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Plasmon mode as a detection of the chiral anomaly in Weyl semimetals

Weyl semimetals are one kind of three-dimensional gapless semimetal with nontrivial topology in the momentum space. The chiral anomaly in Weyl semimetals manifests as a charge imbalance between the Weyl nodes of opposite chiralities induced by parallel electric and magnetic fields. We investigate the chiral anomaly effect on the plasmon mode in both intrinsic and doped Weyl semimetals within the random phase approximation. We prove that the chiral anomaly gives rise to a different plasmon mode in intrinsic Weyl semimetals. We also find the chiral anomaly leads to some exotic properties in the plasmon dispersion in doped Weyl semimetals. Consequently, the unconventional plasmon mode acts as a signature of the chiral anomaly in Weyl semimetals, by which the spectrum of plasmon provides a proper way to detect the Lifshitz transition.

Weyl semimetals are one kind of three dimensional gapless semimetals with nontrivial topology in the momentum space. The chiral anomaly in Weyl semimetals manifests as a charge imbalance between the Weyl nodes of opposite chiralities induced by parallel electric and magnetic fields. We investigate the chiral anomaly effect on the plasmon mode in both intrinsic and doped Weyl semimetals within the random phase approximation. We prove that the chiral anomaly gives rise to a new plasmon mode in intrinsic Weyl semimetals. We also find the chiral anomaly leads to some exotic properties in the plasmon dispersion in doped Weyl semimetals. Consequently, the unconventional plasmon mode acts as a signature of the chiral anomaly in Weyl semimetals, by which the spectrum of plasmon provides a proper way to detect the Lifshitz transition. Weyl semimetals 1 (SMs) are a new class of gapless topological phase, which can be seen as three dimensional (3D) analogs of graphene. Weyl fermions emerge from the band degenerate points-the Weyl nodes-in the momentum space, which are characterized by their chirality. Due to the fermion doubling theorem 2 , Weyl nodes with opposite chirality always appear in pairs. Each Weyl node behaves like a magnetic monopole in the momentum space, which acts as the source/drain for the Berry curvature field 3 . It has been predicted that this nontrivial momentum-space topology of Weyl nodes gives rise to a number of novel electromagnetic responses [4][5][6][7][8][9][10][11][12][13][14][15] . On the material side, Weyl SMs have been proposed for strongly correlated iridates 16 , semiconductor heterostructures 17,18 , and other materials [19][20][21][22] . In addition, 3D Dirac materials have recently been realized in both Cd 3 As 2 23 and Na 3 Bi 24 , which could greatly facilitate the search for Weyl SMs.
A remarkable phenomenon associated with Weyl nodes is the so-called chiral anomaly 25,26 , in which the application of a pair of parallel electric field E and magnetic field B induces a charge imbalance between the two Weyl nodes with opposite chirality. This chiral anomaly can be utilized to detect 3D Weyl SMs in experiments. For example, a large longitudinal megnetoconductivity was proposed as a consequence of the chiral anomaly 2 , which, however, is difficult to identify unambiguously in magnetotransport data 27 . Recently, nonlocal transport 28 , optical conductivity 29,30 , and optical absorption 31 measurements have also been proposed to probe the chiral anomaly in 3D Weyl SMs.
In this Rapid Communication, we propose an alternative detection method of the chiral anomaly by employing the plasmon mode in 3D Weyl SMs. We show that the chiral anomaly would lead to a new plasmon mode in intrinsic Weyl SMs. The chiral anomaly causes a redshift of the frequency of plasmon mode in doped Weyl SMs. Once the small Fermi surface crosses the Weyl node that corresponds to the Lifshitz transition (LT) point, the frequency turns out to be a violetshift. Therefore, the plas-mon mode can be regarded as a signature of the chiral anomaly in 3D Weyl SMs. We also show how to extract the information of the LT point from the plasmon dispersion.
We begin with a low-energy effective Hamiltonian for Weyl fermions in the vicinity of the Weyl node of chirality χ = ±: where v F is the Fermi velocity, σ = (σ x , σ y , σ z ) refers to the three Pauli matrices, and µ χ stands for the chiralitydependent chemical potential given by a superposition of the equilibrium carrier density and the pumped carrier density originating from the chiral anomaly. The latter grows linearly with time but the large momentum internode scattering would counteract this imbalance of carriers between two Weyl nodes. Eventually the system reaches a non-equilibrium steady state characterized by an internode relaxation time τ v . Consequently, the density of electrons pumped into or out of the neighborhood of Weyl node χ can be expressed as 2 We also define several chirality-dependent quantities: the fermi wave vector k 3 Fχ = 6π 2 n χ , the chemical potential µ χ = v F k F χ , and the charge density n χ = n + ∆ρ χ . When the two Weyl nodes are equally populated, the corresponding Fermi wave vector and the chemical potential become k 3 F = 6π 2 n, µ = v F k F . For convenience, we restrict our discussion to E · B > 0. For the undoped case with vanishing equilibrium chemical potential, µ χ depends only on the pumped charge associated with the chiral anomaly: Meanwhile for the doped case with a finite chemical potential µ, we obtain the corresponding chirality-dependent chemical potential as where we have introduced a dimensionless ratio between the pumped charge and the equilibrium charge It follows from Eq. (5) that by tuning the external fields the system undergoes a chirality-dependent LT at γ = ±1, i.e., the change of the topology of the chiralitydependent Fermi surface. In the following we shall work in the weak magnetic field limit, thus neglect the Landau level structure of Weyl nodes [28][29][30][31] . In addition, we will focus on the n-doped case with a finite positive equilibrium chemical potential µ > 0 throughout this paper (the discussion of the p-doped case is similar). It has been demonstrated that no plasmon exists in undoped Weyl or Dirac SMs 2,32 within the random phase approximation (RPA). However, when the chiral anomaly occurs, the anomalous charge transfer between the two Weyl nodes forces the Fermi surfaces to move away from their equilibrium position in opposite directions as shown in Fig.1. Thus the chemical potentials of the two Weyl nodes are µ + and µ − satisfying the relation µ + = −µ − ≡ µ > 0. In principle, the metallic nature of intrinsic Weyl SMs with chiral anomaly would support plasmon modes. In the following, we present an exact and general proof of the existence of the novel plasmon due to the chiral anomaly in undoped Weyl SMs.
The general form of the wave vector q-and frequency ω-dependent dielectric function within the RPA is given by where V (q) = 4πe 2 /κq 2 is the Fourier transform of the 3D Coulomb interaction with κ being the effective dielectric constant. Let us consider one of the Weyl nodes. The noninteracting polarization function Π(q, ω) reads where g is the number of pairs of Weyl nodes, η is a positive infinitesimal, and s, s ′ = ± are the band indices. The overlap of eigenstates F ss ′ (k, k ′ ) is given by where θ kk ′ is the angle between the 3D wave vectors k ′ and k with k To proceed with the theoretical details, we assume zero temperature T = 0 K. The Fermi distribution function f (x) turns into a simple step function θ(µ − x). Because of the general relation of the polarization functionh Π(q, −ω) = [Π(q, ω)] * , we can restrict our discussion to the positive frequency case ω > 0. In the rest of the calculation, we will set = v F = 1, which immediately implies the relation of µ = k F . The polarization function Π(q, ω) of the other node with opposite chirality can be written as The particle-hole symmetry of Weyl nodes enables us to relabel the energy dispersions in the following way:ǫ ks → −ǫ ks and writẽ Utilizing the property of the Fermi distribution function f (x) +f (−x) = 1 leads tõ One can immediately observe the relation It is obvious that [Π (q, −ω)] * is nothing but the definition of the polarization function with a positive chemical potential µ. Recalling the general property of Π (q, −ω) = [Π (q, ω)] * , we arrive at the result Π(q, ω) =Π(q, ω).
In general, for a system with particle-hole symmetry, its polarization function depends only on the magnitude of chemical potential, namely, Π (q, ω, µ) = Π (q, ω, −µ) = Π (q, ω, |µ|). Therefore, we come to a conclusion that the polarization function of undoped Weyl SMs with the chirality-dependent chemical potentials µ ± is identical to that of doped Weyl SMs with chemical potential µ = |µ ± |. Next we set out to find the plasmon dispersion, which can be obtained within the RPA by finding zeros of the dielectric function 34 , where Γ is the decay rate of the plasmon. For weak damping, Eq. (11) reduces to the following approximate equation Re ε(q, ω) = 0.
To order q 0 , the real part of the dielectric function has the form where the function κ * (ω) is defined as with α κ = e 2 /κ the effective fine structure constant. Substituting Eq. (14) into Eq. (13) yields Neglecting the logarithmic corrections in Eq. (15), we can obtain the lowest plasmon frequency ω 0 ≈ 4gακµ 2 3π . Two remarks are in order here. First, the linear dependence of ω 0 in µ also holds for the Driac SMs 32 and Weyl SMs in the absence of the chiral anomaly 2 . Second, recalling the chirality-dependent chemical potential in Eq. (4), one immediately finds that ω 0 ∝ |B| 1/3 . This is different from the results of ω 0 ∝ |B| 1/2 in previous work 11,35 , where the authors have focused on the contribution of the zeroth Landau level in a large magnetic field.
Taking into account the leading order contribution of q, we get Re ε(q → 0, ω) To gain some insight into the long wavelength plasmon dispersion, we write down an approximate expression from Eqs. (13) and (17) as For comparison, we compute the exact solution, the long wavelength solution, and the approximate solution, respectively, which are plotted in Fig.2. In the long wavelength regime, all the three solutions are in good agreement with each other. It should be noted that the lower branch of the approximate solution is fully in the intraband single particle excitation (SPE) region, which is merely an artifact due to the weak damping approximation of Eq. (11). Now we turn to investigate the effect of the chiral anomaly on the plasmon mode in doped Weyl SMs. Simultaneously turning on the parallel electric field E and magnetic field B, the amount of electrons transferred from one Weyl node to the other is equal to ∆ρ χ , with the result that the Fermi surface of one Weyl node shifts upward and the other shifts downward (see Fig.3). We assign the chirality-dependent chemical potential µ ± for the large and small Fermi surfaces with µ + > |µ − |. In the long wavelength approximation q ≪ ω ≪ |µ − | ≤ µ + , to order q 2 , the real part of the polarization function takes the form To obtain an approximate behavior of the long wavelength plasmon dispersion, one can arrive at an expression from Eqs. (13) and (19) as where the notations κ * (ω) and ω 0 are given by which can be traced back to the counterpart of Eq. (16) in the undoped case by taking µ + = µ − . The plasmons can also be revealed as sharp peaks in the energy loss function (ELF), defined as the imaginary part of the inverse dielectric function, i.e., Im [1/ε(q, ω)] that can be probed in various spectroscopy experiments, such as the electron energy-loss spectroscopy. As shown in Fig.4(a), in the presence of the chiral anomaly the plasmon exhibits some exotic features in the ELF spectrum. As long as the ratio γ gradually increases from 0 to 1, the plasmon frequency with chiral anomaly ω ch has a redshift with respect to the frequency without chiral anomaly ω eq and finally reaches a minimum ω min . On the other hand, once the small Fermi surface crosses the Weyl node, i.e., γ > 1, the plasmon frequency becomes larger than ω min and then has a continuous violetshift. The behavior of plasmons of doped Weyl SMs under the influence of the chiral anomaly can be captured by our long wavelength expressions in Eq. (21) and summarized as follows ω min < ω ch < ω eq , 0 < γ < 1; Therefore, the unique features of undamped plasmon mode can clearly characterize the chiral anomaly in doped Weyl SMs. It should be emphasized that compared with other methods 28-31 , our method possesses the advantage that we can directly determine the position of the chirality-dependent LT point from the plasmon dispersion, as shown in Fig.4(b), which coincides with the minimal frequency of plasmon mode. Actually, when |γ − 1| → 0, the small Fermi level is close to the Weyl node point, such that small energy or momentum could induce interband transition and lead to a large number of electron-hole excitations. The plasmon mode will be damped by these electron-hole excitations. Increasing q will broaden the damping region (see Fig.4(b)). Hence, the fate of the plasmon mode indeed connects with the chirality-dependent LT.
In summary, we investigated the chiral anomaly effect on the plasmon mode in 3D Weyl SMs within the RPA. We proved that a new plasmon mode would emerge in undoped Weyl SMs due to the chiral anomaly. We also demonstrated the unusual properties of the plasmon mode in doped Weyl SMs and further pointed out that the plasmon can be taken as a fingerprint of the chiral anomaly. Finally, we showed how to identify the chirality-dependent LT point from the plasmon dispersion. Our work sheds light on the probing of the chiral anomaly in 3D Weyl SMs via the plasmon mode. The tunability of plasmons due to the chiral anomaly also makes Weyl SMs promising candidates for plasmonics 37 .
We where Λ is the cutoff. Using the Dirac identity 1 x±iη = P 1 x ∓ iπδ(x) one can get where the notation P means the principal value of the integral. It is straightforward to calculate the imaginary part of the intrinsic polarization function In fact, there are two different methods to calculate the real part of the polarization function. One is to directly carry out the integral. The other is to apply the Kramers-Krönig relation.
Since the cutoff Λ is much larger than both q and ω, it is instructive to take a look at the expression of Re Π − (q, ω) in large Λ limit. Making use of the limits lim t q →∞ t 3 q log t + q t − q − 2t 2 = 2q 2 3 , lim