Magneto-transport phenomena of type-I multi-Weyl semimetals in co-planar setups

TL;DR

This paper investigates the unique magneto-transport properties of multi-Weyl semimetals with topological charge n>1, revealing how their transport coefficients depend on n and temperature, and verifying results through numerical lattice models.

Contribution

It provides analytical and numerical analysis of transport phenomena in multi-Weyl semimetals, highlighting the dependence on topological charge n and temperature effects.

Findings

LMC and PHC vary cubically with n at zero temperature

Finite temperature corrections scale with (n + n^2)T^2

TECs scale quadratically with n, PNC varies non-monotonically with n

Abstract

Having the chiral anomaly induced magneto-transport phenomena extensively studied in single Weyl semimetal (WSM) as characterized by topological charge $n=1$, we here address the transport properties in the context of multi-Weyl semimetals (m-WSMs) where $n>1$. Using semiclassical Boltzmann transport formalism with the relaxation time approximation, we investigate several intriguing transport properties such as longitudinal magneto-conductivity (LMC), planar Hall conductivity (PHC), thermo-electric coefficients (TECs) and planar Nernst coefficient (PNC) for m-WSMs in the co-planar setups with external magnetic field, electric field and temperature gradient. Starting from the low-energy model, we show analytically that at zero temperature both LMC and PHC vary cubically with topological charge as $n^3$ while the finite temperature ($T \neq 0$) correction is proportional to $(n+n^2)T^2$. Interestingly, we find that both the longitudinal and transverse TECs vary quadratically with topological charge as $n^2$ and the PNC is found to vary non-monotonically as a function of $n$. Our study hence clearly suggests that the inherent properties of m-WSMs indeed show up distinctly through the chiral anomaly and the chiral magnetic effect induced transport coefficients in two different setups. Moreover, in order to obtain an experimentally realizable picture, we simultaneously verify our analytical findings through the numerical calculations using the lattice model of m-WSMs.

Equations

$$H_n\left(\mathbf{k}\right)={\alpha }_n{k}_{\perp }^n\left[\cos \left(n{\varphi }_k\right){\sigma }_x+\sin \left(n{\varphi }_k\right){\sigma }_y\right]+v{k}_z{\sigma }_z$$

$${ϵ}_{\mathbf{k}}^{\pm }=\pm \sqrt{{\alpha }_n^2{k}_{\perp }^{2n}+v^2{k}_z^2}$$

$${\Omega }_a^m(\mathbf{k})=(-1{)}^m\frac{1}{4|n_{\mathbf{k}}{|}^3}{ϵ}_{abc}{\mathbf{n}}_{\mathbf{k}}\cdot \left(\frac{\partial {\mathbf{n}}_{\mathbf{k}}}{\partial k_b}\times \frac{\partial {\mathbf{n}}_{\mathbf{k}}}{\partial k_c}\right).$$

$${\Omega }_{\mathbf{k}}^{\pm }=\pm \frac{1}{2}\frac{nv{\alpha }_n^2{k}_{\perp }^{2n-2}}{{ϵ}_{\mathbf{k}}^3}\left(k_x,k_y,n{k}_z\right).$$

$$v_{\mathbf{k}}=\frac{1}{{ϵ}_{\mathbf{k}}}(k_{x}n{\alpha }_n^2{k}_{\perp }^{2(n-1)},k_{y}n{\alpha }_n^2{k}_{\perp }^{2(n-1)},v^2{k}_z).$$

$$H={\mathbf{N}}_{\mathbf{k}}\cdot \mathbf{\sigma }.$$

$$\cos(k_{0})=\frac{t_{0}}{t_{z}}\bigl{[}\frac{m_{z}}{t_{0}}+\cos k_{x}+\cos k_{y}-2\bigr{]}.$$

$$\cos(k_{0})=\frac{t_{0}}{t_{z}}\bigl{[}\frac{m_{z}}{t_{0}}-(6+\cos 2k_{x}+\cos 2k_{y}-4\cos k_{x}-4\cos k_{y})\bigr{]}.$$

$$J_{\alpha }=L_{\alpha \beta }^{11}{E}_{\beta }+L_{\alpha \beta }^{12}(-{\nabla }_{\beta }T),$$

$$Q_{\alpha }=L_{\alpha \beta }^{21}{E}_{\beta }+L_{\alpha \beta }^{22}(-{\nabla }_{\beta }T),$$

$$\left(\frac{\partial }{\partial t}+\dot{\mathbf{r}}\cdot {\nabla }_{\mathbf{r}}+\dot{\mathbf{k}}\cdot {\nabla }_{\mathbf{k}}\right)f_{\mathbf{k},\mathbf{r},t}=I_{coll}\{f_{\mathbf{k},\mathbf{r},t}\},$$

$$(\dot{\mathbf{r}}\cdot {\nabla }_{\mathbf{r}}+\dot{\mathbf{k}}\cdot {\nabla }_{\mathbf{k}})f_{\mathbf{k}}=\frac{f_0-f_{\mathbf{k}}}{\tau (\mathbf{k})},$$

$$\dot{\mathbf{r}}=D(\mathbf{B},{\mathbf{\Omega }}_{\mathbf{k}})[{\mathbf{v}}_{\mathbf{k}}+\frac{e}{\hslash }(\mathbf{E}\times {\mathbf{\Omega }}_{\mathbf{k}})+\frac{e}{\hslash }({\mathbf{v}}_{\mathbf{k}}\cdot {\mathbf{\Omega }}_{\mathbf{k}})\mathbf{B}],$$

$$\hslash \dot{\mathbf{k}}=D(\mathbf{B},{\mathbf{\Omega }}_{\mathbf{k}})[e\mathbf{E}+\frac{e}{\hslash }({\mathbf{v}}_{\mathbf{k}}\times \mathbf{B})+\frac{e^2}{\hslash }(\mathbf{E}\cdot \mathbf{B}){\mathbf{\Omega }}_{\mathbf{k}}].$$

$${\sigma }_{yx}$$

$${\sigma }_{xx}$$

$${\alpha }_{yx}$$

$${\alpha }_{xx}$$

$$\nu =\frac{E_y}{-dT/dx}=\frac{{\alpha }_{xy}{\sigma }_{xx}-{\alpha }_{xx}{\sigma }_{xy}}{{\sigma }_{xx}^2+{\sigma }_{xy}^2}.$$

$${\sigma }_{xx}(\mathrm{CA})$$

$${\sigma }_{xx}$$

$${\sigma }_{yx}(\mathrm{CA})$$

$${\sigma }_{yx}=\eta B^2\cos \gamma \sin \gamma [n^3{\mu }^{-2/n}+(n+n^2)T^2{\mu }^{-2-2/n}]$$

$${\alpha }_{xx}(\mathrm{CME})=\rho n^2{B}^2{\cos }^2\gamma T{\mu }^{-1-2/n}+\mathcal{O}(T^3{n}^3)$$

$${\alpha }_{xx}=\rho T[-\mu n+B^2{\mu }^{-1-2/n}{n}^2{\cos }^2\gamma ]+\mathcal{O}(T^3{n}^3)$$

$${\alpha }_{yx}(\mathrm{CME})=\rho \cos \gamma \sin \gamma n^2{B}^{2}T{\pi }^2{\mu }^{-1-2/n},$$

$${\alpha }_{yx}=\rho T{B}^2{n}^2\cos \gamma \sin \gamma {\mu }^{-1-2/n}+\mathcal{O}(T^3{n}^3).$$

$$\nu =B^2\cos \gamma \sin \gamma (f_1(n,T,\mu )+f_2(n,T,\mu )),$$

$$eE{v}_x+e^{2}BE\cos \gamma ({\mathbf{v}}_{\mathbf{k}}.{\mathbf{\Omega }}_{\mathbf{k}})\frac{\partial f_0}{\partial ϵ}+eB(-v_z\sin \gamma \frac{\partial }{\partial k_x}+(v_x\sin \gamma -v_y\cos \gamma )\frac{\partial }{\partial k_z}+v_z\cos \gamma \frac{\partial }{\partial k_y})f_k=\frac{f_0-f_k}{D\tau }$$

$$f_k-f_0=(eDE\tau v_x+e^{2}DBE\tau \cos \gamma ({\mathbf{v}}_{\mathbf{k}}.{\mathbf{\Omega }}_{\mathbf{k}})+v.\Gamma )$$

Full text

I Introduction

In the field of three-dimensional (3D) topological systems, Weyl semimetal (WSM) has emerged as a prime topic of interest. In condensed matter physics, Weyl fermion appears as a low energy excitation of gapless chiral fermion near the touching of a pair of non-degenerate bands Murakami_2007 ; Peskin_1995 ; Murakami2:2007 ; Yang:2011 ; Burkov1:2011 ; Burkov:2011 ; Volovik ; Wan_2011 ; Xu:2011 . The non-trivial topological properties of the WSMs appear due to Weyl nodes. The Weyl node describing the singularity in k-space acts as a source or sink of the Berry curvature. According to no-go theorem, the Weyl nodes always come in pairs of positive and negative topological charges (also referred to as chirality) and total topological charge in the Brillouin zone vanishes Nielsen:1981 ; Nielsen:1983 . In order to have a topological charge (designated by $n$) associated with the Weyl node, WSM has to break either time-reversal symmetry (TRS) or the space inversion symmetry (IS) Burkov:2011 ; Volovik ; Wan_2011 ; Xu:2011 . The topological charge whose strength is related to the Chern number is quantized to integer values Xiao_2010 .

The WSM phase has been realized experimentally in several inversion asymmetric compounds (TaAs, MoTe2, WTe2) without breaking TRS Lv_2015 ; Huang_2015 ; Hasan_2015 ; Wu_2016 ; Jiang_2017 ; Yan_2017 . However, all of these materials mentioned above belong to single Weyl semimetal, whose energy dispersions are linear in wave vectors and topological charge equals to $\pm 1$. Recently, it has been proposed that the multi-Weyl fermions can also be realized in condensed matter systems Xu:2011 ; bernevig12 ; hasan16 ; Nagaosa_2014 . The multi-Weyl semimetals (m-WSMs) are referred to those materials which contain Weyl nodes with topological charge higher than 1 (i.e. $n>1$). The quasi-particle dispersion for $n>1$ shows natural anisotropy in dispersion. The double WSM ($n=2$) and triple WSM ($n=3$) show linear dispersion along one symmetry direction and quadratic and cubic energy dispersion relations for the other two directions respectively. From the density functional theory (DFT) calculations, it has been suggested that HgCr2Se4 and SrSi2 can be the candidate materials for double WSM Xu:2011 ; bernevig12 ; hasan16 whereas A(MoX)3 (with $A=Rb$, $TI$; $X=Te$) kind of materials can accommodate triple-Weyl points zunger17 . Discrete rotational symmetry in a lattice imposes a strict restriction that only the Weyl nodes with topological charge $n\leq 3$ can be permitted in real materials bernevig12 ; Nagaosa_2014 . Moreover, the single WSM can be viewed as 3D analogue of graphene whereas the double WSM and triple WSM can be represented as 3D counterparts of bilayer falko06 and ABC-stacked trilayer graphene peres06 ; macdonald08 , respectively.

The single WSM exhibits several fascinating transport properties in the presence as well as the absence of external fields. Negative longitudinal magnetoresistance (LMR) and planar Hall effect (PHE) are the two most important transport properties which appear due to the non-conservation of separate electron numbers of opposite chirality for relativistic massless fermions, an effect known as the chiral or Adler-Bell-Jackiw anomaly Goswami:2013 ; Adler:1969 ; Bell:1969 ; Nielsen:1981 ; Nielsen:1983 ; Aji:2012 ; Zyuzin:2012 ; Volovik ; Wan_2011 ; Xu:2011 ; Moore_2015 . This is in contrast to the chiral magnetic effect (CME) which refers to an electric current flowing along the direction of the applied magnetic field triggered by the chirality imbalance in the Weyl nodes without any electric field. In recent years, these magneto-transport properties in Dirac and Weyl SM are extensively studied both theoretically and experimentally Kim:2014 ; Son:2013 ; Fiete_2014 ; He:2014 ; Liang:2015 ; CLZhang:2016 ; QLi:2016 ; Xiong ; Hirsch ; Sharma:2016 ; Tewari_2017 ; Vladimir_2017 ; Ma_2019 ; Burkov_jpcm ; Pavan_2013 ; Burkov_2017 ; Nandy_2017 ; Nandy_2018 ; Spivak_2016 ; Das_2018 ; Yip_2015 ; Jia_2016 ; Xu_2016 ; Erfu_2016 ; Li_2018 ; Liang_2018 ; Wang_2018 ; deng19 ; Chen_2018 ; Singha_2018 ; kumar18 . It is noteworthy that the chiral anomaly (CA) induced PHE, observed for a coplanar arrangement of electric and magnetic fields, is characteristically different from the Lorentz force mediated conventional Hall effect where transverse arrangement between the above fields is required. Although, the transport properties in the presence as well as absence of external magnetic field have recently been studied in m-WSMs using both the low-energy model and the lattice model Roy_2016 ; park_2017 ; Roy_2018 ; Gorbar_2017 ; Gorbar_2018 ; Rodrigo_2020 ; Roy_2020 ; Wang_2019 ; Sengupta_2019 ; Liu_2018 ; Fiete_2016 , planar Hall conductivity (PHC) has not been studied in m-WSM so far. In particular, the effects of enhancement of the density of states, anisotropic nonlinear energy dispersion, and modified spin-momentum locking structure on PHE in m-WSMs remain unexplored.

The thermo-electric phenomena such as Peltier coefficient, Nernst effect and longitudinal magneto-thermal conductivity are well studied in the context of regular Dirac and Weyl SMs using semiclassical Boltzmann theory Vladimir_2017 ; Sharma:2016 ; sharma17 ; Fiete_2014 ; Zyuzin_2017 ; Saha_2018 ; Chernodub_2018 ; Spivak_2016 ; Trivedi_2017 ; Nandy1_2017 and are also recently observed in experiments Watzman_2018 ; Liang_2017 ; Rana_2018 ; Hess_2018 . Moreover, the thermo-electric transport properties in m-WSMs have been studied Gorbar_2017 ; Gorbar_2018 ; Fiete_2016 . There exist a plethora of theoretical works studying mainly the anomalous or conventional Nernst response in WSMs Vladimir_2017 ; Sharma:2016 ; Fiete_2014 ; Spivak_2016 ; sharma17 ; Zyuzin_2017 ; Saha_2018 ; Chernodub_2018 ; Trivedi_2017 ; Nandy1_2017 ; Gorbar_2017 ; Gorbar_2018 ; Fiete_2016 . An anomalous Nernst effect requires the presence of Berry curvature in a direction perpendicular to both the applied temperature gradient and the induced voltage whereas in the case of conventional Nernst effect, external magnetic field has to be applied perpendicular to both $\nabla{\mathbf{T}}$ and induced voltage. In this work, we study an unconventional Nernst effect, namely, the planar Nernst effect (PNE) which is different from the conventional Nernst effect as well as the anomalous Nernst effect, is known to occur in ferromagnetic systems Avery_2012 ; Pu_2006 ; Back_2013 . We consider a situation where both the applied thermal gradient $\mathbf{\nabla T}$ and magnetic field $\mathbf{B}$ are in-plane but not parallel to each other. This situation generates an in-plane transverse voltage and the corresponding phenomenon is referred to as the PNE. Actually, PNE is the thermal counterpart of the PHE where the applied electric field $\mathbf{E}$ is replaced by ${\nabla\mathbf{T}}$. Therefore, it is now natural question to ask how PNE behaves in m-WSMs. Moreover, in the planar Nernst setup (i.e. both the applied thermal gradient $\mathbf{\nabla T}$ and magnetic field $\mathbf{B}$ are in-plane but not parallel to each other), the response of thermo-electric coefficients in the context of m-WSMs has not been explored yet.

In this paper, we study several intriguing transport coefficients in m-WSMs considering the co-planar setups. Using the low-energy model of m-WSMs, we first analytically calculate longitudinal magneto-conductivity (LMC), planar Hall conductivity (PHC), longitudinal thermo-electric coefficient (LTEC) and transverse thermo-electric coefficient (TTEC) (usually referred to as the Peltier coefficient). Interestingly, we find that both LMC and PHC go as $n^{3}$ at zero temperature while the finite temperature correction is ${\mathcal{O}}((n+n^{2})T^{2})$. On the other hand, both LTEC and TTEC follow $n^{2}T$ dependence. Moreover, we find that LMC and LTEC show $B^{2}\cos^{2}\gamma$ dependence whereas PHC and TTEC are proportional to $B^{2}\sin\gamma\cos\gamma$ in m-WSMs. Here, $\gamma$ is the angle between applied $\mathbf{B}$ and $\mathbf{E}$ for the measurement of PHE or between applied $\mathbf{B}$ and $\mathbf{\nabla T}$ for the measurement of thermo-electric coefficients. Secondly, using the thermo-electric tensor and the charge conductivity tensor, we are able to calculate the functional form of planar Nernst coefficient (PNC). We find that PNC, which is proportional to $B^{2}\sin\gamma\cos\gamma$, does not show any monotonic dependence on topological charge as compared to LMC, PHC or TECs. Finally, in order to get a complete picture and verify our analytical findings, we numerically investigate the magnetic field dependence, and angular dependence of electrical conductivity, thermo-electric coefficients and PNE considering the lattice models of m-WSMs.

The rest of the paper is organized as follows. In Sec. II, we introduce the low-energy Hamiltonian as well as TRS breaking lattice Hamiltonian for m-WSMs. Sec. III is devoted to the general expressions of LMC, PHC, TECs and PNC. In Sec. IV, analytical expressions (Sec. IV.1) using low-energy model and numerical results (Sec. IV.2) considering the lattice model of m-WSMs are presented for different transport properties such as LMC, PHC, TECs and PNC. The analytical calculations are given in detail in the Appendix B and C. Finally, we summarize our results and discuss possible future directions in Sec. V.

II Model Hamiltonian

II.1 Low-energy Hamiltonian

The low-energy effective Hamiltonian describing the Weyl node with topological charge $n$ can be written as Xu:2011 ; bernevig12 ; Nagaosa_2014 ; Roy_2017

[TABLE]

where $k_{\bot}=\sqrt{k_{x}^{2}+k_{y}^{2}}$ and $\phi_{k}={\rm arctan}(k_{y}/k_{x})$. Here, $\alpha_{n}$ bears the connection to the Fermi velocity. For example, $\alpha_{1}$ has the dimension of Fermi velocity, while $\alpha_{2}$ has the dimension of mass. $v$ is equivalent to the velocity associated with $z$-direction. Here, $\sigma_{i}$’s $\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ are the Pauli matrices representing the pseudo-spin indices. The Hamiltonian given in Eq. (1) can be written in a compact form as $H=\mathbf{n}_{\mathbf{k}}\cdot\boldsymbol{\sigma}$ with $\mathbf{n}_{\mathbf{k}}=(\alpha_{n}k^{n}_{\bot}\cos\left(n\phi_{k}\right),\alpha_{n}k^{n}_{\bot}\sin\left(n\phi_{k}\right),vk_{z})$. The energy dispersion of the Weyl node is given by

[TABLE]

where $\pm$ represents conduction and valence bands respectively. It is clear from the Eq. (2) that the topological charge determines not only the topological nature of the wave function but also the anisotropic energy dispersion of the system. The single Weyl dispersion $\epsilon_{\mathbf{k}}=v\sqrt{k^{2}_{x}+k^{2}_{y}+k^{2}_{x}}$ can be obtained by setting $n=1$ and $\alpha_{1}=v$ in Eq. (2). Therefore, it is clear that the dispersion around a Weyl node with $n=1$ is isotropic in all momentum directions. On the other hand, for $n>1$, we find that the dispersion around a double Weyl node ($n=2$) becomes quadratic along both $k_{x}$ and $k_{y}$ directions whereas varies linearly with $k_{z}$. Substituting $n=3$ in Eq. (2), it is easy to see that the dispersion around a triple Weyl ($n=3$) node is cubic along both $k_{x}$ and $k_{y}$ directions and becomes linear in $k_{z}$ direction. We additionally note that in this study we restrict ourselves to type-I m-WSM (Eq. 1) where a single multi-Weyl node, separated from the opposite chirality multi-Weyl node in momentum space, is presented with the absence of the tilt parameter.

The Berry curvature of the m${}^{\textrm{th}}$ band for a Bloch Hamiltonian $H_{n}({\mathbf{k}})$, defined as the Berry phase per unit area in the ${\mathbf{k}}$ space, is given by  Xiao_2010

[TABLE]

The explicit form of different Berry curvature components associated with the multi-Weyl node are given by

[TABLE]

The Berry curvature of a single WSM can be easily obtained from Eq. (4) by setting $n=1$ and $\alpha_{1}=v$ which gives ${\Omega}_{\mathbf{k}}^{\pm}=\pm\mathbf{k}/\epsilon^{3}_{\mathbf{k}}$ with $\mathbf{k}=(k_{x},k_{y},k_{z})$. Therefore, the Berry curvature is isotropic in all momentum directions for single Weyl case. On the other hand, it is clear from Eq. (4) that the Berry curvature becomes anisotropic for WSMs with $n>1$ i.e., for double WSM ($n=2$) and triple WSM ($n=3$) due to the presence of $k^{2n-2}_{\bot}$ factor and monopole charge $n$. In particular, we find that $\Omega_{z}$ is algebraically dependent on $n$ in a quadratic manner while $\Omega_{x}$ and $\Omega_{y}$ bear linear algebraic dependence on $n$. Therefore, the multi-Weyl nature can indeed modify Berry curvature induced transport properties in double and triple WSMs as compared to single Weyl case.

The components of the quasi-particle velocity ($v_{\mathbf{k}}=\frac{\partial\epsilon_{\mathbf{k}}}{\partial\mathbf{k}}$) associated with the multi-Weyl node are given by

[TABLE]

It is clear from Eq. (5) that the velocity for a single WSM is $v_{\mathbf{k}}=v(k_{x},k_{y},k_{z})/\epsilon_{\mathbf{k}}$ which shows the isotropic nature of the velocity in all momentum directions. One can figure out from the same equation that the velocity is no longer isotropic if we consider WSMs with $n>1$ compared to the single WSM. In particular, since the energy dispersion becomes anisotropic in double and triple WSMs as described in Eq. (2), the $x$ and $y$ components of the velocity vary with different power in $k_{x}$ and $k_{y}$ due to the factor $k_{\bot}^{2(n-1)}$ in these cases while $v_{z}$ remains unaltered (varies linearly with $k_{z}$) irrespective of the value of $n$.

II.2 Lattice Hamiltonian

We now discuss a prototype lattice model for type-I m-WSM that breaks TRS but remains invariant under inversion. The corresponding lattice model can be written as Roy_2017

[TABLE]

For the single WSM with $n=1$, the momentum-dependent form factors $\mathbf{N}_{\mathbf{k}}$ (setting the lattice constant $a=1$) takes the form $N_{x}=t\sin k_{x}$, $N_{y}=t\sin k_{y}$ and $N_{z}=t_{z}\cos k_{z}-m_{z}+t_{0}(2-\cos k_{x}-\cos k_{y})$. In this model, the Weyl nodes are located at ${\mathbf{k}}=(0,0,\pm k_{0})$ with

[TABLE]

On the other hand, in the case of a double WSM ($n=2$), the form of $N_{k}$ becomes $N_{x}=t(\cos k_{x}-\cos k_{y})$, $N_{y}=t\sin k_{x}\sin k_{y}$ and $N_{z}=t_{z}\cos k_{z}-m_{z}+t_{0}(6+\cos 2k_{x}+\cos 2k_{y}-4\cos k_{x}-4\cos k_{y})$. The lattice model of double WSM contains two Weyl nodes at $(0,0,\pm k_{0})$ with

[TABLE]

Similarly, for a triple-WSM with the topological charge $n=3$, one should replace $N_{k}$ by $N_{x}=t\sin k_{x}[1-\cos k_{x}-3(1-\cos k_{y})]$, $N_{y}=-t\sin k_{y}[1-\cos k_{y}-3(1-\cos k_{x})]$ and $N_{z}=t_{z}\cos k_{z}-m_{z}+t_{0}(6+\cos 2k_{x}+\cos 2k_{y}-4\cos k_{x}-4\cos k_{y})$. Here, the Weyl points appear at ${\mathbf{k}}=(0,0,\pm k_{0})$ with $k_{0}$ followed by the Eq. (8). The energy dispersions of single, double and triple WSMs along various high-symmetry directions are shown in Fig. 1. We note that one can obtain low-energy Hamiltonian (1) from the above-mentioned lattice Hamiltonian by suitably expanding around the gap closing momentum $k_{0}$. The energy dispersion and the Berry curvature, obtained from lattice model, are shown in Fig. 1 and Fig. 2, respectively.

III Semiclassical formalism for calculating transport coefficients

It has been shown that in the presence of electric field and magnetic field, transport properties get substantially modified due to the presence of non-trivial Berry curvature which acts as a fictitious magnetic field in the momentum space Xiao_2010 . In this section, using semiclassical Boltzmann transport theory, we present general expression of some specific transport properties, namely, LMC, PHC and TECs that could generally be observed in all Dirac and Weyl semimetals. In this regime, we consider $T\ll\sqrt{B}\ll\mu$, where $\mu$ is the chemical potential, measured from the band-touching point and ignore the Landau quantization of the energy levels.

In the presence of external perturbative fields (for example, electric field $\mathbf{E}$ and temperature gradient $\mathbf{\nabla T}$), the charge current $\mathbf{J}$ and thermal current $\mathbf{Q}$ from linear response theory, can be written as

[TABLE]

[TABLE]

where $\alpha$ and $\beta$ are spatial indices running over $x$, $y$, $z$. Here, $L_{\alpha\beta}^{11}$ and $L_{\alpha\beta}^{12}$ define the charge conductivity tensor and thermo-electric tensor respectively. The tensors $L_{\alpha\beta}^{12}$ and $L_{\alpha\beta}^{21}$ are related to each other by the Onsager’s relation : $L_{\alpha\beta}^{21}$=T $L_{\alpha\beta}^{12}$. In the low temperature regime, the transport coefficients obey the Mott relation Mermin as $L_{\alpha\beta}^{12}=-\frac{\pi^{2}}{3e}k^{2}_{B}T\frac{\partial L_{\alpha\beta}^{11}}{\partial\mu}$, where $e$ is the electronic charge and $k_{B}$ is the Boltzmann constant.

The Boltzmann transport equation in its’ phenomenological form can be written as John_2001

[TABLE]

where the right side $I_{coll}\{f_{\mathbf{k},\mathbf{r},t}\}$ is the collision integral which incorporates the effects of electron correlations and impurity scattering. We are interested in computing the electron distribution function which is given by $f_{\mathbf{k},\mathbf{r},t}$. Under the relaxation time approximation with the parameter $\tau$ that quantifies the average time between two successive collisions, the steady-state Boltzmann equation can be written as

[TABLE]

where $f_{0}$ is the equilibrium Fermi-Dirac distribution function. In this work, we ignore the momentum dependence of $\tau$ for simplifying the calculations and assume it to be a constant Son:2013 ; Kim:2014 ; Fiete_2014 ; Sharma:2016 . Now we shall revisit the semiclassical equations of motion for an electron in presence of Berry curvature Son_2012 ; Duval_2006

[TABLE]

[TABLE]

Here, $D(\mathbf{B,\Omega_{k}})=(1+\frac{e}{\hbar}(\mathbf{B}.\mathbf{\Omega_{k}}))^{-1}$ is the phase space factor as the Berry curvature $\mathbf{\Omega_{k}}$ modifies the phase space volume element $dkdx\rightarrow D(\mathbf{B,\Omega_{k}})dkdx$ Duval_2006 . Hereafter, we denote $D(\mathbf{B,\Omega_{k}})$ by $D$. The term $(\mathbf{E}\times\mathbf{\Omega_{k}})$ represents the anomalous velocity perpendicular to the applied electric field. On the other hand, the third term of Eq. 13 $(\mathbf{v_{k}}\cdot\mathbf{\Omega_{k}})\mathbf{B}$ represents the chiral magnetic effect (CME). This leads to interesting signature of transport phenomena in Weyl semimetals and can appears for $\mathbf{E}=0$ (i.e., $\mathbf{E}\cdot\mathbf{B}=0$) Son_2012 ; Franz_2013 ; Yin_2012 ; Chen_2013 ; Kenji_2008 . Moreover, the term $(\mathbf{E}\cdot\mathbf{B})\neq 0$ is responsible for chiral anomaly which arises in axion-electrodynamics of WSM.

III.1 Setup 1: Longitudinal Magneto-Conductivity and Planar Hall Conductivity

The PHE is defined through an induction of in-plane transverse voltage when the co-planar electric and magnetic fields are not perfectly aligned with each other. In order to get the general expression for PHC and LMC, we consider that the electric field is applied along the $x-$axis and the magnetic field is rotated in $x-y$ plane at a finite angle $\gamma$ from the $x-$axis, i.e. $\mathbf{B}=B\cos\gamma\hat{x}+B\sin\gamma\hat{y}$, $\mathbf{E}=E\hat{x}$. The corresponding setup 1 is shown in Fig. 3(a).

Plugging the equations of motion described in Eq. (13) and Eq. (14) into the Boltzmann equation, the general expression of the PHC $\sigma_{yx}$ and LMC $\sigma_{xx}$ in the above configuration can be written as Fiete_2014 ; Sharma:2016 ; Nandy_2017 ; Nandy_2018

[TABLE]

and

[TABLE]

We would like to point out that we use $\approx$ sign in Eq. (15) and Eq. (16) as we ignore the contribution from the correction factors arising due to the presence of external magnetic field. In particular, for the semiclassical regime, it is sufficient to retain only the leading order terms in the distribution function $f_{k}$ as the contribution from the correction factors are several order of magnitude smaller than the leading order terms Nandy_2017 (see Appendix A). The important point to note here is that $B\cos\gamma$ factor associated with $\mathbf{v_{k}}\cdot\mathbf{\Omega_{k}}$ in Eq. (15) and Eq. (16) bears the signature of chiral anomaly (${\mathbf{E}}\cdot{\mathbf{B}}$) which we shall investigate below in detail.

III.2 Setup 2: Thermo-Electric Coefficient and Planar Nernst Coefficient

In order to compute the planar TECs, we apply the temperature gradient $\nabla T$ along the $x$-axis and the magnetic field is rotated in the $x-y$ plane in the absence of electric field i.e. $\mathbf{B}=B\cos\gamma\hat{x}+B\sin\gamma\hat{y}$, $\mathbf{\nabla T}=\nabla T\hat{x}$, $\mathbf{E}=0$. The planar thermo-electric setup (setup 2) is shown in Fig. 3(b). Using the equations of motion and semiclassical Boltzmann equation, one can write the TTEC $\alpha_{yx}$ and LTEC $\alpha_{xx}$ in this setup as Fiete_2014 ; Sharma:2016

[TABLE]

and

[TABLE]

Similar to LMC and PHC, in the expression of LTEC and TTEC, we ignore the contribution from the correction factors due to presence of external magnetic field. While passing by we can comment that $\mathbf{v_{k}}\cdot\mathbf{\Omega_{k}}$ is the key ingredient for chiral magnetic effect and the associated $B\cos\gamma$ factor in Eq. (17) and Eq. (18) is coming from ${\mathbf{B}}\cdot\nabla{\mathbf{T}}$.

Now we will formulate the planar Nernst effect (PNE) which is characterized by coplanar $\nabla{\mathbf{T}}$, ${\mathbf{E}}$ and ${\mathbf{B}}$. In setup 2, the longitudinal temperature gradient $\nabla_{x}{\mathbf{T}}$ produces a transverse electric field $E_{y}$ as a result of the coplanar component of the ${\mathbf{B}}$ field; this is known as PNE. Unlike the conventional and anomalous Nernst effects, PNE appears when the ${\nabla}{\mathbf{T}}$ and the magnetic field $\mathbf{B}$ are not aligned with each other. Using the charge conductivity tensor $\sigma$ and thermo-electric tensor $\alpha$, the PNC $\nu$ can be written as Sharma:2016

[TABLE]

In general, the generation of a transverse electric field in the presence of a transverse temperature gradient refers to the Nernst effect. The conventional Nernst effect appears due to Lorentz force in a system in the presence of an external magnetic field ${\mathbf{B}}$ applied perpendicular to the temperature gradient $\nabla{\mathbf{T}}$. The anomalous Nernst effect appears only due to the anomalous velocity of the quasiparticle generated by the non-trivial Berry curvature in the absence of external magnetic field. Actually, the conventional (anomalous) Nernst effect requires the finite magnetic field (Berry curvature) in a direction perpendicular to the plane of applied $\nabla{\mathbf{T}}$ and the induced voltage. On the other hand, this setup will generate an in-plane induced voltage normal to applied in-plane $\nabla{\mathbf{T}}$ and the induced electric field, applied $\nabla{\mathbf{T}}$ and ${\bm{B}}$ all lie in the same plane. Therefore, one can infer that the PNE is fundamentally different from the conventional as well as anomalous Nernst effects.

IV Results

In this section, we study several intriguing transport properties such as LMC, PHC, TECs and PNC using the low-energy model as well as the lattice model of m-WSM. Using the low energy model, we first calculate these transport coefficients analytically within the semiclassical regime and after that we verify our analytical findings by considering the TRS breaking lattice model.

IV.1 Analytical Results Using Low Energy Model

In order to study LMC in m-WSMs, we first breakdown the complete expression of LMC as given in Eq. (16) into three terms, (1) $\sigma^{(1)}_{xx}$ $\sim v_{x}^{2}$, (2) $\sigma^{(2)}_{xx}$ $\sim({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})^{2}$, and (3) $\sigma^{(3)}_{xx}$ $\sim v_{x}({\bm{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})$. From the Eq. (16), it is clear that the terms (2) and (3) give together CA induced LMC in m-WSMs. This is due to the fact that both (2) and (3) contain the prefactor $B\cos\gamma$ originated from the CA (${\mathbf{E}}\cdot{\mathbf{B}}$). Since it is known that $\sigma^{(2)}_{xx}$ is the most dominant contribution to CA induced LMC in single WSMs without tilt Nandy_2017 , we will now refer $\sigma^{(2)}_{xx}$ term as $\sigma_{xx}(\rm CA)$ for the rest of the work.

We now analytically calculate each term of LMC as well as the total LMC using the low-energy model of m-WSMs. The detailed calculations are shown in the Appendix C. The CA term $\sigma_{xx}^{(2)}$ and the total LMC are given by

[TABLE]

[TABLE]

where $\eta=\frac{v\tau e^{4}\alpha^{2/n}_{n}}{16\hbar^{2}\pi^{3/2}}\frac{\Gamma(2-1/n)}{\Gamma(5/2-1/n)}$.

We shall now examine the LMC in detail as a function of $n$, $\mu$, and $T$. It is clear from the Eq. (LABEL:eq_xx2t_1) and Eq. (21) that both $\sigma_{xx}(\rm CA)$ and $\sigma_{xx}$ vary as $n^{3}B^{2}\cos^{2}\gamma$ at zero temperature. The first term $\sigma_{xx}^{(1)}$ (containing velocity part only) yields $B$-independent contribution (generally referred to as the Drude contribution) to LMC and it varies linearly with the topological charge. Therefore, the magnitude of LMC increases as $n^{3}$ for WSMs with higher $n$ Roy_2018 . We also find that $\sigma_{xx}(\rm CA)$ is the most dominant contribution to LMC in all WSMs ($n=1,2,3$). Moreover, the magnitude of chiral anomaly induced LMC decreases slowly with the chemical potential $\mu$ in double (scales as $\mu^{-1}$) and triple (scales as $\mu^{-2/3}$) WSMs compared to single WSM (scales as $\mu^{-2}$) whereas the Drude contribution increases with $\mu^{2}$ in all WSMs. The multi-Weyl nature thus enters into the LMC through the monopole charge. On the other hand, at finite temperature, both CA contribution and Drude contribution to LMC follow $T^{2}$ dependence for all WSMs. Interestingly, the temperature dependent CA induced LMC is proportional to $(n+n^{2})\mu^{-2-2/n}$ while the Drude part becomes $\mu$ independent and linearly proportional to $n$.

To investigate the PHC in m-WSMs, we now break the expression of PHC (as given in Eq. (15)) similar to the LMC in following form: (1) $\sigma^{(1)}_{yx}$ $\sim v_{x}v_{y}$, (2) $\sigma^{(2)}_{yx}$ $\sim({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})^{2}$, (3) $\sigma^{(3)}_{yx}$ $\sim v_{x}({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})$ and (4) $\sigma^{(4)}_{yx}$ $\sim v_{y}({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})$. It is clear that the terms (2) and (4), containing $B\cos\gamma$ factor, yield the CA induced PHC as the current flows in $y$-direction. Since, in the case of type-I WSM without tilt, term (2) is the most dominant contribution to PHC, we below refer $\sigma^{(2)}_{yx}$ as $\sigma_{yx}(\rm CA)$ for the rest of the paper. Using the low-energy model of m-WSMs, we analytically calculate each term of PHC as well as the total PHC. Please see the Appendix C for the detailed calculations. Now, the $\sigma_{yx}(\rm CA)$ and the total PHC are given by

[TABLE]

[TABLE]

From Eq. (22) and Eq. (23), we find that both $\sigma_{yx}({\rm CA})$ and total PHC $\sigma_{yx}$ show $n^{3}B^{2}\sin\gamma\cos\gamma$ dependence at $T=0$ in m-WSMs. Unlike LMC, the $B$-independent Drude contribution is zero in the case of PHC. Therefore, it is clear that the total contribution of PHC is coming from chiral anomaly in m-WSMs. The magnitude of PHC increases as $n^{3}$ at zero temperature. Similar to the case of LMC, the PHC at $T=0$ decreases slowly with doping as we go from single WSM ($n=1$) to triple WSM ($n=3$). On the other hand, the temperature dependent contribution to PHC varies as $(n+n^{2})T^{2}\mu^{-2-2/n}$.

We have also calculated LMC ($\sigma_{zz}$) and PHC ($\sigma_{yz}$) using the low-energy model when the electric field is applied in the $\hat{z}$-direction. Comparing $\sigma_{zz}$ and $\sigma_{xx}$ side by side, we find that the qualitative behavior of both $\sigma_{zz}$ and $\sigma_{xx}$ are dictated by CA. Hence, their functional dependence with $B$ and $\gamma$ remain unaltered for all WSMs (i.e. for $n=1,2,3$). Interestingly, unlike $\sigma_{xx}$ where the velocity term ($\sigma_{xx}^{(1)}$) is proportional to $n\mu^{2}$ at $T=0$, in the case of $\sigma_{zz}$ the same term becomes $\propto n^{-1}\mu^{2/n}$. This leads to some quantitative differences between $\sigma_{zz}$ and $\sigma_{xx}$ in WSMs with $n>1$. This is due to the presence of anisotropy in energy dispersion (i.e., $\epsilon_{k}$ is linear with momentum along $z$ direction whereas becomes quadratic and cubic along $y$ direction for double and triple WSMs respectively) as well as Berry curvature in these systems. Moreover, we also find quantitative differences between $\sigma_{yz}$ and $\sigma_{yx}$ in m-WSMs with $n>1$. One can find that $\sigma_{yz}^{(1)}$ varies as $\mu^{2-2/n}$ while $\sigma_{yx}^{(1)}$ goes as $\mu^{-2/n}$. The complete calculations of $\sigma_{zz}$ and $\sigma_{yz}$ are presented in Appendix C.

Next, we study the thermo-electric responses in m-WSMs using the low-energy model. We will follow the same prescription for the term-wise breakdown of LTEC and TTEC. $\alpha_{xx}^{(1)}$ contains quadratic velocity term $v_{x}^{2}$, $\alpha_{xx}^{(2)}$ contains $({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})^{2}$ coming from chiral magnetic effect, and $\alpha_{xx}^{(3)}$ involves $v_{x}({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})$. We shall hereafter refer $\alpha_{xx}^{(2)}$ as $\alpha_{xx}({\rm CME})$ due to the fact that the dominant contribution in $\alpha_{xx}$ is coming from the bare CME term $({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})^{2}$. The CME term $\alpha_{xx}^{(2)}$ and the total LTEC are given by

[TABLE]

[TABLE]

where $\rho=\frac{v\tau e^{3}\alpha^{2/n}_{n}{\Gamma(2-1/n)}K_{B}^{2}\pi^{1/2}}{48{\Gamma(5/2-1/n)}\hbar^{2}}$. The detailed calculations are presented in Appendix C.

The term containing linear power of $n$ in Eq. (25) is coming from the $\alpha_{xx}^{(1)}$ consisting only the velocity factors. It is clear from the above equations that both the $\alpha_{xx}$(CME) and total $\alpha_{xx}$ vary as $n^{2}B^{2}\cos^{2}\gamma$. $\alpha_{xx}$(CME) decreases with $\mu$ in a $n$ dependent manner such that for $n=1$, it falls off more rapidly (as $\mu^{-3}$) than that of for $n=2$ whereas it becomes the most slowly decreasing function of $\mu$ (as $\mu^{-5/3}$) for $n=3$ among all the WSMs. Interestingly, we find that although the magnitude of $\alpha_{xx}$ enhances with the topological charge similar to electrical conductivities, the scaling of $\alpha_{xx}$ with topological charge ($\propto n^{2}$) is different compared to $\sigma$ ($\propto n^{3}$).

Moreover, LTEC is a more rapidly decaying function of $\mu$ than LMC. Both of the above observations can be understood using Mott relation between Eq. (21) and Eq. (25) in the limit $T\rightarrow 0$. Therefore, it is clear that their origins are characteristically different.

Similar to PHC, one can notice that $\alpha_{yx}^{(2)}\sim({\bm{\Omega}_{\bm{k}}}.{\bm{v}_{\bm{k}}})^{2}$ bears the maximum contribution of TTEC as compared to all the other remaining terms $\alpha_{yx}^{(1)}\sim v_{x}v_{y}$ $\alpha_{yx}^{(3)}\sim v_{x}({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})$ and $\alpha_{yx}^{(4)}\sim v_{y}({\mathbf{\Omega}_{\bm{k}}}.{\mathbf{v}_{\bm{k}}})$. The CME term $\alpha_{yx}^{(2)}$ and the total TTEC are given by

[TABLE]

[TABLE]

It is clear from the above equations that the total TTEC does not have any $B$-independent contribution and is dominated by the CME term. We find that the transverse component of $\alpha$ varies similarly (varies as $n^{2}B^{2}$) compared to the longitudinal component except the angular part which goes as $\sin\gamma\cos\gamma$ as given in Eq. (27). Using the Mott relation, one can obtain the TTEC (Eq. 27) from PHC (Eq. 23). We also calculate LTEC ($\alpha_{zz}$) and TTEC ($\alpha_{yz}$) when the temperature gradient is along $z$-direction (see Appendix C). We find that there exists quantitative difference between $\alpha_{xx}$ and $\alpha_{zz}$ as well as between $\alpha_{yx}$ and $\alpha_{yz}$ due to anisotropic energy dispersion of WSMs with $n>1$.

We now compute the functional dependence of the PNC as we already obtained $\sigma$ and $\alpha$. Using Eq. (19), the functional form of PNC $\nu$ can be written as

[TABLE]

with $f_{1,2}$ being complicated functions of $n,T$ and $\mu$. We present the detailed form of $f_{1}(n,T,\mu)$ and $f_{2}(n,T,\mu)$ in the Appendix C. Unlike $\sigma$ and $\alpha$, it is clear from the functional form of $\nu$ that the topological charge dependence is not monotonous in this case. This is due to the fact that both the numerator and the denominator of $f_{1}$ and $f_{2}$ have non-linear products consisting of $\mu$, $n$ and $T$. Hence, one can expect that the behavior of $\nu$ for different $n$ would strongly depend on the values of $\mu$ and $T$. The angular dependence of PNC is same as the transverse transport coefficients.

In summary, we find that the magnitude of all the transport coefficients such as LMC, PHC, TECs and PNC increases with the topological charge. In particular, the electrical conductivities $\sigma$ (LMC and PHC) enhance with $n^{3}$ while the thermo-electric coefficients (LTEC and TTEC) increase with $n^{2}$. In the presence of external magnetic field, the longitudinal transport coefficients i.e., both electric $\sigma$ and thermo-electric $\alpha$, show $B^{2}\cos^{2}\gamma$ dependence whereas the transverse transport coefficients follow $B^{2}\sin\gamma\cos\gamma$ in type-I m-WSMs without tilt. Moreover, unlike single WSM, there exists quantitative difference between longitudinal transport coefficients (e.g. $\sigma_{xx}$ and $\sigma_{zz}$) in the case of double and triple WSMs when an external field is applied along the anisotropic direction (e.g. $x$-direction) of the underlying energy dispersion.

Interestingly, we find that for single WSM, the magnitude of transport coefficients decreases rapidly (as $\mu^{-2}$ for $\sigma$ at $T=0$ and $\mu^{-3}$ for $\alpha$) with doping compared to double and triple WSMs where the magnitude drops as $\mu^{-2/n}$ at $T=0$ for $\sigma$ and $\mu^{-2/n-1}$ for $\alpha$. All of these $n$-dependent scaling come from the fact that the energy dispersion (Eq. (2)), Berry curvature (Eq. (4)) and the velocity (Eq. (5)) significantly change with $n$. Therefore, by looking at the scaling of transport coefficients with the monopole charge, one can distinguish a double and triple WSMs from a single WSM. We would like to note that unlike $\mu$, the functional dependence of different transport coefficients in m-WSMs on temperature $T$ (scales as $T^{2}$ for LMC and PHC and $T$ for LTEC and TTEC) remains unaltered as compared to the single Weyl case. Furthermore, it has been shown that in m-WSMs, the number of Fermi arc is given by the monopole charge $n$ Roy_2020 . Hence, it is expected that transport properties due to surface states would get enhanced in WSMs with $n>1$ (double and triple WSMs) compared to single WSM as the number of available conducting states increases. Therefore, our results suggest that regardless of having similar angular and magnetic field dependencies of different magneto-transport properties in all cases ($n=1,2,3$), there is a lot of new physics popping up for double and triple WSMs compared to single WSM.

We would like to point out that all anomalous (in the absence of magnetic field) thermoelectric coefficients in the limit of zero temperature and small chemical potential are found to be proportional to the integer topological charge of the Weyl nodes in m-WSMsGorbar_2017 . However, our results (i.e. the scaling dependence with $n$) are very different and can not be compared with Ref. Gorbar_2017 because CA as well as CME which are the origin for the transport properties discussed in this work are absent in Ref.Gorbar_2017 .

IV.2 Numerical results in Lattice Model

In order to discuss transport properties in a physical multi-Weyl system, it is always good to consider a lattice model of Weyl fermions with the lattice regularization providing a physical ultra-violet smooth cut-off to the low-energy Dirac spectrum carbotte16 ; yago17 . Here, we consider tight binding lattice model of m-WSMs as discussed in Sec. II.2 to study electric and thermo-electric responses as a function of $B$ and angle $\gamma$.

At the outset, we would like to mention while showing the variation of the total contribution of transport quantities with $B$ that we consider $\bar{\sigma}_{ij}=\sigma_{ij}(B)-\sigma_{ij}(B=0)$ and $\bar{\alpha}_{ij}=\alpha_{ij}(B)-\alpha_{ij}(B=0)$ with $i=j$ for longitudinal and $i\neq j$ for transverse coefficients. For the angular variation, in the case of longitudinal component we study the following quantities: $\bar{\sigma}_{ii}=\sigma_{ii}(\gamma)-\sigma_{ii}(\gamma=\pi/2)$ and $\bar{\alpha}_{ii}=\alpha_{ii}(\gamma)-\alpha_{ii}(\gamma=\pi/2)$, while the transverse components, $i\neq j$, are designated by $\bar{\sigma}_{ij}=\sigma_{ij}(\gamma)-\sigma_{ij}(\gamma=0)$ and $\bar{\alpha}_{ij}=\alpha_{ij}(\gamma)-\alpha_{ij}(\gamma=0)$. We now mention that $B$, $T$ and $\mu$ are measured in the units of Tesla, Kelvin and eV respectively. The conductivities are measured in the units of (Ohm.m)-1. We present all the transport coefficients in the normalized version for the sake of convenience.

IV.2.1 Setup 1: Longitudinal Magneto-conductivity and Planar Hall Conductivity

Magnetic Field: The behavior of total LMC $\bar{\sigma}_{xx}$ as a function of magnetic field $B$ for single, double and triple WSMs are shown in Fig. 4(b). It is clear from the Fig. 4(b) that LMC increases quadratically with magnetic field for all cases. The magnitude of LMC also increases with the topological charge $n$ of the WSMs. For detailed investigation, we compute each term of $\sigma_{xx}$ and find that the leading $B^{2}$-dependent contribution is coming from the CA term which is plotted as a function of $B$ in Fig. 4(a). The variation of total PHC $\sigma_{yx}$ as a function of magnetic field is shown in Fig. 4(d) for $n=1$, $2$ and $3$. One can see that PHC increases in non-linear fashion with $n$ for a particular magnetic field and also shows $B^{2}$ dependence for all WSMs. The dominating CA term $\sigma_{yx}({\rm CA})$ is shown in Fig. 4(c). It is evident from this numerical study that the CA is the origin for the appearance of PHC in m-WSMs. Therefore, one can infer that our results for LMC and PHC using lattice model qualitatively agree with the results obtained from low-energy m-WSM model (see Eq. (21) and Eq. (23)).

Angle: We will now discuss angular dependence of both LMC and PHC for m-WSMs. The LMC and PHC as a function of the angle $\gamma$ for a particular magnetic field are depicted in Fig. 5(a)-(d). We find that the CA term of LMC ($\sim({\mathbf{\Omega}_{k}}.{\mathbf{v}_{k}})^{2}$) as well as the total LMC $\bar{\sigma}_{xx}$ show $\cos^{2}\gamma$ dependence whereas the CA term of PHC $\sigma_{yx}({\rm CA})$ and total PHC $\bar{\sigma}_{yx}$ exhibit $\sin\gamma\cos\gamma$ dependence irrespective of the value of n. The magnitude of both the conductivities increases with the topological charge associated with the Weyl node. These numerical findings are in full congruence with the analytical results. We would like to point out that the oscillation amplitude of $\bar{\sigma}_{yx}$ for single and double WSMs almost coincides with each other whereas the magnitude for triple WSM is much greater than these two.

Scaling with ${\mathbf{n}}$: The variation of LMC and PHC with topological charge is shown in Fig. 6. We consider Fig. 4(b), (d) and Fig. 5(b), (d) to investigate the $n$ dependence of LMC and PHC for a given value of $B$ and $\gamma$. It is clear from the figure that both LMC and PHC follow cubic variation ($\sim n^{3}$) for WSMs with $2\leq n\leq 3$ while for WSMs with $1\leq n\leq 2$, a deviation from $n^{3}$ dependence is visible. The underlying reason might be related to CA term which controls both LMC and PHC maximally for $n>2$. On the other hand, velocity term might be responsible for this deviation for $n<2$. Therefore, it is clear that due to lattice effects, there is a deviation between the numerical results based on lattice model and analytical results based on low-energy model on monopole charge dependence of LMC and PHC.

IV.2.2 Setup 2: Thermo-electric coefficients and planar Nernst coefficient

Magnetic field: The CME contribution as well as the total contribution of LTEC and TTEC are shown in Fig. 7. We find that both LTEC and TTEC vary quadratically with the magnetic field. This observation can be verified using the low-energy model (see Eq. (25) and Eq. (27)). The important point to note here is that CME is the main origin for the magnetic field dependence of TECs. Interestingly, for a given value of $B$, ${\bar{\alpha}}_{xx}$ and ${\bar{\alpha}}_{yx}$ increase non-linearly with $n$ which reflects the multi-Weyl nature in these coefficients.

Angle: We first plot the CME term of TECs in Fig. 8, in particular $\alpha_{xx}$(CME) in Fig. 8(a) and $\alpha_{yx}$(CME) in Fig. 8(c), respectively. The numerical findings again satisfy the analytical results based on the low-energy model, i.e., $\alpha_{xx}({\rm CME})$ $\propto$ $\cos^{2}\gamma$ and $\alpha_{yx}({\rm CME})$ $\propto$ $\sin\gamma\cos\gamma$. The behavior of total TECs are then shown in Fig. 8(b) and Fig. 8(d) where both ${\bar{\alpha}}_{xx}$ and ${\bar{\alpha}}_{yx}$ exhibit $\cos^{2}\gamma$ and $\sin\gamma\cos\gamma$ dependence, respectively. The multi-Weyl character is reflected in the non-linear enhancement of the amplitude of oscillation for ${\bar{\alpha}}_{xx}$ and ${\bar{\alpha}}_{yx}$ with $n$.

Scaling with $\mathbf{n}$: In Fig. 9, we show that longitudinal and transverse TECs, $\alpha_{xx}$ and $\alpha_{yx}$, respectively, vary quadratically with $n$. This reflects the fact that the low-energy model is able to capture the underlying physics of TECs more quantitatively as compared to LMC and PHC. It is clear that the quadratic dependence with $n$ of $\alpha$’s (Fig. 6) is more clear than the cubic dependence with $n$ of $\sigma$’s ((Fig. 9)) for $1\leq n\leq 2$. To be precise, the CME term is the main origin for $n^{2}$ dependence of both TTEC and LTEC while the velocity term, which scales as $n$, contributes sub-dominantly to LTEC in m-WSMs.

Planar Nernst Coefficient: We shall now compute the planar Nernst coefficient $\nu$ using the lattice model of m-WSMs ($n=1$, $2$ and $3$). We find that $\nu$ varies quadratically with $B$ for all $n$ as shown in Fig. 10(a). This behavior is similar to the behavior of all the transport coefficients in both setups. On the other hand, the angular dependence of the planar Nernst coefficient appears to be $\sin\gamma\cos\gamma$ which is similar with the behavior of PHC and TTEC (see Fig. 10(b)). This behavior is consistent with the analytical result as given in Eq. (28). Interestingly, we find that unlike $\sigma$ and $\alpha$, $\nu$ does not exhibit a monotonic behavior with topological charge $n$ at a fixed $B$ and $\gamma$. This non-monotonic dependence of PNC on monopole charge can be explained from analytical functional form obtained from $f_{1}$ and $f_{2}$ in Eq. (28). It is also clear from the Fig. 10 that the multi-Weyl nature is clearly reflected in the PNC since $\nu$ for $n=1$ is distinctly different from double and triple WSMs. We note that PNC is an admixture of $\sigma$, mediated by CA, and thermo-electric coefficients $\alpha$, caused by CME. The dependence of PNC on the external parameters such as ${\bm{B}}$ and ${\gamma}$ can thus be related to the PHC and TTEC. On the other hand, the anisotropic dispersion imprints its signature on PNC via the topological charge and chemical potential; however, their functional forms in PNC are different from that of observed in PHC and TTEC.

V Conclusions

In this work, we study several intriguing transport properties such as LMC, PHC, TECs and PNC for type-I m-WSMs without tilt, characterized by the topological charge $n$ being more than unity, using semiclassical Boltzmann transport theory with the relaxation time approximation. It is clear that anisotropic non-linear dispersion in m-WSMs causes enhanced transport behavior as compared to the isotropic linear single WSMs. Interestingly, the non-uniform responses, depending on the orientation of the applied fields, can in general be obtained due to the anisotropy. Furthermore, there exist more number of conducting Fermi arc states in m-WSMs than the single WSMs leading to enhanced transport in m-WSMs. We here mainly focus on the co-planar setups where external magnetic field ${\mathbf{B}}$ and electric field ${\mathbf{E}}$ or temperature gradient $\nabla{\mathbf{T}}$ lie in the same plane. Using the ${\mathbf{E}}$-${\mathbf{B}}$ arrangement, one can observe electric coefficients $\sigma$’s such as PHE and LMC while $\nabla{\mathbf{T}}$-${\mathbf{B}}$ setup is considered for the measurement of TECs $\alpha$’s and subsequently PNC can be investigated using $\sigma$ and $\alpha$. We validate our low-energy model based analytical results through the numerical lattice calculations. We emphasize that our work can stimulate experimental efforts to uncover the multi-Weyl nature, specially, the monopole charge dependence of different transport coefficients as the setup considered here are realizable in experimentsJia_2016 ; Xu_2016 ; Erfu_2016 ; Li_2018 ; Liang_2018 ; Wang_2018 ; deng19 ; QLi:2016 ; Chen_2018 ; Singha_2018 ; kumar18 ; Watzman_2018 ; Liang_2017 ; Rana_2018 ; Hess_2018 . It is important to note that the transport properties as derived in this work not only become finite but also are expected to show similar magnetic and angular dependencies in an time-reversal symmetric but inversion broken low energy model of m-WSM. However, to predict the correct experimental behavior of the transport properties in an inversion broken m-WSM, one needs to study the lattice model which is an open interesting question and we leave it for future study. At the same time, we note that it would be really interesting to investigate these transport properties for type-II m-WSM which we leave for future study.

In the presence of co-planar electric and magnetic fields, not perfectly aligned with each other (with $\gamma$ being the angle between ${\mathbf{E}}$ and ${\mathbf{B}}$), we derive analytical expressions for both LMC and PHC using the low-energy model. Interestingly, we find that at zero temperature PHC goes as $n^{3}B^{2}\sin\gamma\cos\gamma$ whereas LMC follows $n^{3}B^{2}\cos^{2}\gamma$ dependence. Therefore, it is clear that the magnitude of LMC and PHC both increase with $n^{3}$ as we go from single WSM $n=1$ to triple WSM $n=3$. This is due to the fact that number of conducting channel increases with $n$. We also find that for single WSM, the magnitude of LMC and PHC both decrease rapidly as $\mu^{-2}$ with doping compared to double and triple WSMs where the magnitude drops as $\mu^{-2/n}$ at $T=0$. On the other hand, at finite temperature, we show that both LMC and PHC receive a quadratic temperature correction (i.e., scales as $T^{2}$) with linear and quadratic topological charge. We emphasize that our numerical findings further support that CA is the key ingredient behind all of these observations.

Moving on to the thermo-electric responses, we investigate TECs and PNC for m-WSMs in a setup where co-planar thermal gradient and magnetic field are not perfectly aligned with each other (with $\gamma$ being angle between ${\mathbf{B}}$ and $\mathbf{\nabla T}$). Interestingly, we find that unlike LMC and PHC, both longitudinal and transverse TECs vary quadratically with monopole charge and linearly with temperature. In particular, the longitudinal TEC varies as $n^{2}B^{2}T\cos^{2}\gamma$ while transverse TEC follows as $n^{2}B^{2}T\sin\gamma\cos\gamma$. Additionally, TECs decay more rapidly with chemical potential as $\mu^{-2/n-1}$ compared to both PHC and LMC. Hence the electric and thermo-electric coefficients have different dependencies on the inherent parameters of m-WSMs. We clearly show using numerical treatment that the CME governs the TECs. Therefore, it is essential to mention that CME and CA imprint distinct signature on the transport properties in m-WSMs. Moreover, unlike single WSM, there exists quantitative difference between longitudinal transport coefficients in the case of double and triple WSMs when an external field ($E/\nabla T$) is applied along the anisotropic direction of the underlying energy dispersion. Finally, we study PNE which is of a very different nature from the conventional Nernst effect and even Berry phase mediated anomalous Nernst effect. We find that although PNC behaves qualitatively in an identical manner with ${\mathbf{B}}$ and $\gamma$ as compared to the transverse transport coefficient (PHC and TTEC), it does not exhibit a monotonic variation with $n$ like all the other transport coefficients. Therefore, by looking at the scaling of transport coefficients with the monopole charge, which can be experimentally verifiable, one can distinguish a double and triple WSMs from a single WSM.

Note added: During the completion of our work, we came across the paper Sharma_2020 , which discusses the planar Nernst coefficient for Dirac and single Weyl semimetals.

Acknowledgements.

We sincerely thank Renato M. A. Dantas for fruitful discussions.

Appendix A Calculational detail of Eq. (15) in planar Hall setup

Here our aim is to achieve a modified distribution function. We can start from the Eq. 12 and using $\mathbf{\dot{r}}$ (Eq. (13)) and $\mathbf{\dot{k}}$ (Eq. (14) ), one can obtain

[TABLE]

where $D(\mathbf{B},\mathbf{\Omega_{k}})=(1+e(\mathbf{B}.\mathbf{\Omega_{k}}))^{-1}$ is the phase factor. Now the ansatz we are following is given below

[TABLE]

where $\Gamma$ is the correction factor due to magnetic field $\mathbf{B}$. Therefore, Eq. (29) takes the form

[TABLE]

Hence, the distribution function $f_{k}$ becomes

[TABLE]

where $c_{x}$, $c_{y}$ and $c_{z}$ are correlation factors which incorporate Berry phase effects and related $\Gamma$. Therefore, the general expression of the PHC in the above configuration from the semiclassical Boltzmann equation can be written as Nandy_2017 ; Nandy_2018 ; Sharma:2016

[TABLE]

with $c_{x}=\Gamma_{x}/eE\tau\sin\gamma$, $c_{y}=\Gamma_{y}/eE\tau\cos\gamma$ and $c_{z}=\Gamma_{z}/eE\tau$. Before that we define $m_{ij}=\partial^{2}H/\partial k_{i}\partial k_{j}$. The detail expression for $\Gamma_{x,y,z}$ are given below:

[TABLE]

and $N=e^{2}EBD\tau$, $M_{1}=-\sin\gamma/m_{xx}+\cos\gamma/m_{xy}+eB\cos\gamma(R\cos\gamma-P\sin\gamma)$ , $M_{2}=1/D\tau-eB\sin\gamma/m_{xz}+eB\cos\gamma/m_{yz}$, $M_{3}=eB\cos\gamma T+1/m_{xz}$ and $M_{4}=eB\sin 2\gamma/m_{xy}-eB\cos^{2}\gamma/m_{yy}-eB\sin^{2}\gamma/m_{xx}$ with $P=\sigma_{x}/m_{xx}+\sigma_{y}/m_{xy}+\sigma_{z}/m_{xz}$, $R=\sigma_{x}/m_{xy}+\sigma_{y}/m_{yy}+\sigma_{z}/m_{yz}$ and $P=\sigma_{x}/m_{xz}+\sigma_{y}/m_{yz}+\sigma_{z}/m_{zz}$. Numerical calculation shows that $c_{x},c_{y},c_{z}\to 0$. Therefore, the modified distribution function would be simply given by

[TABLE]

which we use in our analysis in computing $\sigma$ and $\alpha$.

Appendix B Calculation of LMC in regular setup

Now we shall compute the LMC and electrical Hall conductivity for the continuum model (1). We present this calculation to clearly mention the calculation details which we follow for Sec. C, LABEL:app3. Here we assume the electric and magnetic field to have the following form: $\mathbf{E}=E\hat{j}$ and $\mathbf{B}=B\hat{j}$. we refer $\frac{\partial f_{0}}{\partial\epsilon}=\tilde{f}_{0}$. This is the coefficient of electric charge current along $j$ direction for an applied electric field in $j$ direction: $\sigma_{jj}$

[TABLE]

We now decompose the above expression term by term to investigate it more rigorously: $\sigma_{jj}=\sigma_{jj}^{(1)}+\sigma_{jj}^{(2)}+2\sigma_{jj}^{(3)}$ where

[TABLE]

We note here that two LMCs are given by $\sigma_{xx}$ and $\sigma_{zz}$, $J_{x}=\sigma_{xx}E_{x}$ and $J_{z}=\sigma_{zz}E_{z}$.

We make resort to cylindrical polar co-ordinate to do the analytical calculation. $\int{d^{3}\mathbf{k}}/{(2\pi)^{3}}=(1/(2\pi)^{3})\int^{\infty}_{0}k_{\bot}dk_{\bot}\int_{-\infty}^{\infty}dk_{z}\int^{2\pi}_{0}d\phi$. We need to compute the following momentum integral for finite temperature; we use the Sommerfeld expansion.

[TABLE]

We use the change of variable $\beta(\epsilon-\mu)=x$ and above integral becomes

[TABLE]

In the first integral we use $x\to-x$ and using the fact that $(e^{-x}+1)^{-1}=1-(e^{x}+1)^{-1}$. We assume $\beta\mu\gg 1$ and obtain

[TABLE]

Now one can expand $h$ around $\mu$ as the integrand decreases exponentially with increasing $x$

[TABLE]

In our case, $h(k)=(k-\mu)k^{\nu}$ and $\tilde{f}(\epsilon)_{eq}=\pi^{2}K_{B}^{2}T^{2}\nu\mu^{\nu-1}$ and when $h(k)=k^{\nu}$ then $\tilde{f}(\epsilon)_{0}=\mu^{\nu}+\pi^{2}K_{B}^{2}T^{2}\nu(\nu-1)\mu^{\nu-2}$.

We shall derive the analytical form of LMC in finite temperature by considering $-\frac{\partial f_{0}}{\partial\epsilon}=\beta f_{0}(1-f_{0})=\tilde{f}(\epsilon)_{0}$.

[TABLE]

Now we perform the variable substitution $k_{z}\rightarrow k_{z}/v$ and $k_{\bot}\rightarrow k_{\bot}\alpha_{n}^{-1/n}$ Hence the energy becomes $\epsilon\to\epsilon^{\prime}=\sqrt{k_{\bot}^{2n}+k^{2}_{z}}$.

[TABLE]

We then use another change of variable $k_{\bot}\to{k}_{\bot}^{1/n}$ and $\epsilon^{\prime}\to\epsilon^{\prime\prime}=\sqrt{k_{\bot}^{2}+k_{z}^{2}}$.

[TABLE]

Finally, one can perform another transformation $k_{\bot}=k\sin{\theta}$ and $k_{z}=k\cos{\theta}$ and hence $\epsilon^{\prime\prime}\to\epsilon^{\prime\prime\prime}=k$. $\int^{\infty}_{0}dk_{\bot}\int_{-\infty}^{\infty}dk_{z}\to\int^{\pi}_{0}d\theta\int^{\infty}_{0}dk$.

[TABLE]

In order to evaluate the integrals we need to perform a series expansion in terms of $eB/\hbar\mu^{2/n}$. We use the series expansion $(1+x)^{-1}=\sum_{i=0}^{\infty}(-x)^{i}$ with $x\ll 1$ for the denominator. Therefore, the integral becomes

[TABLE]

The leading order terms are given by

[TABLE]

Similarly, $\sigma^{(2)}_{zz}$ is given by

[TABLE]

Similarly, $\sigma^{(3)}_{zz}$ is given by

[TABLE]

Therefore, the CA term in $\sigma_{zz}$ is proportional to $B^{2}(n^{3}\mu^{-2/n}+n^{2}T^{2}\mu^{-2-2/n})$. The complete $\sigma_{zz}$ is also proportional to $\mu^{2/n}/n+T^{2}/n^{3}+B^{2}(n^{3}\mu^{-2/n}+n^{2}T^{2}\mu^{-2-2/n})$.

Similarly, $\sigma^{(1)}_{xx}$ is given by

[TABLE]

Similarly, $\sigma^{(2)}_{xx}$ is given by

[TABLE]

Similarly, $\sigma^{(3)}_{xx}$ is given by

[TABLE]

Therefore, the CA term in $\sigma_{xx}$ is proportional to $B^{2}(n^{3}\mu^{-2/n}+n^{2}T^{2}\mu^{-2-2/n})$. The complete $\sigma_{xx}$ is also proportional to $n\mu^{2}+nT^{2}+B^{2}(n^{3}\mu^{-2/n}+n^{2}T^{2}\mu^{-2-2/n})$.

Appendix C Calculations for LMC, PHC, TECs and PNC in coplanar setup

Having discussed the LMC in normal regular setup, we shall now turn our attention to PH setup. Here we compute two main quantities $\sigma_{jj}$ and $\sigma_{ij}$. The magnetic field here is assumed to have the form: $\mathbf{B}=B\cos\gamma\hat{j}+B\sin\gamma\hat{y}$ and $\mathbf{E}=E\hat{j}$ with $j=x,z$. Now using the same procedure mentioned in Appendix B, the CA term in $\sigma_{xx}$ is proportional to $B^{2}\cos^{2}\gamma(n^{3}\mu^{-2/n}+n^{2}T^{2}\mu^{-2-2/n})$. The complete $\sigma_{xx}$ is proportional to $n\mu^{2}+nT^{2}+B^{2}\cos^{2}\gamma(n^{3}\mu^{-2/n}+n^{2}T^{2}\mu^{-2-2/n})$. On the other hand, the CA term in $\sigma_{zz}$ is proportional to $B^{2}\cos^{2}\gamma(n^{3}\mu^{-2/n}+nT^{2}\mu^{-2-2/n})$. The complete $\sigma_{zz}$ is also proportional to $\mu^{2/n}/n+T^{2}\mu^{2/n-2}/n^{3}+B^{2}\cos^{2}\gamma(n^{3}\mu^{-2/n}+n^{2}T^{2}\mu^{-2-2/n})$. Similarly, we find that the CA term in $\sigma_{yx}$ is proportional to $B^{2}\cos\gamma\sin\gamma(n^{3}\mu^{-2/n}+nT^{2}\mu^{-2-2/n})$. The complete $\sigma_{yx}$ is also proportional to $B^{2}\cos\gamma\sin\gamma(n^{3}\mu^{-2/n}+nT^{2}\mu^{-2-2/n})$. Now, the CA term in $\sigma_{yz}$ is proportional to $B^{2}\cos\gamma\sin\gamma(n^{3}\mu^{-2/n}+nT^{2}\mu^{-2-2/n})$. The complete $\sigma_{yz}$ is proportional to $B^{2}\cos\gamma\sin\gamma(n^{3}\mu^{2-2/n}+nT^{2}\mu^{-2/n}+n^{3}\mu^{-2/n}+nT^{2}\mu^{-2-2/n})$. We note that $\gamma=0$ corresponds to the regular setup where ${\mathbf{B}}$ and ${\mathbf{E}}$ lie in two different planes. The above results reduce to the regular result for LMC and PHC if $\gamma$ is set to zero.

We shall now present the thermo-electrical coefficients for transport. The magnetic field here is assumed to have the same form as mentioned above: $\mathbf{B}=B\cos\gamma\hat{j}+B\sin\gamma\hat{y}$ and $\mathbf{\nabla T}=\nabla T\hat{j}$ with $j=x,z$. Therefore, the CME term in $\alpha_{xx}$ is proportional to $n^{2}B^{2}T\cos^{2}\gamma\mu^{-1-2/n}$. The complete $\alpha_{xx}$ is proportional to $T(-\mu n+B^{2}\mu^{-1-2/n}n^{2}\cos^{2}\gamma)$.

On the other hand, one can find that the CME term in $\alpha_{zz}$ is proportional to $n^{2}B^{2}T\cos^{2}\gamma\mu^{-1-2/n}$. The complete $\alpha_{zz}$ is proportional to $T(-\mu^{2/n-1}/n^{2}+B^{2}\mu^{-1-2/n}n^{2}\cos^{2}\gamma)$. We also find that the CME term and complete $\alpha_{yx}$ are both proportional to $TB^{2}n^{2}\cos\gamma\sin\gamma\mu^{-1-2/n}$. Interestingly, the CME tern in $\alpha_{yz}$ is proportional to $TB^{2}n^{2}\cos\gamma\sin\gamma\mu^{-1-2/n}$ whereas the complete $\alpha_{yz}\propto n^{2}TB^{2}\cos\gamma\sin\gamma(\mu^{-1-2/n}+\mu^{1-2/n})$. We would like to point out that $\gamma=0$ corresponds to the regular setup where ${\mathbf{B}}$ and ${\mathbf{\nabla}T}$ lie in two different planes. The above results reduce to the regular result for LTEC and TTEC if $\gamma$ is set to zero. After the full calculation, one can obtain the Nernst coefficient to be

[TABLE]