current is of great order in the ω expansion.P R Then, in the limit of eB → 0,we importance to maintain Lorentz symmetry [42,83–85], 2 2 can replace ð1=S⊥Þ n;l dk⊥=ð2πÞ [as we noted below which is also associated with the side-jump effect [86–88]. Eq. (13)]. The phase space integration leads to For recent discussions on the side-jump effect within the Z framework of the chiral kinetic theory, see Refs. [89–92]. d3k k2 μ2ω jz ≃ ω z δðμ − ε Þ¼ : ð Þ The most interesting observation in the calculation of 5 3 2 k 2 48 ω ð2πÞ εk 6π Eq. (47) is that the energy derivative appears in the small limit from a shift in l for opposite ð↑; ↓Þ. This derivative is One may think that this result has a wrong coefficient by already seen in the original field-theoretical computation 1=3 as compared to the CVE formula, but the calculation is [25], but our mode decomposed description gives a plain

014045-10 MODE DECOMPOSED CHIRAL MAGNETIC EFFECT AND … PHYS. REV. D 102, 014045 (2020) explanation of the microscopic origin of the ω derivative. systems. In our mode decomposed formulation, a mode Now, we could have reproduced the correct CVE formula, with a fixed l is specifically considered, and our analysis, in but this requires extra discussions on the magnetic con- this sense, is regarded as the canonical approach in which N tribution to the angular momentum. In fact, it would be an for finite density is directly fixed. The grand canonical extremely interesting question how the CVE formula would approach fixes a finite μ for finite density and a finite ω for be modified by nonzero eB. One might think that Eq. (47) finite rotation. We note that the canonical approach to fix l already gives the answer, but even for numerical analysis, and the grand canonical approach to fix ω cannot be more careful treatments (e.g., imposing a proper boundary equivalent since the causality bound prevents us from condition at r ¼ R to discretize the transverse momenta; taking the thermodynamic limit. One application of the see Ref. [46]) are crucial. Since these vital improvements mode decomposed description is the analysis on the density are technically involved, and since this is a tremendously redistribution ∝ ω · B as found in Ref. [45]. The favored important problem on its own, we will report all such LLLs at finite B for particles and antiparticles are directed details in another publication. in opposite orientations, and a finite rotation induces imbalance between particles and antiparticles. We also VI. SUMMARY checked that the CVE formula apart from the magnetization current results from the weighted sum of higher Landau In this work, we introduced the mode decomposed levels. Noncentral heavy-ion collisions produce hot and description of fermionic wave functions at finite magnetic dense matter with strong magnetic field and huge vorticity. B field . In this framework of the mode decomposed Our formulation can potentially have a tight connection to description, fermion fields are presented with quantum heavy-ion collisions at finite impact parameter. numbers corresponding to the helicity and the orbital In closing, let us mention a prospective possibility of l angular momentum . Using these wave functions, we implementing electron vortex beams. The mode decom- obtained explicit forms of the mode-by-mode contributions posed formulation as presented in this paper is to be to the density, the chirality density, the vector current, and translated into the theory of electron vortex beams for the axial vector current. We found a remarkable relation which wave functions with fixed l are concerned. Phase zð↑;↓Þ between the vector current mode jn;l;k and the chirality vortices are receiving increased attention in particle colli- ð↑;↓Þ density ρ . We would call this surprisingly simple sion physics [62]. Applying our formulation to this topic 5n;l;k could be an intriguing possibility. Furthermore, we would jzð↑;↓Þ ¼ρð↑;↓Þ relation, n;l;k 5n;l;k, the mode decomposed CME. say that the idea could be directly tested in an optical setup; We would emphasize noteworthy advantages of the mode a particularly interesting example is the mode decomposed decomposed CME as follows: first of all, not the chiral CME manifested by the helicity projection as discussed in chemical potential but the chirality density is directly this work. If relativistic electron beams with a certain associated with the vector current. In this way, the mode helicity selected out are prepared, a possible realization of decomposed CME can evade the problematic chiral chemi- the mode decomposed CME should be the generation of cal potential which may cause controversial interpretations. finite chirality density in response to the imposed helicity We confirmed that an appropriate superposition of modes projected current through Eq. (34). In other words, one can weighted with the chiral chemical potential can correctly handle the chirality density by manipulating the helicity reproduce the usual CME from the mode decomposed projected current and vice versa. This situation of electron formula. Second of all, the mode decomposed CME does beams makes a sharp contrast to that in heavy-ion collision hold for all Landau levels and any mass. We note that a experiments. In the latter, a finite chirality density could be similar relation between the axial current and the density is created only locally in each collision event, while only derived only for the LLLs. Such an elegant relation reminds averaged quantities such as fluctuations are measurable. us that the CME is anomaly protected, i.e., the coefficient in We would emphasize an attractive point of electron front of the magnetic field is unaffected by any infrared vortex beams. Now, nonrelativistic beams produce vortex scales. It would be an intriguing future problem to study beams with l separated, and relativistic beams are going to how the mode decomposed CME may undergo a change by be available in the near future. The impact is immense; it is interaction effects. Probably only the LLL part is anomaly still quite difficult to control rotation in experiments. The protected as the CME is, and yet, the mode decomposed heavy-ion collision can certainly generate relativistic swirl, CME makes sense as long as the quasiparticle picture is but rotation itself is not really controllable. It is even valid (e.g., in the mean-field approximation in which challenging to control rotation in tabletop experiments. A interaction effects are renormalized in a dressed mass). large l instead of a large ω could be an alternative method Furthermore, we applied the mode decomposed formu- to investigate rotating fermions. For the purpose to estab- lation to rotating fermion systems. It is known that a finite lish more reliable predictions for possible experiments rotation shifts the one-particle energy by a cranking term, especially when a finite magnetic field coexists, it is ω · J, which has analogy with the μN term in finite-density indispensable to develop a theoretical framework in which

014045-11 FUKUSHIMA, SHIMAZAKI, and WANG PHYS. REV. D 102, 014045 (2020) electromagnetic angular momenta are properly taken into ACKNOWLEDGMENTS account (see, e.g., Ref. [93] for significance of electro- We thank Xu-Guang Huang and Kazuya Mameda for magnetic contributions). All these issues including useful discussions. K. F. was supported by Japan Society for optical applications for chiral physics deserve further the Promotion of Science KAKENHI Grants No. 18H01211 investigations. and No. 19K21874.

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