Nonlinear Chiral Transport in Dirac Semimetals

We study the current of chiral charge density in a Dirac semimetal with two Dirac points in momentum space, subjected to an externally applied time dependent electric field and in the presence of a magnetic field. Based on the kinetic equation approach, we find contributions to the chiral charge current, that are proportional to the second power of the electric field and to the first and second powers of the magnetic field, describing the interplay of the chiral anomaly and the drift motion of electrons moving under the action of electric and magnetic fields.


I. INTRODUCTION
The Weyl and Dirac semimetals are recently discovered materials, whose conduction and valence bands with linear energy dispersion touch at a number of Weyl or Dirac points in the Brillouin zone [1][2][3][4][5][6]. These systems belong to the Fermi point universality class of fermionic vacua [2] and possess nontrivial topology of the electronic band structure. The non-degenerate Weyl point might be described as monopole sink or source of the Berry curvature and assigned with the topological charge, an integral of the Berry curvature over the surface enclosing the point. Since net topological charge is zero, Weyl points always appear in pairs of opposite charge. The Dirac point might be composed of two Weyl points with topological charges of opposite sign. In certain classes of three-dimensional semimetals such Dirac points occur in pairs separated along a rotation axis of the crystal provided unbroken both time-reversal and inversion symmetries [7][8][9][10].
One of the distinct properties of Weyl and Dirac semimetals is the chiral anomaly, which is a nonconservation of chiral charge induced by the externally applied parallel electric and magnetic fields [11,12]. The presence of the chiral charge imbalance leads to a number of phenomena such as for example the chiral magnetic effect -charge current driven along the magnetic field [13], chiral electric separation effect -the flow of chiral charge imbalance along the electric field [14], the quantum and classical negative magnetoresistance [15][16][17][18][19], and contributions to the nonlinear optical response [20,21]. Another anomalous transport phenomena, although unrelated to the chiral anomaly, is the chiral separation effect, which describes the flow of fermions with opposite chiral charges in opposite directions along with the external magnetic field [22,23]. The progress in the topological semimetals is reviewed in Ref. [24].
Recently, a question of the interplay of the chiral anomaly and the nonlinear chiral transport was addressed for a ferromagnetic Weyl semimetal [25]. Based on the kinetic equation approach [26,27], it was shown that the chiral anomaly might lead to the quadratic in electric field corrections to the chiral charge current.
Here we study the chiral charge current driven by the time-dependent electric field in the presence of magnetic field in the Dirac semimetal, with a pair of Dirac points in it's band structure. Besides the chiral charge imbalance, the chiral anomaly generates a spin imbalance in each Dirac valley, such that the total spin polarization in the system is zero, although the staggered spin polarization is induced. We show that the chiral charge current as well as the current of staggered spin polarization is proportional to the second power of the electric field and is described by joint action of the chiral anomaly and the electron motion in the presence of the electric and magnetic fields. We also find that under curtain conditions the nonlinear corrections to the chiral separation effect, which describes the electron heating effect of the electric field, might dominate the transport.

II. MODEL
Let us consider a model of the inversion and time reversal symmetric gapless Dirac semimetal with two Dirac points separated in momentum space on the crystal rotation axis, having in mind Cd 3 As 2 and Na 3 Bi as particular material candidates. The system is described by the Hamiltonian where m(k z ) = m 1 k 2 z − m 0 in which m 0 m 1 > 0, σ and s are the vectors composed of the three Pauli matrices denoting the pseudo-spin and spin degrees of freedom (we set = 1). Two Dirac points are separated by a distance 2 m 0 /m 1 along z-axis in momentum space. Note that the Hamiltonian is block diagonal in spin space, and hence one can introduce sign s = ± to label the eigenvalues of s z .
To proceed, we consider spherical Fermi surface, set 2 √ m 0 m 1 ≡ v, and linearize the Hamiltonian around each Dirac point where the momentum in each valley is now measured relatively to the corresponding Dirac point, which is labeled by η = ±, as k z → k z − η m 0 /m 1 . We note that each Dirac point is composed of two Weyl points of opposite topological charge, which are related by the time reversal symmetry and determined by the spin eigenvalues. The Berry curvature for each of four Weyl points is given by Ω η,s = ηsk/2k 2 , wherek = k/k is the unit vector in the direction of momentum.
In the absence of the spin-flip processes the s zcomponent of the spin is conserved, allowing us to introduce the topological charge for the spin-up and spin-down electrons C η,+ − C η,− , with C η,s = S dS·Ω η,s /2π, where integral is over some surface S enclosing the Weyl node. While the total topological charge η (C η,+ + C η,− ) is zero, the staggered spin charge is finite η η(C η,+ − C η,− )/2 = 2 [7][8][9][10]. The model bears a resemblance to the antiferromagnet with two oppositely-directed magnetic sublattices.
Hence, in the situation where a magnetic field is applied to the semimetal, one expects the chiral separation effect. Turning on an electric field in addition to the magnetic field leads to a chiral anomaly with pronounced nonlinear corrections to the chiral charge current. This is in contrast to the chiral electric separation effect studied in Refs. [14,22,23], being linear in power of electric field.
Having established the model of the Dirac semimetal, let us now find the electric field induced corrections to the chiral charge current in the system.

III. BOLTZMAN EQUATION
Let us analyze the chiral charge current in the Dirac semimetal within the kinetic equation approach focusing on the zero temperature limit. This approach has been described extensively in the literature and here we briefly outline the key points [26,27]. We assume spatially homogeneous time-dependent electric field and magnetic field B applied to the system (we will comment on the effect of the wave-vector of the electromagnetic field later in the conclusions).
To proceed, we consider the case of electron doped semimetal, in which the chemical potential, µ > 0, is in the conduction band, neglect the Zeeman effect of a magnetic field compared to its orbital effect, and focus on the response quadratic in powers of electric field. In the regime of long inter-valley and spin-flip relaxation times we neglect the anisotropy of the Fermi surface in the presence of the fields and define the energy of the wave-packet as ε = vk.
The kinetic equation for the distribution function of the wave-packet f η,s (t, k) reads It is supplemented by the solutions of equations of motion, which contain contributions from the Berry curvaturek where v = ∂ε/∂k, D η,s = 1 + e c (B · Ω η,s ), and e < 0. For the collision integral in 4, we assume that the intervalley scattering rate is exponentially suppressed with respect to the intra-valley scattering rate. We then note that the Hamiltonian in Eq. 1 is block-diagonal in spinspace and the z-component of the particle's spin is a conserved quantity in the absence of the spin-flip processes. Turning on the spin-flips we adopt the model, in which the intra-valley spin-flip relaxation time is much longer than the intra-valley spin-conserving relaxation time. We also assume magnetic length v/ √ ω c µ (where ω c = −ev 2 B/cµ is the cyclotron frequency) to be much larger than the correlation radius of the scattering potential. Hence the relaxation times can be considered magnetic field independent [28]. These assumptions allow us to simplify the collision integral and separate the intra-valley spin-conserving contribution where I in [f η,s ] describes the energy relaxation processes, the valley-flip and spin-flip elastic scattering processes are described by the functional in which τ V (ε) and τ ′ V (ε) are the inter-valley spinconserving and spin-flip scattering times, τ ′ 0 (ε) is the intra-valley spin-flip scattering time, and is the momentum relaxation rate of a particle with energy ε, in which τ 0 (ε) describes the intra-valley spinconserving scattering processes. We consider the hierarchy of the elastic scattering times τ 0 (ε) < (τ ′ 0 (ε), τ V (ε)) < τ ′ V (ε), where the inter-valley spin-flip length vτ ′ V (µ) is assumed to be smaller than the system size.
The triangle brackets ... mean integration over the directions of momentum, taking into account the change of the phase space in the presence of the magnetic field [26,27], such that f η,s ≡ dΘ 4π D η,s (k)f η,s (t, k).
Finally, let us define the quantity of interest -the chiral charge current density in the Dirac semimetal The first term depends on the signs η and s. It might give finite contribution provided the staggered spin polarization is induced due to chiral anomaly. The second and third terms in the integrand are the corrections to the motion of the wave-packet due to nontrivial Berry curvature. These are independent of the signs η, s and describe the chiral separation and inverse Faraday effects, respectively.
Let us now study the chiral charge response of the Dirac semimetal taking into account the inter and intra valley scattering processes. To reveal the topologically nontrivial contributions we consider the limit of weak magnetic field, in which the cyclotron frequency is much smaller than the electron momentum relaxation rate.
We search for the approximate solution of Eq. 4, keeping contributions to the distribution function up to second order of the electric field. Similarly to Ref. [29], we expand distribution function in powers of the incident electric field +f (2) η,s (k) + where f (0) (ε) is the equilibrium distribution function, f (1) (ω, k) is the first order correction, and f η,s (k), f η,s (2ω, k) are the second order corrections at the zeroth and double frequencies, respectively. Following Perel' and Pinskii [30], we set additional constrain on the second order correction η,s meaning that the second order solution does not change the concentration of particles. The kinetic equation for the correction to the distribution function in n-th (n > 0) power of the electric field is given by Physically, the solution to the equation linear in electric field (with n = 1 in Eq. 13) describes the elastic scattering in the system, while the nonlinear solution should also account for the energy relaxation. Hence, neglecting the inelastic scattering in the collision integral, the solution of the first order differential equation Eq. 13 for f (1) η,s (ω, k) is given by where parameter κ η,s (ε, ω) = ωcµ εDη,s τ (ε) 1−iωτ (ε) is introduced for brevity,k ⊥ is the unit vector in the direction of momentum lying in the plane transverse to the direction of magnetic fieldB = B/B. The last term in 14 absorbs the contribution from the cyclotron part of the Lorentz force. It is worth to note here that the cyclotron frequency in κ η,s (ε, ω) is renormalized with the Berry curvature.
Multiplying 14 with D ν,s and integrating over the directions of momentum, one obtains equation for f (1) η,s in the form Using relation 9 one finds that the chiral anomaly con-tribution η,s comes from the inter-valley spin-conserving and intravalley spin-flip scattering processes and describes the emergence of non-equilibrium chiral charge as well as staggered spin accumulations. In the static limit ω → 0 the coefficient in 16 is real and proportional to long spinflip relaxation time. The sum η,s f η,s gives standard expression for the first order solution, which satisfies η,s f (1) η,s =0. The exact solution in the second order is rather cumbersome. However, we are interested in the limit of weak magnetic field ω c < ω and keep up to quadratic in powers of ω c τ corrections to the distribution function. The solution can be formally written as where we neglect Λ[f (2) η,s ], since η,s ηs f (2) η,s ∝ ω 2 c µ 2 (E 0 · B) 2 ≪ 1 is beyond the validity of our assumptions. The difference between the momentum relaxation rates of the first and second harmonics is neglected.
To determine the contribution to f η,s (2ω) ∝ E 2 0 , which describes the system heating, one has to take into account the inelastic processes and apply a condition of a constant concentration of particles in the presence of the electric field, given by Eq. 12. For the collision integral, describing the inelastic processes, we consider the simplest form where τ in is the energy relaxation time of the electron due to coupling to some thermal bath. Taking into account only the terms, which give dominant contribution to the chiral charge current, we find We obtain the second order correction in the form where we substitute f η,s with a solution of Eq. 15. The solution forf (2) η,s (k) can be found from 20 with the substitutions ω = 0, E · E → |E| 2 , and E 0 f η,s (ω, k) . We are now in the position to calculate the chiral charge current density.

IV. CHIRAL CHARGE CURRENT
Similarly to the distribution function 11, the current density can be expanded in powers of electric field The first terms describes the chiral charge current in the absence of the electric field. The second term describes second order correction at the zero frequency. The sum is over first two harmonics.
In the zeroth order one recovers the chiral separation effect (for a review see Ref. [14]) This electric field non-dissipative chiral current density exists in the equilibrium state of massless chiral fermions in a magnetic field [22,23] and flips it's sign for the hole doped system, in which µ < 0. The electric field induced corrections to 22 vanish in the linear response, J (1) (ω) = 0, as required by the presence of the spatial inversion symmetry. In the second order one obtains magnetic field independent zeroth harmonic contribution, which originates from the third term in 5 and describes the inverse Faraday effect where τ ≡ τ (µ) is the momentum relaxation rate at the Fermi energy.
At ωτ < 1, we separate main contributions in first and second powers of magnetic field to the response at double frequency J (2) 2 (2ω). The linear in magnetic field contribution is given by where the model dependent coefficient G(ω) is determined by the scattering potential The first term describes the heating effect of the electric field. At low frequencies, it is controlled by the energy relaxation processes. This contribution to the current is parallel to the magnetic field. The second term in 24 describes the effect of the chiral anomaly, being pronounced in the case of parallel electric and magnetic fields. It is worth noting that at ω = 0, the signs of two terms in 24 are opposite.
In the model of intra-valley short range potential, where τ 0 (ε) ∝ ε −2 , assuming other scattering times to be energy independent, one estimates G(0) ∝ 1 − τ /τ 0 , The chiral anomaly contribution might be finite at zero frequency, although, it is much smaller than the heating effect of the electric field.
The interplay of the chiral anomaly and the Hall effect is described by the contribution quadratic in magnetic field. At small frequencies ωτ < 1 we find The physical meaning of the chiral anomaly contributions to J (2) (2ω) can be understood if we consider a two-step process. The first step contains long inter-valley scattering, which equilibrates the spin imbalance generated due to chiral anomaly ∝ (E 0 ·B). The second step is the drift motion of particles within the valleys under the joint action of the electric and magnetic fields, described by the Lorentz force eE+ e c [v×B]. This is in contrast to the chiral anomaly generated charge current being determined by the inter-valley scattering processes [16,17].
Finally, we note that the photocurrentJ (2) can be found from 24 by taking the limit ω → 0 and performing a substitution

V. DISCUSSION AND CONCLUSIONS
It is interesting to compare our results with the collisionless limit of the system. In this case one has to take into account the effect of the magnetic field on the spectrum of quasiparticles, which leads to where the ω → 0 divergence is cut by the inelastic relaxation rate. We find that J (2) (2ω) vanishes for the parallel orientation of electric and magnetic field. The angular dependence in Eq. 27 is similar to the one derived for the strain induced non-equilibrium spin current in the Dirac semimetal [31]. It is also instructive to comment on the chiral charge response of the Dirac semimetal on the spatial dependent electric component of the field E(t, r) = E 0 exp(−iωt + iq · r) + c.c. At ωτ < 1, we find that the dominant contribution to the chiral charge current is due to chiral anomaly It is proportional to the first powers of electric and magnetic fields and also linearly proportional to the wavevector q.
Probing the chiral anomaly contribution to the chiral charge current might be straightforward via nonlocal measurements [32,33]. However, one needs to be able to extract it from the total signal, which also contains large, although electric field and frequency independent contribution, described by 22 and the contribution independent on the direction of the electric field, described by the first term in 24.
To conclude, we have calculated the nonlinear in the electric field corrections to the chiral charge current in the Dirac semimetal. These are proportional to the second power of the externally applied electric field and consist of contributions, which are proportional to the first and second powers of the magnetic field. We have also commented on the chiral anomaly generated staggered spin accumulation, i.e. the nonequilibrium spin polarization in each Dirac valley of the semimetal, with vanishing net spin polarization.