Magneto-optical conductivity of double Weyl semimetals

TL;DR

This paper studies the magneto-optical response of double Weyl semimetals, revealing anisotropic conductivity and resonant peaks that scale with magnetic field strength, considering various physical effects.

Contribution

It provides a detailed analysis of the anisotropic magneto-optical conductivity in double Weyl semimetals, including the effects of chemical potential, temperature, impurities, and symmetry breaking.

Findings

Resonant peaks in optical conductivity scale linearly with magnetic field strength.

Optical conductivity exhibits anisotropy with distinct background behaviors.

Chemical potential, temperature, and impurities significantly influence the optical response.

Abstract

We investigate the magneto-optical response of double Weyl semimetals whose energy dispersion is intrinsically anisotropic. We find that in the presence of a magnetic field, the most salient feature of the optical conductivity is a series of resonant peaks with the corresponding frequencies scaling linearly with the strength of the magnetic field. In addition, the optical conductivity is found to be anisotropic, with two of the three longitudinal components residing at a linear background and the remaining one at a constant background. The effects of chemical potential, temperature, impurity scattering, and particle-hole symmetry breaking on the optical conductivity are also studied.

Equations

$$H_{\chi }(\mathbf{k})=\lambda (k_x^2-k_y^2){\tau }_x+2\lambda k_x{k}_y{\tau }_y+\chi v{k}_z{\tau }_z,$$

$$E_{\pm ,\chi }(\mathbf{k})=\pm \sqrt{v^2{k}_z^2+{\lambda }^2(k_x^2+k_y^2{)}^2},$$

$$H_{\chi }(\mathbf{\Pi })=\lambda ({\Pi }_x^2-{\Pi }_y^2){\tau }_x+\lambda ({\Pi }_x{\Pi }_y+{\Pi }_y{\Pi }_x){\tau }_y+\chi v{\Pi }_z.$$

$$a=\frac{l_B}{\sqrt{2}}({\Pi }_x-i{\Pi }_y),a^{†}=\frac{l_B}{\sqrt{2}}({\Pi }_x+i{\Pi }_y),$$

$$\displaystyle H=\left(\begin{array}[]{cc}\chi vk_{z}&\frac{2\lambda}{l_{B}^{2}}a^{2}\\ \frac{2\lambda}{l_{B}^{2}}{a^{\dagger}}^{2}&-\chi vk_{z}\\ \end{array}\right),$$

$$\displaystyle E_{n,s,\chi}(k_{z})=\left\{\begin{array}[]{cc}sE_{n},&n\geq 2,\\ -\chi vk_{z},&n=0,1,\end{array}\right.$$

$${\alpha }_{n,s,\chi }=s\sqrt{\frac{E_{n,s,\chi }+\chi v{k}_z}{2E_{n,s,\chi }}},{\beta }_{n,s,\chi }=\sqrt{\frac{E_{n,s,\chi }-\chi v{k}_z}{2E_{n,s,\chi }}}.$$

$${\sigma }_{\mu \nu }(\omega )$$

$$j_{x,\chi }$$

$$j_{y,\chi }$$

$$j_{z,\chi }$$

$$\mathrm{Re}({\sigma }_{\mathrm{xx}}(\omega ))$$

$$\mathrm{Re}({\sigma }_{\mathrm{zz}}(\omega ))$$

$$\mathrm{Im}({\sigma }_{\mathrm{xy}}(\omega ))$$

$$\mathrm{Re}({\sigma }_{\mathrm{xx}}(\bar{\omega }))$$

$$\mathrm{Re}({\sigma }_{\mathrm{zz}}(\bar{\omega }))$$

$$\mathrm{Im}({\sigma }_{\mathrm{xy}}(\bar{\omega }))$$

$$\displaystyle\xi(\bar{\omega})=\left\{\begin{array}[]{cc}\lfloor\frac{\bar{\omega}^{2}}{2\sqrt{\bar{\omega}^{2}-1}}\rfloor,&\text{if }\bar{\omega}>1,\\ \infty,&\text{otherwise.}\end{array}\right.$$

$$\displaystyle H=\left(\begin{array}[]{cc}\chi vk_{z}+sb&\frac{2\lambda}{l_{B}^{2}}a^{2}\\ \frac{2\lambda}{l_{B}^{2}}{a^{\dagger}}^{2}&-\chi vk_{z}+sb\\ \end{array}\right),$$

$$E_{n,s,\chi }(k_z)$$

$$E_{1,s,\chi }(k_z)$$

$$E_{0,s,\chi }(k_z)$$

Full text

I Introduction

Recently, topological semimetals have been actively studied because of the existence of nontrivial band crossings protected by topology and symmetries Chiu et al. (2016). According to the dimension and degeneracy of the band crossings, the currently most studied topological semimetals can be classified into three classes: Weyl semimetals Murakami (2007); Wan et al. (2011); Burkov and Balents (2011); Weng et al. (2015); Huang et al. (2015); Xu et al. (2015a); Lv et al. (2015); Lu et al. (2015), Dirac semimetals Young et al. (2012); Wang et al. (2012, 2013); Neupane et al. (2014); Xu et al. (2015b); Liu et al. (2014); Borisenko et al. (2014) and nodal line semimetals Burkov et al. (2011); Bzdušek et al. (2016); Chen et al. (2017); Yan et al. (2017); Bi et al. (2017). For the Weyl semimetals, the band crossings are isolated points (zero dimension) with two-fold degeneracy, known as the Weyl points, and play the role of monopoles in the Brillouin zone. As the net charge of the monopoles in the Brillouin zone must be zero, the number of Weyl points with opposite monopole charge must be equal Nielsen and Ninomiya (1981).

The monopole charge of a Weyl point is equal to the number of Berry fluxes passing through a closed surface enclosing the target Weyl point only. So far, most studies on Weyl semimetals have focused on the case with the lowest monopole charge, $C=\pm 1$ (we will refer to Weyl semimetals with $C=\pm 1$ as single Weyl semimetals for notational simplicity) Hosur and Qi (2013); Yan and Felser (2017); Lu and Shen (2016); Hasan et al. (2017); Burkov (2017). The reason that this case has attracted special research interest is because the energy dispersion away from the Weyl points is linear in all directions and thus an analog of the elementary Weyl fermions in particle physics, furthermore, it is the one found in condensed matter experiments to date Hasan et al. (2017). However, the much more unexplored Weyl points with higher monopole charge Xu et al. (2011); Fang et al. (2012); Huang et al. (2016); Lai (2015); Jian and Yao (2015); Zhang et al. (2016); Wang et al. (2016); Chen and Fiete (2016); Ahn et al. (2016, 2017); Hayata et al. (2017); Gupta (2017); Roy and Sau (2015); Li et al. (2016) are also of fundamental interest in at least two aspects. First, the increase of monopole charge no doubt will enhance a series of effects predicted or observed in single Weyl semimetals, like the anomalous Hall effect Yang et al. (2011); Burkov (2014); Chan et al. (2016); Yan and Wang (2016), and the ones related to chiral anomaly Son and Spivak (2013); Liu et al. (2013); Aji (2012); Zyuzin and Burkov (2012); Wang and Zhang (2013); Hosur and Qi (2015); Kim et al. (2013); Parameswaran et al. (2014). Second, the energy dispersion will become intrinsically anisotropic for higher monopole charge, which consequently will induce novel physics absent in single Weyl semimetals. For instance, it is found that Weyl points with higher monopole charge exhibit anisotropic screening to the Coulomb interactions Lai (2015); Jian and Yao (2015); Zhang et al. (2016); Wang et al. (2016).

In this work, we will investigate the magneto-optical conductivity of Weyl semimetals with $C=\pm 2$, the so-called double Weyl semimetals, of which HgCr2Se4 Xu et al. (2011); Fang et al. (2012) and SrSi2 Huang et al. (2016) were predicted as candidate materials. In the presence of a magnetic field, the continuum spectrum of the Weyl Hamiltonian will transform to a series of discrete Landau levels whose extrema have divergent density of states. When optical transition between two Landau levels takes place at their extrema, a resonant peak will show up in the real part of the optical conductivity. In experiments, from these resonant peaks, information, like the energy gap and the Fermi velocity, of the underlying band structure can be extracted Jiang et al. (2007); Deacon et al. (2007); Plochocka et al. (2008); Schafgans et al. (2012); Orlita et al. (2015); Chen et al. (2015); Akrap et al. (2016). For instance, it is known that when the frequencies of the resonant peaks ($\omega$) and the magnetic field ($B$) follow an $\omega\propto\sqrt{B}$ law, the energy dispersion of the Hamiltonian is gapless and linear, like those in the Dirac semimetals and single Weyl semimetals Gusynin et al. (2007); Ashby and Carbotte (2013); Malcolm and Nicol (2014). For the double Weyl semimetals concerned in this paper, we find the resonant peaks of the optical conductivity exhibit several distinctive features which can thus be applied to determine whether a material is double Weyl semimetal or not in experiments.

The paper is organized as follows. In Sec. II, we give the Hamiltonian and the formula of the optical conductivity. In Sec. III, numerical results of the conductivity are given. In Sec. IV, we give our estimation of the conditions needed for the magneto-optical experiments. Discussions and conclusions are presented in Sec. V.

II Formalism

As Weyl points are crossings of two bands without degeneracy, the low-energy effective Hamiltonian of double-Weyl semimetals is given by ($\hbar=c=k_{B}=1$)

[TABLE]

where $\tau_{i}$ are Pauli matrices, $\lambda$ is a constant parameter with the dimension of inverse mass, $v$ refers to the velocity in the direction of linear dispersion and $k_{x,y,z}$ refer to the momenta relative to the Weyl points; $\chi=\pm 1$ denote two kinds of chirality. Without loss of generality, $v$ and $\lambda$ are assumed to be positive. The energy spectra can be readily obtained,

[TABLE]

which are linear along the $z$ direction and quadratic in the $x$-$y$ plane.

Now we consider a magnetic field that is exerted along the $z$ direction, ${\bf B}=B\hat{z}$. Following the standard minimal coupling, i.e., ${\bf k}\rightarrow\mathbf{\Pi}={\bf k}+e{\bf A}$ with $A=(0,Bx,0)$, the Hamiltonian in Eq. (1) will be rewritten as

[TABLE]

Note that $\Pi_{x}$ and $\Pi_{y}$ do not commute, so we have adopted the symmetric ordering for operators. The magnetic field will discretize the continuum energy spectra into a series of Landau levels (or Landau bands), whose form can be easily obtained by introducing the following ladder operators,

[TABLE]

where $l_{B}=1/\sqrt{eB}$ is the magnetic length. Bring Eq. (4) into Eq. (3) and rewrite the Hamiltonian into a matrix form,

[TABLE]

it is readily found that the Landau levels are given by

[TABLE]

where $E_{n}(k_{z})\equiv\sqrt{v^{2}k_{z}^{2}+4n(n-1)(\lambda eB)^{2}}$; $s=+$ and $s=-$ correspond to the particle branch and hole branch, respectively; and when $n=0$ or $1$, $s=+$ is implicitly assumed. A graph illustration of the Landau levels is shown in Fig. 1. Before proceeding, we make two remarks on the Landau levels: (i) $E_{n}(0)\propto B$, such a linear dependence is distinct from the $E_{n}(0)\propto\sqrt{B}$ law found in the Dirac and single Weyl semimetals Gusynin et al. (2007); Ashby and Carbotte (2013); (ii) there are two degenerate chiral Landau levels ($n=0,1$), with the number equating to the monopole charge of the Weyl points. As we will show in Sec. III.4, when the degeneracy is lifted, optical transition between the two chiral Landau levels will take place.

For $n\geq 2$, we denote the eigenvectors corresponding to $E_{n,s,\chi}(k_{z})$ as $|n,s,\chi\rangle=(\alpha_{n,s,\chi}|n-2\rangle,\beta_{n,s,\chi}|n\rangle)^{T}$, where $|n\rangle$ satisfies $a|n\rangle=\sqrt{n}|n-1\rangle$ and $a^{{\dagger}}|n\rangle=\sqrt{n+1}|n+1\rangle$. The coefficients $\alpha_{n,s,\chi}$ and $\beta_{n,s,\chi}$ can be determined straightforwardly, with

[TABLE]

For $n=0,1$, the eigenvectors are simply $(0,1)^{T}$.

The optical conductivity can be obtained from the Kubo formula, which is

[TABLE]

where $f_{n,s,\chi}=1/[\exp{(E_{n,s,\chi}-\mu)/T}+1]$ is the Fermi-Dirac distribution function, with $\mu$ the chemical potential and $T$ the temperature; $\Gamma$ denotes the impurity scattering rate, in this work, we assume that all Landau levels share the same $\Gamma$ for simplicity; $j_{\mu,\chi}$ ($\mu=x,y,z$) denote the current operators, whose explicit forms are

[TABLE]

In the clean limit, i.e., $\Gamma=0$, the dissipative components which correspond to the absorption of light are found to be

[TABLE]

[TABLE]

where we have defined $f_{n,\pm}\equiv 1/[\exp{(\pm E_{n}-\mu)/T}+1]$ for simplicity of notation.

Note that $\rm{Re}(\sigma_{yy})=\rm{Re}(\sigma_{xx})$ due to rotational symmetry. From Eq. (12) and Eq. (21), it is readily found that the other two transverse components $\sigma_{xz}$ and $\sigma_{yz}$ are zero. The selection rules of the optical transition can be immediately read out from the $\delta$-functions in Eq. (22). The $\delta$-functions in the first line of $\rm{Re}(\sigma_{xx})$ and $\rm{Im}(\sigma_{xy})$ denote the transitions between nearest-neighbor Landau levels within the same branch, i.e., the selection rules are $\Delta n=n_{p,f}-n_{p,i}=1$ and $\Delta n=n_{h,f}-n_{h,i}=-1$, where $n_{p(h),i}$ and $n_{p(h),f}$ denote the indices of the initial particle (hole) Landau level and the final particle (hole) Landau level of the transition process, respectively (see the caption of Fig. 1). The $\delta$-functions in the second line of $\rm{Re}(\sigma_{xx})$ and $\rm{Im}(\sigma_{xy})$ indicate that the transitions can also occur between the particle-branch Landau levels and the hole-branch Landau levels, with the selection rules $\Delta n=n_{p,f}-n_{h,i}=\pm 1$. The $\delta$-function in $\rm{Re}(\sigma_{zz})$ indicates that the transitions can only occur between the particle-branch Landau levels and the hole-branch Landau levels, with the selection rule $\Delta n=n_{p,f}-n_{h,i}=0$.

After performing the integration over $k_{z}$, we find that

[TABLE]

where $\bar{\omega}\equiv\omega l_{B}^{2}/2\lambda$, $\bar{\mu}\equiv\mu l_{B}^{2}/2\lambda$ and $\bar{T}\equiv Tl_{B}^{2}/2\lambda$ are dimensionless quantities, and the upper bound of the summation over $n$, $\xi(\bar{\omega})$, is given by

[TABLE]

The expressions of the optical conductivity are too complicated to obtain analytical results. Thus, we will resort to numerical calculations in the following discussions.

III Magneto-optical conductivity

III.1 Longitudinal component

Fig. 2 shows the real part of the longitudinal conductivity at the neutrality condition, i.e., $\mu=0$. For both $\text{Re}(\sigma_{xx})$ and $\text{Re}(\sigma_{zz})$, the most prominent feature is a series of resonant peaks originated from the optical transitions between Landau levels. Due to the difference in selection rules, it is found that the resonant peaks for $\text{Re}(\sigma_{xx})$ are located at $\omega=2[\sqrt{n(n+1)}+\sqrt{n(n-1)}]\lambda eB$, while for $\text{Re}(\sigma_{zz})$, they are located at $\omega=4\sqrt{n(n-1)}\lambda eB$. It is immediately seen that the frequencies of the resonant peaks, for both $\text{Re}(\sigma_{xx})$ and $\text{Re}(\sigma_{zz})$, follow an $\omega\varpropto B$ law, which is distinct from the $\omega\varpropto\sqrt{B}$ law obeyed by the Dirac semimetals and single Weyl semimetals Gusynin et al. (2007); Ashby and Carbotte (2013). In experiments, the frequency ratio between neighboring peaks is an important quantity for figuring out the Landau level indices related to the peaks, and further extracting information about the underlying band structure, e.g., whether an energy gap exists or not, and the value of Fermi velocity Jiang et al. (2007); Deacon et al. (2007); Orlita et al. (2015); Chen et al. (2015); Akrap et al. (2016). Here the frequency ratio between the $(n+1)$-th and $n$-th (counting from the left) resonant peaks for $\text{Re}(\sigma_{xx})$ is given by $[\sqrt{(n+2)(n+1)}+\sqrt{n(n+1)}]/[\sqrt{n(n+1)}+\sqrt{n(n-1)}]$, and for $\text{Re}(\sigma_{zz})$ it is $\sqrt{n+2}/\sqrt{n}$. Furthermore, the energy offset between the $(n+1)$-th and $n$-th peaks is found to decrease with the increase of $n$ and saturates to $4\lambda eB$ for large $n$, such a behavior is also distinct from the Dirac and single Weyl semimetals, in which the energy offset will monotonically decrease to zero Gusynin et al. (2007); Ashby and Carbotte (2013).

Besides the difference in the locations of the resonant peaks, another distinctive feature between $\text{Re}(\sigma_{xx})$ and $\text{Re}(\sigma_{zz})$ is that the resonant peaks of the former sit on a linear background, while those of the latter sit on a constant background (see insets of Fig. 2). Both observations agree with the fact that without a magnetic field, the optical conductivity $\text{Re}(\sigma_{xx})$ is linear in frequency whereas $\text{Re}(\sigma_{zz})$ is a constant independent of frequency Ahn et al. (2017). Such distinction originates from the apparent discrepancy of dispersion relations between $x$-$y$ plane and $z$ direction and is not present in single Weyl semimetals for which all longitudinal components should reside at a linear background Ashby and Carbotte (2013).

Fig. 3 further illustrates the effect of chemical potential $\mu$ to the longitudinal conductivity. For $\text{Re}(\sigma_{xx})$, when $\mu$ is tuned away from the neutrality condition but still only crosses the chiral Landau levels, i.e., $\bar{\mu}<\sqrt{2}$ (due to particle-hole symmetry, we will only consider the $\mu>0$ case), the relatively weak peak at $\bar{\omega}=\sqrt{2}$ shown in Fig. 2(a) will be split into two absorption edges which are located at $\bar{\omega}_{1}=\sqrt{\bar{\mu}^{2}+2}-\bar{\mu}$ and $\bar{\omega}_{2}=\sqrt{\bar{\mu}^{2}+2}+\bar{\mu}$, as shown in Fig. 3(a). For $\text{Re}(\sigma_{zz})$, when $\mu$ only crosses the chiral Landau levels, however, the change of $\mu$ has no effect to the optical conductivity, this is because the selection rule for $\text{Re}(\sigma_{zz})$ is $\Delta n=n_{p,f}-n_{h,i}=0$ with $n_{p(h)}\geq 2$. When $\mu$ is tuned to cross the $n$-th particle Landau level with $n\geq 2$, we find that for $\text{Re}(\sigma_{xx})$, a new resonant peak will appear at $\bar{\omega}=\sqrt{n(n+1)}-\sqrt{n(n-1)}$ (the case with $n=2$ is shown in Fig. 3(a)). The new peak corresponds to the intra-branch transition between the $n$-th and $(n+1)$-th particle Landau levels. The transition is now allowed because the $n$-th particle Landau level gets occupied while the $(n+1)$-th particle Landau level remains empty. For $\text{Re}(\sigma_{zz})$, when $\mu$ is tuned to cross the $n$-th particle Landau level, the resonant peak at $\bar{\omega}=2\sqrt{n(n-1)}$ will disappear and an absorption edge will appear at $\bar{\omega}=2\bar{\mu}$ (the case with $n=2$ is shown in Fig. 3(b)).

III.2 Transverse component

The transverse (Hall) conductivity $\text{Im}(\sigma_{xy})$ at $\mu=0$ vanishes for all frequencies, this fact can be read directly from the expression of $\text{Im}(\sigma_{xy})$ in Eq. (23). A more physical explanation is that the transverse conductivity has four contributions: I, the transition from the $n$-th hole-branch Landau level to the $(n+1)$-th particle-branch Landau level; II, the transition from the $(n+1)$-th hole-branch Landau level to the $n$-th particle-branch Landau level; III, the transition from the $(n+1)$-th hole-branch Landau level to the $n$-th hole-branch Landau level; IV, the transition from the $n$-th particle-branch Landau level to the $(n+1)$-th particle-branch Landau level. Importantly, the contributions from I and III have a sign difference to the ones from II and IV, respectively. Moreover, at $\mu=0$, due to particle-hole symmetry the contributions exactly cancel each other out; thus, $\text{Im}(\sigma_{xy})$ vanishes for all frequencies in this case. Away from the neutrality condition, particle-hole symmetry is broken and consequently the transverse conductivity $\text{Im}(\sigma_{xy})$ will take nonzero values in certain frequency regions, as shown in Fig. 4(a).

In experiments, the quantities $\sigma_{\pm}\equiv\sigma_{xx}\pm i\sigma_{xy}$ are also of interest because they can be used to determine the polarization of lights. Concretely, $\sigma_{+}$ corresponds to right-handed polarized light, and $\sigma_{-}$ corresponds to left-handed polarized light. The absorption parts of $\sigma_{\pm}$ are shown in Fig. 4(b). At the lower frequency regime, i.e., $\omega\lesssim 2\mu$, $\text{Re}(\sigma_{-})$ vanishes as the longitudinal conductivity makes a cancellation with the transverse conductivity. The picture is similar to that of the single Weyl semimetals Ashby and Carbotte (2013).

III.3 Temperature effect and impurity scattering effect

The effect of finite temperature to the optical conductivity enters through the Fermi-Dirac distribution function. Fig. 5(a) shows the temperature effect to $\text{Re}(\sigma_{xx})$ at $\mu=0$, it is readily seen that with the increase of temperature, the height of the resonant peaks will be suppressed, and resonant peaks corresponding to intra-branch transition which are absent at zero temperature will show up (see the blue dashed line and the green dashed line). The effect of impurity scattering to $\text{Re}(\sigma_{xx})$ is shown in Fig. 5(b). It is readily seen that the main effect of the impurity scattering is just a suppression of the resonant peaks.

For $\text{Re}(\sigma_{zz})$ under the neutrality condition, finite temperature and impurity scattering mainly induce a suppression to the resonant peaks, no new peaks will show up (not shown here).

III.4 Effect of particle-hole symmetry breaking

Next we will explore the effect of particle-hole symmetry breaking since in real materials there is no constraint to preserve this symmetry. To achieve this, we add a diagonal quadratic term of the form $\frac{k_{x}^{2}+k_{y}^{2}}{2m}\tau_{0}$ to the original Hamiltonian in Eq. (1), with $m\gtrsim\lambda^{-1}$. The Hamiltonian with standard minimum coupling in matrix form thus changes to

[TABLE]

with the symmetry-breaking term $sb=\omega_{c}(a^{{\dagger}}a+\frac{1}{2})$, where $\omega_{c}=eB/m$. Correspondingly, the Landau levels become

[TABLE]

As the symmetry-breaking term does not change the selection rules (this can be simply confirmed by bringing the current operators into the Kubo formula), the positions of the resonant peaks can be readily figured out. For $\text{Re}(\sigma_{zz})$, the positions of the resonant peaks do no change. While for $\text{Re}(\sigma_{xx})$, the main resonant peaks will undergo splitting and position shift. Concretely, the inter-branch $\omega=2[\sqrt{n(n+1)}+\sqrt{n(n-1)}]\lambda eB$ peaks will be split to $\omega=\pm\omega_{c}+2[\sqrt{(n+1)n}+\sqrt{n(n-1)}]\lambda eB$, and the intra-branch $\omega=2[\sqrt{n(n+1)}-\sqrt{n(n-1)}]\lambda eB$ peaks will be split to $\omega=\pm\omega_{c}+2[\sqrt{(n+1)n}-\sqrt{n(n-1)}]\lambda eB$.

More interestingly, as the degeneracy of the two chiral Landau levels is lifted by the symmetry-breaking term, an optical transition within the two chiral Landau levels is allowed. As the energy offset between the two chiral Landau levels is fixed to $\omega_{c}$ for arbitrary $k_{z}$, an absorption peak at $\omega=\omega_{c}$ is expected. Compared to the transitions between chiral Landau levels and non-chiral Landau levels, this transition is remarkable in the sense that it will not be influenced by the change of the chemical potential as long as the chiral Landau levels cross the Fermi level (if we restrict ourselves to the continuum model, this is always the case). Note that this effect is exclusive to multi-Weyl semimetals as there is no chiral Landau level degeneracy to be lifted for singe-Weyl semimetals.

Although we have chosen a simple diagonal term to discuss the effect of particle-hole symmetry breaking, the above conclusions will qualitatively hold for more complicated symmetry-breaking terms. From the results above, we can see that although the particle-hole symmetry breaking will affect the frequencies of the resonant peaks, the influence is in fact marginal and can be analyzed and regulated in real experiments. The linear scaling dependence on the magnetic field strength and the anisotropic behavior of the longitudinal conductivity still hold.

III.5 Generalization to higher monopole charge

The above discussions could be systematically generated to multi-Weyl semimetals with arbitrary monopole charge $Q$. Specifically, the number of chiral Landau levels equals $Q$, and the Landau levels behave as $E_{n}(0)\propto B^{Q/2}$; while the selection rules for optical conductivity remains unmodified.

Thus for $Q\geq 3$, the scaling dependence of the resonant-peak frequencies on the magnetic field will follow the law $\omega\propto B^{Q/2}$, and the longitudinal conductivity will exhibit more anisotropic behaviors. This general observation can be utilized to determine the monopole charge of multi-Weyl semimetals.

IV Experimental estimation

Now we give a brief experimental estimation. We take the material candidate HgCr2Se4 Xu et al. (2011); Fang et al. (2012) as a concrete example. It was predicted that there are only two double-Weyl points in HgCr2Se4 which are very close to the Fermi energy Xu et al. (2011); Fang et al. (2012), thus, it is an ideal platform to test the above predictions on the optical conductivity of double-Weyl semimetals.

According to the band structure calculated in Ref. Fang et al. (2012), we find $\lambda(\hbar\pi/a)^{2}\approx 0.4$ eV with the lattice constant $a=10.753$ Å Baltzer et al. (1965), which gives $\lambda^{-1}\approx 1.63m_{e}$. Then the interval between two near-neighbor resonant peaks is estimated to be $\Delta\omega\approx 4\lambda\hbar eB\approx 0.28$ meV for $B=1$ T. To make the resonant peaks well separated in energy, the magnetic field can be chosen to the order of $10$ T. In accordance with the magnetic field, the adequate light for the magneto-optical experiment falls into the mid-infrared regime. As the energy scale is of the order of several meV, the sample should better be very clean and the experimental temperature should better be lower than $1$ meV, i.e., $T<10$ K.

V Discussions and conclusions

To conclude, in this paper, we studied the magneto-optical response of double Weyl semimetals. We found that the longitudinal components of the optical conductivity have a series of resonant peaks with their corresponding frequencies scaling linearly with the strength of the magnetic field, and their frequency ratios following a special law determined by the underlying Landau level structure. Note that the linear scaling law is distinct from the characteristic square-root scaling law obeyed by the Dirac semimetals and single Weyl semimetals Gusynin et al. (2007); Ashby and Carbotte (2013). We also found that the optical conductivity is quite anisotropic, with two of the three longitudinal components residing at a linear background and the rest one at a constant background. The anisotropy of the optical conductivity simply originates from the anisotropy of the energy dispersion of double Weyl semimetals. Because transitions related to the chiral Landau levels are strongly dependent on the positions of the Fermi energy, it is found that the tuning of chemical potential duly modifies the corresponding resonate peaks of the optical conductivity.

Furthermore, we found that when the degeneracy of the chiral Landau levels is lifted by a particle-hole symmetry breaking term, a new transition channel within the chiral Landau levels is opened for $\text{Re}(\sigma_{xx})$ , with the transition frequency directly controlled by symmetry breaking parameter and irresponsive to the change of chemical potential. The effect of finite temperature and impurity scattering to the optical conductivity is mainly a suppression of the resonant peaks, which indicates that our study is quite robust against these subtleties.

Among above results, the linear scaling law between the frequencies of the resonant peaks and the magnetic field, and the anisotropic behavior of the longitudinal conductivity are two most distinctive features which can be applied to determine whether the predicted candidate materials, like HgCr2Se4 Xu et al. (2011); Fang et al. (2012) and SrSi2 Huang et al. (2016), are truly double Weyl semimetals or not.

VI Acknowledgments

We are grateful to Zhongbo Yan and Phillip E. C. Ashby for their helpful discussions. This work is supported by NSFC under Grant NO. 11375168.