Weyl semimetal is a three-dimensional Dirac material whose low energy dispersion is linear in momentum. Adding a quadratic (Schrödinger) term to the Weyl node breaks the original particle-hole symmetry and also breaks the mirror symmetry between the positive and negative Landau levels in present of magnetic field. This asymmetry splits the absorption line of the longitudinal magneto-optical conductivity into a two peaks structure. It also results in an oscillation pattern in the absorption part of the Hall conductivity. The two split peaks in Re*σ _{xx}* (or the positive and negative oscillation in Im

*σ*) just correspond to the absorptions of left-handed (

_{xy}*σ*

_{−}) and right-handed (

*σ*

_{+}) polarization light, respectively. The split in Re

*σ*and the displacement between the absorption of

_{xx}*σ*

_{+}and

*σ*

_{−}are decided by the magnitude of the quadratic term and the magnetic field.

## INTRODUCTION

Weyl semimetal, a three-dimensional (3D) analog to graphene, has a pair of non-degenerate bands crossing at isolated points (Weyl nodes) within the Brillouin zone.^{1–4} Around the Weyl nodes, the dispersion is linear and the low energy physics is described by the 3D two-component Dirac Hamiltonian *H* = ± *ħ v _{F}*

**σ**⋅

**k**, where

*v*is the Fermi velocity,

_{F}**σ**is the vector of the Pauli matrices,

**k**is the momentum, and ± defines the chirality of the Weyl node. Weyl semimetals can only exist in system with either time-reversal or inversion symmetry broken.

^{5–8}When both of the symmetries are present, the Weyl nodes must be degenerate,

^{5,9}and the system is thus described by four-component Dirac Hamiltonian. Such a material is referred to 3D Dirac semimetal. Na

_{3}Bi

^{10,11}and Cd

_{3}As

_{2}

^{12–14}have been predicted and verified to be 3D Dirac semimetals. Recently a series of experiments have also realized Weyl semimetal state in TaAs.

^{15–18}A Zeeman field

**B**along the primary axes will break the time-reversal symmetry of the Dirac semimetal and split the Dirac node (degenerate Weyl nodes with different chirality) into two chiral Weyl nodes in the momentum space along the Zeeman field’s direction.

^{9}

When a material is subjected to an external magnetic field, Landau levels directly related to the nature of the fermion will form in the electronic density of state. The absorption lines in the optical conductivity, which result from the transitions between these Landau levels, thus provide a useful method to investigate the nature of the fermion. Researches have pointed out that there is a series of asymmetric peaks lying on top of the linear background in the optical conductivity of Weyl semimetal in present of magnetic field.^{19} For two-dimensional (2D) Dirac materials like graphene and the surface state of 3D topological insulator, the results are similar but the background is flat.^{20–22} Adding a finite Schrödinger term to the surface state of 3D topological insulator breaks the particle-hole symmetry and splits the magneto-optical absorption line into two peaks structure.^{22,23} In fact, particle-hole asymmetry also exists in Weyl semimetals and Dirac semimetals.^{10,12} We often neglect the quadratic term in momentum because the effective mass is very large. If we consider larger energy region and larger magnetic field, it will be more practical to include the quadratic term.

Besides, it is easier to measure the optical conductivity in Weyl semimetal than only the surface of a 3D topological insulator as it needs to remove the effect of the bulk. We wonder whether a small quadratic term in Weyl semimetal will influence the magneto-optical absorption lines or not. In addition, owing to one more dimension, the dispersive nature of the Landau levels in Weyl semimetal may produce richer results than 2D case. So in this contribution, we first calculate the energies of the Landau levels and the density of state for single Weyl node with quadratic term in momentum in present of magnetic field. And then we study in detail the absorption part of the magneto-optical conductivity of the system. Our results can be applied to 3D fermion system with both linear (Dirac) and quadratic (Schrödinger) terms.

## METHODOLOGY

We begin with the Hamiltonian for single isolated Weyl node with isotropic dispersion and including the quadratic term in momentum with effective mass *m*

Because of the lack of data for Weyl semimetal, we refer to the four-component Hamiltonian for Dirac semimetal Na_{3}Bi^{10} and Cd_{3}As_{2}^{12}

where $ \u03f5 0 k = C 0 + C 1 k z 2 + C 2 k x 2 + k y 2 $, *k*_{±} = *k _{x}* ±

*ik*, and $M k = M 0 + M 1 k z 2 + M 2 k x 2 + k y 2 $. For the sake of simplicity, we neglect the large out-of-plane anisotropy, and adopt the parameters for Na

_{y}_{3}Bi,

^{10}

*C*

_{2}= 8.4 eV ⋅ Å

^{2}and

*A*= 2.46 eV ⋅ Å to obtain the effective mass and Fermi velocity, respectively. So we have

*m*= 0.45

*m*(

_{e}*m*is the electron mass) and

_{e}*v*= 3.74 × 10

_{F}^{5}m/s in our following numerical calculation. In present of a uniform magnetic field

**B**along

*z*direction, the Hamiltonian can be obtained through

**k**→

**Π**=

**k**+

*e*

**A**/

*ħ*, with

**A**the vector potential and in Landau gauge $A= \u2212 B y , 0 , 0 $. Introducing the raising and lowering operators

with the magnetic length $ l B = \u0127 / e B $, which satisfy the commutation rule [*a*, *a*^{†}] = 1, we can have the Hamiltonian

Then we can solve the eigenvalue problem *Hψ* = *Eψ*. For *n* = 1, 2, 3... and *s* = + / −, we have the eigen-energies

with $ \epsilon 0 = \u0127 2 m l B 2 = \u0127 e B m $ related to the equivalent space of the Landau levels in pure Schrödinger fermion system, and $ \epsilon 1 = 2 \u0127 2 v F 2 / l B 2 = v F 2 \u0127 e B $ related to the Landau levels in pure Dirac fermion system. Just like Ref. 22, we can introduce $P= \epsilon 1 2 / \epsilon 0 2 $ to measure the “Diracness” of the system. When *P* → ∞ (or 0), the system is pure Dirac (or Schrödinger) system. The corresponding eigenstates have the form

with $ n $ the usual Fock state, and

For special case *n* = 0, we have

and

Figure 1(a) shows the Landau levels’ energies at *k _{z}* = 0 as a function of the magnetic field. Owing to the large mass

*m*= 0.45

*m*we consider here, our system is more “Dirac” than “Schrödinger” like. But the mirror symmetry between positive and negative branches is visibly broken for large magnetic field and high index Landau levels. Figure 1(b) show the dispersive Landau levels along

_{e}*k*direction. Unlike 2D case, the dispersive structure can lead to more possible optical transitions (indicated by vertical arrows) and richer results. Using these eigen-energies, we calculate the density of state in magnetic field

_{z} The results are displayed in Fig. 2. We also calculate the density of state for system without magnetic field, $ D 0 \u0127 \omega = m \pi 2 \u0127 3 v F 2 + \u0127 \omega / m v F 2 + 2 \u0127 \omega / m \u2212 v F $ and $ D 0 \u0127 \omega , m = \u221e = \u0127 \omega 2 2 \pi 2 \u0127 v F 3 $ for comparison. We can see that the density of state for the Landau level system is a series of peaks on top of the zero magnetic field background. Compared with Fig. 2(b), (a) shows that the quadratic term strikingly breaks the particle-hole symmetry, and it has a stronger effect in the negative energy region than the positive one. In 2D case, the *n* = 0 Landau level lies exactly at zero energy for pure Dirac system or near zero-energy even consider the quadratic term.^{22} Thus, there is a peak near zero energy for the 2D density of state. In 3D case, however, the density of state around *ħ ω* = 0 is rather small though not exactly zero because the *n* = 0 Landau level is dispersive in *k _{z}* instead of a constant.

## RESULTS AND DISCUSSION

The magneto-optical conductivity of the Landau level system in the clean limit can be obtained through the Kubo formula

where $ f E n s = 1 exp E n s \u2212 \mu / k B T + 1 $ is the usual Fermi distribution function with chemical potential *μ* and temperature *T*. The current operator *j*_{α} is related to the velocity operator *v*_{α} and is given as follows.

And the matrix elements are

with *λ*_{1} = *λ*_{2} = 1 if *α* = *x*, and *λ*_{1} = − *λ*_{2} = − *i* if *α* = *y*. So the optical selection rule *n* − *n*′ = ± 1 still exits for the Landau level transition. As we are only interested in the absorptive part of the optical conductivity, i.e., the real part of the longitudinal conductivity Re*σ _{xx}* and the imaginary part of the transverse one Im

*σ*, by using the matrix elements above and $ 1 x + i 0 + =P 1 x \u2212i\pi \delta x $, we can derive

_{xy}where

We consider low temperature *T* = 4 K and numerical calculate Re*σ _{xx}* and Im

*σ*in present of magnetic field

_{xy}*B*= 5 T with chemical potential falling between

*n*= 0 and positive

*n*= 1 Landau level at

*k*= 0 (just as the gray dotted line shown in Fig. 1(b1)). The results are shown in Fig. 3(a). And the gray dotted line is the well-known conductivity $Re \sigma x x 0 = e 2 \omega 24 \pi \u0127 v F \theta \u0127 \omega / 2 \u2212 \mu $ for 3D Dirac fermion system

_{z}^{24}without magnetic field as background. Comparing the results of (a

_{1}) with (a

_{2}), we find that except the first two increasing platforms related to the

*n*= 0 Landau level transitions, all other peaks in Re

*σ*are split into two peaks if considering the quadratic mass term. Each peak corresponds to certain inter-Landau level transition, shown schematically by the vertical arrows in Fig. 1(b1) and typed next to the peak of Re

_{xx}*σ*. And Im

_{xx}*σ*is no longer zero, but shows a striking oscillation with a positive peak first and then a negative one at the same position of the two split peaks in Re

_{xy}*σ*. These results are similar to that for the surface state of 3D topological insulator,

_{xx}^{22,23}and can be directly explained from the symmetry broken between the positive and negative branches of Landau level by the quadratic term. Because of this asymmetry, the transition from negative

*n*to positive

*n*+ 1 Landau level require more energy than the transition from negative

*n*+ 1 to positive

*n*Landau level. This energy difference will split the original degenerate peaks for pure Dirac system into two.

For further understanding, we also calculate the optical conductivity for circularly polarized light. It is defined as *σ*_{±} = *σ _{xx}* ±

*iσ*for right-handed (

_{xy}*σ*

_{+}) and left-handed (

*σ*

_{−}) polarization. So the absorptive part is $Re \sigma \xb1 \u0127 \omega =Re \sigma x x \u0127 \omega \u2213Im \sigma x y \u0127 \omega $. The results are shown in Fig. 3(b). There is no peak splitting behavior. However, the peaks of Re

*σ*

_{+}and Re

*σ*

_{−}with quadratic mass term are displaced in energy. Compared with the results without mass term, the peaks of Re

*σ*

_{+}are shifted to higher energy while the peaks of Re

*σ*

_{−}to lower energy. The combination of Re

*σ*

_{+}and Re

*σ*

_{−}just results in the two peaks structure in Re

*σ*shown in (a

_{xx}_{1}). So we know that the two kinds of transition correspond to different handedness of polarizations. Right-handed polarization (

*σ*

_{+}) light only cause transitions from negative

*n*to positive

*n*+ 1 Landau level, while left-handed polarization (

*σ*

_{−}) only cause transitions from negative

*n*+ 1 to positive

*n*Landau level, shown by blue and red vertical arrows in Fig. 1(b), respectively. That also explains why there is no split in the first two increasing platforms in (a

_{1}) because each platform only relates to one kind of transition.

We also notice that the position of the first two increasing platforms in Fig. 3(a) is totally decided by the chemical potential’s position owing to the Pauli exclusion principle. And as the Landau levels are dispersive in *k _{z}*, the absorption line has increasing platforms instead of an abrupt peak in 2D case. To further study the chemical potential’s influence, we have similar figures like Fig. 3 but with chemical potential falling between positive

*n*= 1 and 2 Landau level at

*k*= 0 (gray dotted line shown in Fig. 1(b2)) in Fig. 4. We notice that the patterns at high energy remain the same as Fig. 3, but are quite different at low energy. Owing to the Pauli block, transition from negative

_{z}*n*= 2 to positive

*n*= 1 Landau level does not exist everywhere but only for larger

*k*(refer to arrow transitions in Fig. 1(b2)). So the corresponding peak disappears and becomes a platform at higher energy in Re

_{z}*σ*and Re

_{xx}*σ*

_{−}. As for Im

*σ*, the first positive oscillation peak disappears. Transition from negative

_{xy}*n*= 1 to

*n*= 0 (or

*n*= 0 to positive

*n*= 1) Landau level requires more energy, so the platforms shift to higher energy. However, Transition from

*n*= 0 to positive

*n*= 1 can also occur with a little energy. There is also an extra intra-branch transition, form positive

*n*= 1 to 2 Landau level, which also needs a little energy (refer to the two blue short arrows in Fig. 1(b2)). And both of the transitions need right-handed polarization light. Therefore, there is a very large absorption peak at lower energy region in Re

*σ*and Re

_{xx}*σ*

_{+}.

For chemical potential much higher than the magnetic energy, the Landau level quantization is no longer important, and this is the semiclassical limit. If we are interested in energy range much less than the chemical potential, only intra-branch transitions are involved. For *E*_{n,+,kz=0} < *μ* < *E*_{n+1,+,kz=0}, with *n* ≫ 1, we then have the cyclotron resonance energy around

And the definitions of ε_{0} and ε_{1} are given after Eq. (4). For pure Dirac system without quadratic mass term $ m \u2192 \u221e $, we have $\u0127 \omega c \u2248 \epsilon 1 2 n $. So in semiclassical limit, the cyclotron resonance energy will displace to higher energy if consider the quadratic term. And the displacement is related to the Landau level’s space for pure Schrödinger fermion system which is determined by the magnitude of the effective mass and the magnetic field. Figure 5(a) gives the semiclassical cyclotron resonance energy as a function of the chemical potential for both with and without mass term. And (b) is the real part of the longitudinal conductivity Re*σ _{xx}* in semiclassical limit. It shows a single resonance absorption peak at low energy which contains most of the spectral weight. Although

*μ*lying between

*E*

_{n,+,kz=0}and

*E*

_{n+1,+,kz=0}, this resonance peak results from a series of transitions from

*E*

_{l,+}to

*E*

_{l+1,+}with $l\u2208 0 , n $ instead of one owing to the dispersive relation along

*k*of the Landau levels. As this resonance peak appears at very low energy, its line shape is easy to be influenced by temperature.

_{z}All results considered above are for *m* = 0.45*m _{e}* and

*B*= 5 T at low temperature

*T*= 4 K. As we know, for smaller effective mass and larger magnetic field, the “Diracness” parameter

*P*becomes smaller, so the system is more Schrödinger like. Under such circumstance, the asymmetry of the positive and negative branches of Landau level is more obvious (refer to Fig. 1(a)). So the peaks split in Re

*σ*and the displacement between Re

_{xx}*σ*

_{+}and Re

*σ*

_{−}will become more dramatic. This can also be used to estimate the magnitude of contribution between the linear (Dirac) and quadratic (Schrödinger) terms.

## CONCLUSION

In summary, we have added a quadratic term in momentum to the linear dispersion of single Weyl node. And we find that this quadratic term breaks the mirror symmetry between the positive and negative branches of Landau level in present of magnetic field. Because of this asymmetry, the transition from negative *n* to positive *n* + 1 Landau level is no longer have the same energy as the transition from negative *n* + 1 to positive *n*. So it leads to a peak splitting in the absorption line of longitudinal conductivity and an oscillation pattern in the absorption part of the Hall conductivity. And the two split peaks in Re*σ _{xx}* correspond to the absorption of left-handed and right-handed polarization light, respectively. So the absorptions for circular polarized light still have single peak structure, but there is a displacement in energy for right-handed and left-handed polarization compare with the results without mass term. Besides, the dispersive nature of the Landau levels often leads to increasing platform, whose position is decided by the chemical potential, instead of abrupt peak at the beginning of the absorption line. For very large chemical potential, intra-branch transitions between the Landau levels become important. And the absorption line becomes a single resonance peak at low energy, whose position is shifted to higher energy by the quadratic term. Our work provides an optical method to study the linear (Dirac) and small quadratic (Schrödinger) contribution to the 3D fermion system, which can be realized experimentally in Weyl semimetal or 3D Dirac semimetal material.

## ACKNOWLEDGEMENTS

This work is supported by the State Key Laboratory of Optoelectronic Materials and Technologies of Sun Yat-sen University.