Chiral vortical effect in relativistic and nonrelativistic systems

We formulate the chiral vortical effect (CVE) and its generalization called generalized vortical effect using the semiclassical theory of wave packet dynamics. We take the spin-vorticity coupling into account, and calculate the transport charge current by subtracting the magnetization one from the Noether local one. We find that the transport charge current in the CVE always vanishes in relativistic chiral fermions. This result implies that it cannot be observed in transport experiments in either relativistic systems such as quark-gluon plasmas or condensed matter systems such as Dirac/Weyl semimetals with the pseudo-Lorentz symmetry. We also demonstrate that the anisotropic CVE can be observed in nonrelativistic systems that belong to the point groups $D_n, C_n (n = 2, 3, 4, 6)$, and $C_1$, such as $n$-type tellurium.


I. INTRODUCTION
Quantum anomalies play important roles in high energy and condensed matter physics. In a relativistic system of chiral fermions, the chiral symmetry in the classical action breaks down when the theory is quantized in the presence of electromagnetic fields, which is known as chiral anomaly 1,2 . Such a system is realized in quarkgluon plasmas in heavy-ion collision experiments [3][4][5][6] . A famous consequence of the chiral anomaly is the chiral magnetic effect (CME) [7][8][9][10][11] ; the charge current j flows parallel to a magnetic field B as j ∝ µ 5 B with the chiral chemical potential µ 5 = (µ L − µ R )/2, where µ L/R are the chemical potentials for the left/right-handed fermions, respectively. However, in equilibrium, the charge current is forbidden by the Bloch-Bohm theorem [12][13][14] . In fact, nonzero µ 5 cannot be realized in equilibrium. Under an applied electric field E as well as a magnetic field, the chiral imbalance proportional to E · B is generated out of equilibrium and causes the negative magnetoresistance (NMR) via the CME 15,16 .
In condensed matter physics, the chiral anomaly has been studied in Dirac/Weyl semimetals 17 . They are three-dimensional materials with gapless electronic excitations protected by topology and symmetry. According to the Nielsen-Ninomiya theorem 15 , such nodes always appear in pairs with the opposite chiralities in lattice systems, and hence the CME does not occur in equilibrium. A simple proof using the periodicity of the Brillouin zone was also given 18 . Instead, the Fermi arc surface states [19][20][21][22][23][24][25][26] and the aforementioned NMR [27][28][29][30] were observed in TaAs and its family soon after the firstprinciples predictions 31,32 , although it was pointed out that the observed NMR may be attributed to the current jetting effect 33,34 .
The chiral vortical effect (CVE) is also an anomalyrelated transport phenomenon 9,35-42 ; the charge current flows parallel to the vorticity ω as j ∝ µµ 5 ω with chemical potential µ = (µ L +µ R )/2. Although the conductivity is proportional to µ 5 , the CVE has been believed to oc-cur in equilibrium in noncentrosymmetric Weyl semimetals 43,44 . Indeed, the Bloch-Bohm theorem is inapplicable to the CVE in equilibrium; the theorem requires the thermodynamic limit, while the system with vorticity must be finite-sized so that the causality constraint is respected 14,36 . Recently in the condensed matter context, an extension called generalized vortical effect (GVE) has been proposed 45 , in which the charge current is induced by a velocity gradient ∂ x j v l as j i = qσ ijl v ∂ x j v l /2, for some noncentrosymmetric materials in the hydrodynamic regime such as a Weyl semimetal WP 2 46 . The GVE includes the CVE as the completely antisymmetric case and the anisotropic CVE discussed later.
In this paper, we formulate the CVE and GVE using the semiclassical theory of wave packet dynamics 47 . This formalism correctly describes the anomalous Hall effect 48,49 , the orbital magnetization 50,51 , the anomalous Nernst effect 51 , and many others. One advantage of using the theory is to separate the effects of the spin-vorticity coupling (SVC), magnetic moment, and Berry curvature at the expense of the explicit Lorentz covariance, which makes the physical origins of the CVE and GVE evident. We take the SVC into account, and calculate the transport charge current by subtracting the magnetization one from the local one 45,51,52 . Regarding the relativistic system of chiral fermions, we find that the transport charge current always vanishes whether the system is in equilibrium or not because the contributions of the SVC and Berry curvature cancel each other. In other words, the CVE cannot be observed in transport experiments. On the other hand, we show that the anisotropic CVE can be observed in nonrelativistic systems that belong to the point groups D n , C n (n = 2, 3, 4, 6), and C 1 . We estimate the induced charge current using the low-energy effective Hamiltonian of n-type tellurium.

II. SEMICLASSICAL THEORY
We consider a moving fluid observed in an inertial frame using the semiclassical theory of wave packet dy-namics. The Hamiltonian of the system isĤ(p), whose eigenvalues and eigenstates are n (p) and |u n (p) . The equations of motion arė We assume that the distribution function f is a function of˜ n (p, x) = n (p) − v(x) · p − ω(x) · s n (p) in the presence of the fluid velocity v(x) and the vorticity ω(x) = ∇ x × v(x)/2. The last term corresponds to the SVC, and s n (p) = u n (p)|ŝ|u n (p) withŝ being the spin operator. This distribution function may or may not be the solution of the collision-free Boltzmann equation, In the latter case, we need to add the collision integral to the right hand side of Eq. (2). Nonetheless, we neglect the collision effect and focus on the intrinsic mechanisms of the CVE and GVE.
Hereafter we compute the charge currents. We define the magnetic moment and Berry curvature as The local charge current is then defined as 47 Since the magnetic moment m n (p) describes the self rotation of a wave packet, it contributes to the local charge current when f is nonuniform. The second term was found in the chiral kinetic theory as well, which is indispensable for maintaining the Lorentz covariance 42 , and the magnetic moment corresponds to the spin alignment of chiral fermions. Indeed, when a wave packet is constructed from the positive energy bands of Dirac fermions, a nonabelian extension of the magnetic moment is related to the expectation value of the spin operator 53 . By substituting Eq. (1a) into Eq. (4) and expanding f up to the first order with respect to v(x) and ω(x), we obtain The coefficients, one of which appears later, are which originate from the SVC, magnetic moment, and Berry curvature, respectively. The local charge current itself cannot be measured in transport experiments. Experimentally measured is the transport charge current j tr (x) = j loc (x) − j mag (x) 45,51,52 , where j mag (x) = ∇ x ×M (x) is the magnetization charge current, and M (x) is the orbital magnetization. The reason is as follows 52 : Considering a cross section of the system, denoted by S, we readily find that the surface integral of j mag (x) over S turns into the vanishing line integral of M (x) over ∂S thanks to the Stokes theorem. Thus, the net charge transport is determined by j tr (x) alone. In equilibrium, the orbital magnetization can be calculated as M = ∂P/∂B 50,51 , where P is the pressure.
When the fluid velocity and vorticity vary slowly, the orbital magnetization can be calculated as if they are uniform 45 , namely, The second term of Eq. (7a) originates from the Berryphase correction of the density of states 50,51 , and Together with Eq. (5), the transport charge current reads The first term is well known in hydrodynamics, and the third term was already obtained in Ref. 45 . The second term was overlooked in condensed matter physics but turns out to be important below. The generalized vortical conductivity, which is defined as In the context of high energy physics, the chiral vortical conductivity is given by the scalar part of S − C.

III. CVE FOR RELATIVISTIC FERMIONS
We apply our results to the relativistic system of chiral fermions.
The unperturbed Hamiltonian iŝ H(p) = σ · p with σ being the Pauli matrices for the spin degrees of freedom.
The eigenvalues are , and the corresponding eigenstates are |u + (p) = [cos θ/2, e iφ sin θ/2] T and |u − (p) = [−e −iφ sin θ/2, cos θ/2] T , where p, θ, and φ are the polar coordinates of p. We obtain s σ (p) = σ( /2)e p , m σ (p) = 2 e p /2p, and Ω σ (k) = −σ 2 e p /2p 2 . The distribution function of˜ σ (p, x) is the equilibrium one, which satisfies the collision-free Boltzmann equation (2). The coefficients in Eq. (6) are then evaluated as S i j = (1/3)σ v δ i j , M kl = (2/3)σ v δ kl , and C kl = (1/3)σ v δ kl . Here, σ v is the chiral vortical conductivity computed with the local charge current, i.e., j i loc = σ v ω i , and reads with f σ (x) = [e (x−σµ)/T + 1] −1 for particles (σ = +1) and antiparticles (σ = −1). In high energy physics, the local charge current (5) is employed to analyze the CVE. Indeed, from Eq. (5) we correctly reproduce the well-known fact that S = (1/3)σ v originates from the SVC, while M = (2/3)σ v from the magnetic moment 42 . Besides, it can be checked that Eq. (5) is also consistent with the chiral anomaly, as follows. According to relativistic hydrodynamics involving the chiral anomaly, the CVE is characterized as j µ = qξω µ with ω µ = µνρσ u ν ∂ ρ u σ /2 and 39 where n = ∂P/∂µ, , and P are the charge density, energy density, and pressure, respectively. Here C = (2π ) −2 is the anomaly coefficient defined by ∂ µ j µ = −q 3 C µνρσ F µν F ρσ /8. We note that Eq. (11) does not capture the nonzero temperature part, which is understood as the gravitational contribution to the chiral anomaly 40,41 or as a global anomaly [54][55][56] . Hence, we focus only on the zero temperature limit. Then, using the thermodynamic relation + P = T s + µn, we obtain ξ = (1/3)Cµ 2 = (1/3)σ v . However, in this analysis based on the hydrodynamics, by construction, divergence-free parts are not taken into account. In the semiclassical theory, hence, the contribution corresponding to Eq. (11) is identified as only the SVC term in Eq. (5), that is, S = (1/3)σ v ; the magnetic moment term in Eq. (5) is always divergence-free no matter what f is. For the same reason, the velocity of the chiral vortical wave, which is a gapless collective mode in rotating chiral fermions 57 , reduces to 1/3. In contrast, the transport charge current (9) vanishes since S = C = (1/3)σ v . We again emphasize that these S and C are evaluated with f satisfying the collision-free Boltzmann equation (2). Therefore, in equilibrium, the chiral vortical current is just the magnetization current, which cannot be measured in transport experiments. The same is true even when the chiral imbalance is dynamically generated, since the transport charge current vanishes regardless of the presence or absence of µ 5 . It is a sharp contrast to the CME, where the charge current is proportional to µ 5 , leading to the NMR 15,16 . We remark here that the CVE in condensed matter systems with the pseudo-Lorentz symmetry comes to the same conclusion even if there exist multiple nodes with different energies.

IV. (ANISOTROPIC) CVE IN NONRELATIVISTIC SYSTEMS
In nonrelativistic systems, a variety of SOCs are allowed dependently on the crystal symmetry. From the viewpoint of symmetry, among the 21 noncentrosymmetric point groups, the possible candidates to yield the (anisotropic) CVE are the 11 chiral point groups; O, T , D n , C n (n = 2, 3, 4, 6), and C 1 . However, the CVE can never take place if the contributions from the SVC and Berry curvature cancel each other. In this section, we analyze the (anisotropic) CVE for the above 11 point groups.
In the point groups O and T , only the isotropic CVE is allowed but cannot be observed experimentally in the low-energy regime. We consider a spinful one-band HamiltonianĤ(p) = g 0 (p) + g(p) · σ. The eigenvalues are σ (p) = g 0 (p) + σg(p) [σ = ±1], and the corresponding eigenstates are |u + (p) = [cos Θ/2, e iΦ sin Θ/2] T and |u − (p) = [−e −iΦ sin Θ/2, cos Θ/2] T , where g, Θ, and Φ are the polar coordinates of g(p). We obtain s σ (p) = σ( /2)g(p)/g(p), and Ω σ (p) = −σ 2 sin Θ∇ p Θ × ∇ p Φ/2. The symmetry allows the forms of g 0 (p) = p 2 /2m and g(p) = αp. The only scalar parts of S and C in Eq. (6) are nonzero and read where f ( ) = [e ( −µ)/T + 1] −1 is the Fermi distribution function. Thus, the transport charge current vanishes since S = C as for relativistic chiral fermions. We comment on Ref. 45 . The authors also show that in arbitrary chiral point groups the transport charge current vanishes. The underlying mechanism in their argument is however S = C = 0; the former is because the SVC was not taken into account, and the latter comes from the assumption of the parabolic dispersion relations. This assumption is not always valid, and in fact C does not vanish. As shown here, the cancellation of the contributions from the SVC and Berry curvature, S = C = 0, is essential for the vanishing transport charge current of the CVE. Indeed, in the following we show that some chiral point groups can admit the nonzero transport charge current of the anisotropic CVE.
In the point groups D n and their subgroups C n and C 1 , the anisotropic CVE can be observed experimentally. As a representative, we consider n-type tellurium that belongs to the point group D 3 . Trigonal tellurium lacks the inversion symmetry and has attracted renewed interest in the context of the spin and orbital Edelstein effects 58,59 and Weyl semimetals [60][61][62][63][64] . The bottom of the conduction bands are close to the H and H points and are described by g 0 (p) = (p 2 x + p 2 y )/2m 1 + p 2 z /2m 2 and g(p) = [v 1 p x , v 1 p y , v 2 p z ] T . The material parameters were experimentally determined to be v 1 = 3.42 × 10 4 m/s, v 2 = 1.08 × 10 5 m/s 65 , m 1 = 0.104m 0 , and m 2 = 0.0697m 0 , where m 0 is the electron mass. The signs of v 1 and v 2 are determined by the spin texture obtained by first-principles calculation 61 . Although it was pointed out in Ref. 61 that the antisymmetric SOC is isotropic, the obtained spin texture is tilted to the p z axis, which indicates v 1 < v 2 .
The coefficients S and C in Eq. (6) are diagonal. In particular, at zero temperature, they are written as .347 meV and 2 = m 2 v 2 2 /2 = 2.32 meV. We have also taken the valley degrees of freedom, namely, the H and H points into account. The symmetry allows the anisotropic CVE characterized by In Fig. 1, we show the in-plane component σ v1 = S 1 − (C 1 + C 2 )/2 as a function of the chemical potential. When the vorticity is in the xy plane, the left-hand side of the Boltzmann equation Eq. (2) is nonzero, so that the system is out of equilibrium. The in-plane component σ v1 attains the negatively maximum value σ v1max = −32 µm −2 near the bottom of the conduction bands in the p x p y plane. On the other hand, the out-ofplane component σ v2 = S 2 − C 1 always vanish. This is because the system is in equilibrium when the vorticity is in the z direction. We emphasize again the importance of the contribution from the SVC.

V. DISCUSSION AND SUMMARY
We discuss the experimental setup to observe the anisotropic CVE obtained in the previous section. In order to realize the situation where a rotating fluid is observed in an inertial frame, we use a surface acoustic wave, which is rotational deformation localized at a surface.
It was theoretically proposed 66 and recently experimentally observed 67 that the spin current can be induced by the surface acoustic wave via the SVC. The induced vorticity is expressed as ω x (t, y, z) = (k 2 0 u 0 /2c t )e −ktz+i(ky−k0t) with k 0 = ξc t k, k t = 1 − ξ 2 k, where u 0 is the amplitude of a mechanical resonator, and ξ is a parameter determined by the transverse sound velocity c t and longitudinal sound velocity c l 68 . The localized ac charge current with the frequency k 0 flows in the x direction. We apply the low frequency about k 0 /2π = 100 MHz to ensure that the vorticity is almost static and uniform. With typical parameters, u 0 = 1 nm, c t = 2000 m/s 66 , we estimate the amplitude to be ω x = k 2 0 u 0 /2c t = 1×10 5 s −1 and the maximum charge current to be j x max = qσ v1max ω x = 0.5 A/m 2 . We can also observe the anisotropic CVE simply by rotating a tellurium crystal with the angular velocity ω = (ω x , 0, 0). In this case, it is useful to adopt the co-rotating frame with the material. Under the transformation from the inertial frame to the rotating frame, physical quantities are in general observed as different values. The charge current parallel to the vorticity (or the angular velocity of the rotating frame), however, is irrelevant to choice of the frame; e.g., the chiral vortical current of chiral fermions is unchanged even in the frame co-rotating with the fluid 69 . Hence, the in-plane CVE in Eq. (14a) can be observed when the contacts rotate together with the material.
To summarize, we have formulated the CVE and GVE using the semiclassical theory of wave packet dynamics. We have taken the SVC into account and calculated the transport charge current by subtracting the magnetization one from the Noether local one 45,51,52 . In a relativistic system of chiral fermions, the transport charge current always vanishes since the contributions from the SVC and the Berry curvature cancel each other. This result implies that the CVE cannot be observed in transport experiments even when the chiral imbalance is generated applying parallel electric and magnetic fields 15,16 . The conclusion is the same in condensed matter systems with the pseudo-Lorentz symmetry. We have also demonstrated that the anisotropic CVE can be observed in nonrelativistic systems that belong to the point groups D n , C n (n = 2, 3, 4, 6), and C 1 , such as n-type tellurium. The charge current induced by a surface acoustic wave is estimated to be of the order of 0.1 A/m 2 .

ACKNOWLEDGMENTS
We thank R. Toshio for discussing the GVE and sharing their manuscript 45 . We also appreciate fruit-ful discussion with M. Hirayama on tellurium and with H. Watanabe on antisymmetric SOCs. This work was supported by the Japan Society for the Promotion of Science KAKENHI (Grant No. JP18K13508).
Appendix A: Anisotropic CVE in n-type tellurium For the effective model of n-type tellurium, the nonzero components of S and C in Eq. (6) are Here, the factor 2 originates from the valley degrees of freedom, namely, the H and H points. We have introduced new variables as |v 1 |p x = sin θ cos φ, |v 1 |p y = sin θ sin φ, and |v 2 |p z = cos θ. At zero temperature, using we obtain Eq. (13).