⇦Chinese Physics Letters, 2020, Vol. 37, No. 11, Article code 117201 Universal Minimum Conductivity in Disordered Double-Weyl Semimetal Zhen Ning (宁震)1, Bo Fu (付博)2, Qinwei Shi (石勤伟)3*, and Xiaoping Wang (王晓平)1,3* Affiliations 1Department of Physics, University of Science and Technology of China, Hefei 230026, China 2Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 3Hefei National Laboratory for Physical Sciences at the Microscale & Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China Received 12 May 2020; accepted 17 September 2020; published online 8 November 2020 Supported by the the National Natural Science Foundation of China (Grant No. 11874337).
*Corresponding authors. Email: phsqw@ustc.edu.cn; xpwang@ustc.edu.cn Citation Text: Ning Z, Fu B, Shi Q W and Wang X P 2020 Chin. Phys. Lett. 37 117201    Abstract We report an exact numerical study on disorder effect in double-Weyl semimetals, and compare exact numerical solutions for the quasiparticle behavior with the Born approximation and renormalization group results. It is revealed that the low-energy quasiparticle properties are renormalized by multiple-impurity scattering processes, leading to apparent power-law behavior of the self-energy. Therefore, the quasiparticle residue surrounding nodal points is considerably reduced and vanishes as $Z_{\rm E}\propto E^{r}$ with nonuniversal exponent $r$. We show that such unusual behavior of the quasiparticle leads to strong temperature dependence of diffusive conductivity. Remarkably, we also find a universal minimum conductivity along the direction of linear dispersion at the nodal point, which can be directly observed by experimentalist. DOI:10.1088/0256-307X/37/11/117201 PACS:72.80.Ng, 72.10.-d, 05.10.Cc © 2020 Chinese Physics Society Article Text Exotic physical properties of topological materials have attracted a great deal of attention in recent years.[1,2,3] An example widely studied is a three-dimensional Weyl semimetal (WSM) which contains isolated band touching points and acts like monopoles in the Brillouin zone (BZ). In the usual WSM (single WSM), the low-energy quasiparticle excitation with linear dispersion in all directions emerges in the vicinity of Weyl nodes which carry topological charge (chirality $\chi = \pm 1$). Recently, single WSMs have been experimentally investigated in $\rm TaAs$[4–6] and $\rm NbAs$,[7] the material realization provides an experiment platform to investigate novel phenomena predicted by the quantum field theory, such as the chiral anomaly[8] and the anomalous Hall effect.[9–12] Beyond the single WSM, there exist Weyl nodes with higher topological charge $\chi=\pm 2$ referred to as the double Weyl semimetal, several candidates for realization of these systems are reported in $\rm HgCr_2Se_4$,[13,14] $\rm SrSi_2$[15] and in optical lattices.[16,17] Double WSMs supports gapless quasiparticle excitation with quadratic dispersion in $k_{x,y}$ plane and linear dispersion along $k_z$ direction. There are many different physical properties in terms of single and double WSMs due to anisotropic band structures.[18–22] For instance, the density of states is linearly proportional to the energy, in contrast to that in the usual Weyl fermion with quadratic energy dependence. The quantum interference correction to the semiclassical conductivity is distinguished,[23] the destructive interference gives a weak anti-localization effect in single WSMs, however, the constructive interference in double WSMs enhances the backscattering and thus leads to weak localization effect. It has been point out that weak short-range interactions are irrelevant perturbation for all three-dimensional Weyl semimetals with arbitrary chirality. However, phase transitions could occur for sufficiently strong interactions which lead to various quantum critical phenomena.[24] In the presence of long-range Coulomb interactions, a random phase approximation and renormalization group (RG) analysis show an anisotropic Coulomb screening at the stable nontrivial fixed point due to the anisotropic dispersions of double-Weyl fermions.[25,26] The disorder effects on noninteracting systems of Weyl fermions have been widely studied by large-scale numerical computation[27–30] and theoretical approaches.[31,32] The RG analysis has shown that the quasiparticle residue of double WSMs vanishes at a special energy scale ($E_{\rm c}$), $Z_{\rm E}\propto \sqrt{\ln(E/E_{\rm c})}$,[32] at which the RG flow stops. Thus, such a sharp reduction of quasiparticle residue near the nodal point is expected to substantially modify the physical quantities such as the classical diffusive conductivity.[33,34] However, the quasiparticle residual obtained by the perturbative renormalization group approach may not be justifiable because of its ignorance of scattering events of multiple impurities. Therefore, a more careful numerical analysis involving all higher order scattering processes is needed. In this work, using the Lanczos method,[35,36] we determine accurately the quasiparticle properties of a disordered double WSM. We construct a power-law function of the self-energy for describing behaviors of quasiparticle and the nontrivial transport properties around the nodal point. This unusual behavior is distinctly different from that predicted by theoretical calculation.[18] In particular, we compute the diffusive conductivity via the Kubo formalism, and show that such a novel quasiparticle self energy leads to strong temperature dependence of classical conductivity. As a rewarding outcome, we find a universal minimum conductivity at the nodal point along the direction of linear dispersion, which is similar to the minimum conductivity at the Dirac point of graphene[37] and could be further studied by transport measurements. Model and Theoretical method. The low-energy effective Hamiltonian for double WSM near a single Weyl point is given by the two-band model as follows:[29,30] $$\begin{alignat}{1} \hat{H_0} = v_{\perp}\sigma_x(\partial^2_{y}-\partial^2_{x})-2v_{\perp}\sigma_y(\partial_{y}\partial_{x})-iv_z\sigma_z\partial_{z},~~ \tag {1} \end{alignat} $$ where $\sigma_i$ are standard Pauli matrices, $v_{\perp}$ and $v_z$ are velocity constants. The model has a conduction band and a valence band with dispersions $\epsilon_{k,\pm} = \pm\sqrt{v_{\perp}(k^2_x+k^2_y)^2+v_zk^2_z}$, and ${\boldsymbol k}$ is the momentum measured from the Weyl nodes. For exploring the disorder effects, we use a tight-binding Hamiltonian on a cubic lattice to conduct numerical simulation, $$\begin{align} \hat{H}&=\hat{H_0}+\hat{V}, \\ \hat{H_0}&=\sum_{r}(-\varPsi^†_{r+i_{x}} t\sigma_{x}\varPsi_{r}+\varPsi^†_{r+i_{y}} t\sigma_{x}\varPsi_{r}+{\rm H.c.})\\ &+\sum_{r} \Big(-\varPsi^†_{r+i_{x}+i_{y}} \frac{t}{2}\sigma_{y}\varPsi_{r}+\varPsi^†_{r+i_{x}-i_{y}} \frac{t}{2}\sigma_{y}\varPsi_{r}+{\rm H.c.}\Big)\\ &+\sum_{r}\Big(\varPsi^†_{r+i_{z}} \frac{it_z}{2}\sigma_{z}\varPsi_{r}+{\rm H.c.}\Big), \\ \hat{V}&=\sum_{r}\varPsi^†_{r} V(r)I\varPsi_{r}.~~ \tag {2} \end{align} $$ Two double-Weyl points are realized at $(0,0,0/\pi)$. $\varPsi^†_{r}=(c^†_{1,r},c^†_{2,r})$ is the two-component spinor, the hopping parameters are $t = v_{\perp}/a^2_{\rm L}$, $t_z = v_z/a_{\rm L}$, and $a_{\rm L}$ is the lattice constant. $V(r)$ represents a short-range disorder and is introduced by the on-site random number distributed within $[-W/2,W/2]$. The energy scale is measured in units of $t$ in this work, and we set $t=t_z=a_{\rm L}=\hbar=1$. We choose a large sample $[L^3=(500)^3]$ and calculate the quasi-particle self-energy $\varSigma(E,\boldsymbol{k})$.[36] In addition, a small damping parameter $10^{-3}$ is used in calculating Green's function. In our simulation, we find that the self-energy is independent of band index. The wave vector $\boldsymbol{k}$ dependence of the self energy is also insensitive, the analytical evidence is presented in the Supplemental Material. Therefore, the self-energy function is approximately dependent only on the energy $\varSigma(E)$.
cpl-37-11-117201-fig1.png
Fig. 1. (a) Imaginary and (b) real parts of the self-energy of the disordered double WSM for different strengths of disorder $W$ (in units of $t$). The open circles are the numerical results while the solid lines are the fitting curves of (a) Eq. (3) and (b) Eq. (5).
cpl-37-11-117201-fig2.png
Fig. 2. The constant plus power-law function of Eq. (3) is plotted on logarithmic coordinates for different strengths of disorder. The circles are numerical results and the solid line is the linear fitting over an energy scale $E \in [0.005,0.2]$ (or $\log_{10} E = [-2.3,-0.7]$).
Quasiparticle Properties. The first-order Born approximation predicts the liner behavior of the imaginary part of the self-energy ${\rm Im}\varSigma(E)=-\pi\frac{\gamma}{2}|E|$, which is in agreement with our numerical results for weak disorder, as shown in Fig. 1(a) and see the Supplemental Material. However, the self-energy ${\rm Im}\varSigma(E)$ gradually deviates from the linear behavior as disorder strength increases, and its value at zero energy ${\rm Im}\varSigma(E=0)$ is raised. These observations inspire us to naturally generate the linear function ${\rm Im}\varSigma(E)$ into a power-law formula[38–40] $$ \begin{aligned} {\rm Im}\varSigma(E)=-\varSigma_0-J|E|^{a}. \end{aligned}~~ \tag {3} $$ The fitting results are shown in Fig. 1(a), the agreement between Eq. (3) and the numerical results is excellent. In order to further demonstrate the constant plus power-law function [Eq. (3)], we plot the results of ${\rm Im}\varSigma(E)$ on a log-log scale within an energy window $E \in [0.005,0.2]$, as shown in Fig. 2. The high precision numerical data is obtained by choosing a large sample $L^3=(500)^3$.
cpl-37-11-117201-fig3.png
Fig. 3. The fitting results (open circles) of parameters by Eq. (3). We plot $a$, $J$ and $\varSigma_0$ versus disorder strength $\gamma$ in (a), (b) and (c), respectively.
Here, we discuss the fitting results of the parameters $a$, $J$ and $\varSigma_0$ in Eq. (3) as a function of disorder strength $\gamma$, as shown in Figs. 3(a), 3(b) and 3(c), respectively. A dimensionless parameter $\gamma=\frac{a^3_{\rm L}W^2}{96\pi v_{\perp}v_z}$ from Fermi's golden rule is defined to characterize the strength of disorder. For weak disorder ($\gamma < 0.01$), we make an approximation that the power-law exponent remains to be $a\approx1$ as shown in Fig. 3(a). Noted that our numerical simulation is based on the lattice model of Eq. (2) with finite lattice size, a more careful investigation of the weak disorder regime ($\gamma < 0.01$) requires a larger sample and more computational resources. The power-law exponent then reduces continuously to $a\approx0.8$ as the strength of disorder increases towards the $\gamma\approx0.03$ beyond which the exponent $a$ increases with the increasing disorder strength. The parameter $J$ is approximately a superlinear function of disorder strength [Fig. 3(b)]. In addition, the parameter $\varSigma_0$ can be interpreted as an energy scale $E_{\rm c}$ that separates the semimetal phase and the diffusive metal phase.[29] Using the one-loop RG approach (see Ref. [32] for detailed derivation), we can find an exponential function to fit the numerical data of $\varSigma_0$, $$\begin{align} \varSigma_0= \varLambda e^{-\frac{1}{u\gamma}}.~~ \tag {4} \end{align} $$ The two constants $\varLambda \approx 2.6$ and $u\approx 6.0$ can be treated as the fitting parameters, and the fitting is shown in Fig. 3(c). The similar expression of the energy scale $E_{\rm c}$ has been noticed in disordered graphene.[34] Moreover, the fitting formula for the real part of the self-energy ${\rm Re}\varSigma(\omega)$ can be obtained via the Kramers–Kronig relation[28] $$\begin{alignat}{1} {\rm Re} \varSigma(E)=\begin{cases} D\,{\rm{sgn}}(E)|E|^{a}+CE,&(a\neq1)\\ -\frac{2\,J}{\pi}E\ln|\frac{\omega_{\rm c}}{E}|, &(a=1)\\ \end{cases}~~ \tag {5} \end{alignat} $$ where $\mathrm{sgn}$ is the signum function, and the constant parameters are $D=-\mathrm{{\tan}}(\frac{\pi}{2}a)J$, $C=-\frac{2\omega_{\rm c}^{a-1}}{\pi(a-1)}J$. If we set $r=1-a$ and $D =-{\rm tan}[\frac{\pi}{2}(1-r)]\varDelta \mathop{=}\limits_{r\rightarrow0} \frac{-2\varDelta}{\pi r}$, then the limit of ${\rm Re}\varSigma(E)$ will become $\frac{-2\varDelta}{\pi}E{\lim\limits_{r\rightarrow0}}\omega^{-r}_{\rm c}\frac{(\omega_{\rm c}/|E|)^r-1}{r} =\frac{-2\varDelta}{\pi}E\ln(\frac{\omega_{\rm c}}{|E|})$. The power-law function returns to the logarithmic function as $a\rightarrow1$. We find that the numerical results of ${\rm Re}\varSigma(E)$ are in very good agreement with the fitting formula, as shown in Fig. 1(b). The finding of the power-law self energy is very important to understand the quasiparticle behavior around a nodal point. The quasi-particle residue is defined as $Z_{\rm E}=1/[1-\partial_{\rm E}{\rm Re} \varSigma(E)]$. The derivative of $\partial_{\rm E}{\rm Re}\varSigma(E)$ contains a constant term $C$ and a singular term $\propto |E|^{a-1}$ (for $a < 1$). This singular term makes the main contribution to $Z_{\rm E}^{-1}$ as $E\rightarrow0$, and we find the power-law behavior of $Z_{\rm E} \propto |E|^{r}$ with $r=1-a$, the system possesses non-Fermi liquid behavior[28] for $a < 1$. However, for weak disorder, the power-law divergence of $Z_{\rm E}^{-1}$ returns to the logarithmic singularity $Z_{\rm E}^{-1} \propto \ln\frac{\omega_{\rm c}}{|E|}$, which is consistent with the calculation from the first order Born approximation, then the logarithmic corrections to the quasi-particle residue is summed up $Z_{\rm E}\propto \sqrt{\ln(E/E_{\rm c})}$[33,34] using the one-loop renormalization, the quasi-particle residue vanishes as $E\rightarrow E_{\rm c}$. As the Fermi energy approaches to the nodal point, the group velocity along the direction of linear dispersion is renormalized by the factor of $Z_{\rm E}$ as $v_g=Z_{\rm E}c_y$, which is sharply reduced in the low energy regime. Conductivity. Since the power-law self-energy [Eqs. (3) and (5)] gives substantial correction to the quasiparticle properties, it should not be ignored in the investigation of transport behavior of the double WSM.[18–20,30] The standard approach to describe the quasi-particle transport is the Kubo–Greenwood formula.[41,42] Here, we express the longitudinal conductivity $\sigma_{\mu\mu}(\mu = x,z)$ as $$\begin{alignat}{1} \sigma_{xx}(E_{\rm F},T) &= \sigma^0_{xx} \int d\omega\Big[-\frac{\partial f(\omega,E_{\rm F})}{\partial\omega}\Big]K_{xx}(\omega),\\ \sigma_{zz}(E_{\rm F},T) &= \sigma^0_{zz}\int d\omega\Big[-\frac{\partial f(\omega,E_{\rm F})}{\partial\omega}\Big]K_{zz}(\omega),~~ \tag {6} \end{alignat} $$ with $$\begin{align} K_{xx}(\omega)&=\frac{\alpha^2+3\eta^2}{\eta},\\ K_{zz}(\omega)&=1+\Big(\frac{\alpha}{\eta} +\frac{\eta}\alpha\Big)\tan^{-1}\Big(\frac{\alpha}{\eta}\Big),~~ \tag {7} \end{align} $$ where $E_{\rm F}$ is the Fermi energy, $f(\omega)=1/[e^{(\omega-E_{\rm F})/(k_{\rm B}T)}+1]$ is the fermi-Dirac distribution function and we define $\alpha=\omega-{\rm Re}\varSigma(\omega)$ and $\eta=-{\rm Im}\varSigma(\omega)$ for simplicity. We note that the conductivity in $k_x$ and $k_y$ directions are identical, $\sigma_{xx}=\sigma_{yy}$. The conductivity along the direction of quadratic and linear dispersion are measured by the two constant $\sigma^0_{xx} = \frac{e^{2}}{6\pi^2v_{z}}$, $\sigma^0_{zz} = \frac{e^{2}v_{z}}{16\pi^2v_{\perp}}$, respectively. We plot the zero-temperature conductivity as a function of Fermi energy $E_{\rm F}$ in Figs. 4(a) and 4(b), the $\omega$ integration in Eq. (6) is completed by replacing the function $-\frac{\partial f(\omega,E_{\rm F})}{\partial\omega}$ with the delta function $-\delta(\omega-E_{\rm F})$. The conductivity along the $\hat{x}$ direction increases linearly with Fermi energy for weak disorder and high Fermi energy. In this case of $\tau E_{\rm F}\gg1$,[43,44] the function $K_{xx}$ in Eq. (7) returns to the results from the Boltzmann transport theory $\sigma_{xx}\propto \frac{E_{\rm F}}{\gamma}$,[18] exhibiting the similar behavior of conventional 2D electron gas. By increasing the temperature, as shown in Figs. 4(c) and 4(e), the conductivity $\sigma_{xx}$ possesses strongly insulating temperature behavior (increases as the temperature increases, ${d}\sigma/{d}T>0$) at low Fermi energy arising from the thermal excitation of carriers, which is suppressed in the high doping regime. These results also qualitatively agree with the Boltzmann transport theory.[19] However, in the case of $\tau E_{\rm F}\ll1$, the discrepancy between our results and the Boltzmann transport theory becomes obvious as the Fermi energy approaches the nodal point. For example, the residual conductivity at the gapless point is directly dependent on strength of disorder $\sigma^{r}_{xx}= 3\sigma^0_{xx}\varSigma_0$, which becomes non-negligible for the strong disorder case. However, along the direction of linear dispersion, there exists a distinct qualitative difference between our results [Eq. (7)] and the Boltzmann transport theory ($\sigma_{zz} \sim \frac{1}{\gamma}$)[18] due to the multi-scattering effects, as plotted in Figs. 4(b), 4(d) and 4(f). As Fermi energy is close to the nodal point, the conductivity $\sigma_{zz}$ displays a sharp dip [see Fig. 4(b)] due to the renormalization of group velocity.[33,34] In the presence of weak strength of disorder ($\gamma=0.02$), as shown in Fig. 4(d), the dip displays strong temperature dependence. For further illustration, we plot the conductivity $\sigma_{zz}$ in Fig. 4(f) as a function of temperature for different Fermi energies. At low Fermi energy (low carrier density), if the temperature is not too small, the conductivity $\sigma_{zz}$ will exhibit insulating-type temperature dependence (${d}\sigma/{d}T>0$). However, in the high doping regime, it shows metallic temperature behavior (${d}\sigma/{d}T < 0$). Therefore, the non-monotonic temperature dependence of $\sigma_{zz}(T)$ is observed in Fig. 4(f). Unlike the other two directions, the residual conductivity $\sigma^{r}_{zz} =2\sigma^0_{zz}$ is independent of disorder. This phenomenon is analogous to the universal minimum conductivity in graphene.[34,37] After considering the vertex correction (see the Supplemental Material for derivation) the residual conductivity $\sigma^{r}_{zz} =2\sigma^0_{zz}/(1-\frac{1}{2}\gamma^2)\approx2\sigma^0_{zz}$[45] is still weakly dependent on disorder strength. Therefore, such a constant conductivity is directly observable in experiments. Since the dip of the conductivity $\sigma_{zz}$ could be smeared by stronger disorder $\gamma=0.05$, as shown in Fig. 4(b), the energy or the temperature dependence of $\sigma_{zz}$ becomes weaker. Therefore, the strong temperature dependence of conductivity $\sigma_{zz}$ can be investigated in the weakly disordered sample of the double WSM.
cpl-37-11-117201-fig4.png
Fig. 4. The conductivity along $k_x$ direction and $k_y$ direction, i.e., $\sigma_{xx}$ (a) $\sigma_{yy}$ (b), at zero temperature plotted versus Fermi energy for different disorder strengths. (c) and (d) The modification of $\sigma_{\mu\mu}(E_{\rm F})$ due to finite temperature. (e) and (f) The temperature dependence of the conductivity $\sigma_{\mu\mu}(T)$ (measured by their values at zero temperature) at different Fermi energy. In (c)–(f), we fix the strength of disorder at $\gamma= 0.02$.
In summary, exploiting the Lanczos method in momentum space, we have investigated the role of disorder in a double Weyl semimetal. The presence of anisotropic dispersion effectively reduces the dimensionality. Therefore, the physical properties of the disordered double Weyl semimetal are reminiscent of the electronic properties of 2D Dirac fermions, the conductivity along the direction of linear dispersion possesses strong temperature dependence and its minimum is analogous to the universal minimum conductivity at the Dirac point in graphene. References Weyl and Dirac semimetals in three-dimensional solidsTopological semimetalsClassification of stable three-dimensional Dirac semimetals with nontrivial topologyObservation of Weyl nodes in TaAsDiscovery of a Weyl fermion semimetal and topological Fermi arcsElectron transport in Dirac and Weyl semimetalsDiscovery of a Weyl fermion state with Fermi arcs in niobium arsenideChiral anomaly and classical negative magnetoresistance of Weyl metalsAnomalous Hall Effect in Weyl MetalsWeak antilocalization and interaction-induced localization of Dirac and Weyl Fermions in topological insulators and semimetalsChiral Current in the Lattice Model of Weyl SemimetalTopological responses from chiral anomaly in multi-Weyl semimetalsChern Semimetal and the Quantized Anomalous Hall Effect in HgCr 2 Se 4 Multi-Weyl Topological Semimetals Stabilized by Point Group SymmetryNew type of Weyl semimetal with quadratic double Weyl fermionsExploring topological double-Weyl semimetals with cold atoms in optical latticesDouble Weyl points and Fermi arcs of topological semimetals in non-Abelian gauge potentialsThermoelectric transport in double-Weyl semimetalsSemiclassical Boltzmann transport theory for multi-Weyl semimetalsOptical conductivity of multi-Weyl semimetalsDoping and tilting on optics in noncentrosymmetric multi-Weyl semimetalsAnomalous thermoelectric phenomena in lattice models of multi-Weyl semimetalsDetecting monopole charge in Weyl semimetals via quantum interference transportInteracting Weyl fermions: Phases, phase transitions, and global phase diagramCorrelation effects in double-Weyl semimetalsCorrelated double-Weyl semimetals with Coulomb interactions: Possible applications to HgCr 2 Se 4 and SrSi 2 Anderson Localization and the Quantum Phase Diagram of Three Dimensional Disordered Dirac SemimetalsAccurate Determination of the Quasiparticle and Scaling Properties Surrounding the Quantum Critical Point of Disordered Three-Dimensional Dirac SemimetalsDirty Weyl semimetals: Stability, phase transition, and quantum criticalityDisordered double Weyl node: Comparison of transport and density of states calculationsQuantum Criticality between Topological and Band Insulators in 3 + 1 DimensionsEffect of weak disorder in multi-Weyl semimetalsEffect of Disorder on Transport in GrapheneElectron transport in disordered grapheneCorrelated electrons in high-temperature superconductorsEvaluation of the Green’s function of disordered grapheneThe electronic properties of grapheneInteger quantum Hall transition: An alternative approach and exact resultsDisorder effects in two-dimensional d -wave superconductorsDisorder effects in two-dimensional Fermi systems with conical spectrum: exact results for the density of statesQuantum Transport in Two-Dimensional Graphite System
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