Topological Magnus responses in two- and three-dimensional systems

Recently, time-reversal symmetric but inversion broken systems with nontrivial Berry curvature in the presence of a built-in electric field have been proposed to exhibit a new type of linear Hall effect in ballistic regime, namely, the Magnus Hall effect. The transverse current here is caused by the Magnus velocity that is proportional to the built-in electric field enabling us to examine the Magnus responses, in particular, Magnus Hall conductivity and Magnus Nernst conductivity, with chemical potential. Starting with two-dimensional (2D) topological systems, we find that warping induced asymmetry in both the Fermi surface and Berry curvature can in general enhance the Magnus response for monolayer graphene and surface states of topological insulator. The strain alone is only responsible for Magnus valley responses in monolayer graphene, while warping leads to finite Magnus response there. Interestingly, on the other hand, strain can change the Fermi surface character substantially that further results in distinct behavior of Magnus transport coefficients as we observe in bilayer graphene. These responses there remain almost insensitive to warping unlike the case of monolayer graphene. Going beyond 2D systems, we also investigate the Magnus responses in three-dimensional multi-Weyl semimetals (mWSMs) to probe the effect of tilt and anisotropic nonlinear energy dispersion. Remarkably, Magnus responses can only survive for the WSMs with chiral tilt. In particular, our study indicates that the chiral (achiral) tilt engenders Magnus (Magnus valley) responses. Therefore, Magnus responses can be used as a tool to distinguish between the untilted and tilted WSMs in experiments. In addition, we find that the Magnus Hall responses get suppressed with increasing the nonlinearity associated with the band touching around the multi-Weyl node.

Figure 1

Evolution of BC and Fermi surface in ML graphene (16) with warping ${\lambda }_1={\lambda }_2={\lambda }_3=\lambda$ and strain $v_2\ne v_1$. Top panel: the BC $\left[{\mathrm{\Omega }}_z^{ML}(\mathit{k},\zeta =+1)\right]$ and $x$ component of the velocity $\left[v_x^{ML}(\mathit{k},\zeta =+1)\right]$ over the Fermi surface are shown for (a) unstrained ML graphene without warping ($v_2=v_1, \lambda =0$) and strained ML graphene ($v_2=2v_1$) with $C_3$ symmetric warping of different strengths (b) $\lambda =0.2$, (c) $\lambda =0.35$, and (d) $\lambda =0.5$ eV ${\AA }^2$. Bottom panel: (e)–(h) depict ${\mathrm{\Omega }}_z^{ML}(\mathit{k},\zeta =-1)$ and $v_x^{ML}(\mathit{k},\zeta =-1)$. The strength of ${\mathrm{\Omega }}_z^{ML}(\mathit{k},\zeta )$ and $v_x^{ML}(\mathit{k},\zeta )$ are represented in the color codes side by side. The parameters (in the units of energy eV) used in the calculations are ${\mathrm{\Delta }}_g=0.06$ eV and $v_1=0.87\mathrm{eV}\AA$. The Fermi surface is plotted for the constant energy $E=-0.28$ eV.

Figure 2

Total valley summed contributions of (a) MH conductivity $\sigma$ (in the unit of ${10}^{-3}{e}^2/\hslash$) and (b) MN conductivity $\alpha$ (in the unit of ${10}^{-5}e{k}_B/\hslash$) in the presence of strain ($v_2=2v_1$) for different warping strengths $\lambda =0.2$, 0.35, and 0.5 eV ${\AA }^2$are shown for ML graphene. Noticeably, the warping can enhance the responses even after valley sum is performed, as it generates asymmetric contributions between valleys. We consider $\mathrm{\Delta }U=0.01$ eV and $k_{B}T=0.001$ eV. All other parameters are kept the same as those in Fig. 1. The chemical potential $\mu$ is chosen in the unit of eV throughout the paper.

Figure 3

Evolution of BC and Fermi surface in BL graphene (23) with strain $w$ for warping ${\lambda }_1={\lambda }_2={\lambda }_3=\lambda =0.001$ eV ${\AA }^2$. Left column: the BC ${\mathrm{\Omega }}_z^{BL}(\mathit{k},\zeta =+1)$ and $v_x^{BL}(\mathit{k},\zeta =+1)$ over the Fermi surface of BL graphene (a) $w=-3m$, (c) $w=-m$, (e) $w=0$, (g) $w=m$, and (i) $w=3m$ are shown. Right column: we repeat the same set of calculations for $\zeta =-1$ valley. The parameters (in the units of eV) used are ${\mathrm{\Delta }}_g=0.06$ eV, $m=0.008$ eV ${\AA }^2$, and $v=0.5\mathrm{eV}\AA$. The Fermi surface is plotted for the constant energy $E=-0.04$ eV.

Figure 4

Valley responses of (a) MH (in the unit of ${10}^2{e}^2/\hslash$) and (b) MN (in the unit of ${10}^{-1}e{k}_B/\hslash$) conductivities in BL graphene with $\mathrm{\Delta }U=0.01$ eV, and $k_{B}T=0.001$ eV for $w=-3m,0,3m$. All other parameters are kept the same as those in Fig. 3. The prominent and asymmetric valley responses in the presence of strain for BL graphene are markedly different from the symmetric responses for ML graphene.

Figure 5

Valley summed (a) MH (in the unit of $10e^2/\hslash$) and (b) MN (in the unit of ${10}^{-2}e{k}_B/\hslash$) conductivities in BL graphene for $w=-3m,0,3m$. The transport behavior changes with strain substantially. We consider the same parameters as those used in Fig. 4.

Figure 6

Evolution of BC and Fermi surface, calculated from Eq. (27), with different warping strengths (a) $\lambda =50$ eV ${\AA }^3$, (b) $\lambda =200$ eV ${\AA }^3$, and (c) $\lambda =400$ eV ${\AA }^3$ are shown. Fermi surface is plotted for $E=-0.05$ eV. We note that, for $\lambda =50$ eV ${\AA }^3$, the Fermi surface remains circular, which gradually evolves to hexagonal shape with increasing $\lambda$. We consider $v=1\mathrm{eV}\AA$ and $E_0=0$ in our calculation.

Figure 7

(a) MH (in the unit of ${10}^{-2}{e}^2/\hslash$) and (b) MN (in the unit of ${10}^{-4}e{k}_B/\hslash$) conductivities as a function of chemical potential for different warping strengths $\lambda =50$, 200, and 400 eV ${\AA }^3$ are depicted. The parameters used are $v=1\mathrm{eV}\AA , \mathrm{\Delta }U=0.01$ eV, and $k_{B}T=0.001$ eV.

Figure 8

Distribution of the BC and the Fermi surfaces, calculated from Eq. (31), in (a) [(d)] single Weyl node with $n=1$, (b) [(e)] double Weyl node with $n=2$, and (c) [(f)] triple Weyl node with $n=3$ for the untilted case, i.e., $C_{+}=0$ [tilted case, i.e., $C_{+}=2.0]$ are shown. The Fermi surface is calculated for $E=-0.05$ eV with $v=1\mathrm{eV}\AA$. The deformation of BC is clearly observed with increasing nonlinearity and anisotropy in the WSM.

Figure 9

(a) MH (in the unit of $10e^2/\hslash$) and (b) MN (in the unit of $e{k}_B/\hslash$) conductivities are shown for tilted and untilted Weyl nodes with $\zeta =+1$. We consider $v=1\mathrm{eV}\AA , \mathrm{\Delta }U=0.01$ eV, and $k_{B}T=0.001$ eV. We observe that the untilted Weyl node with $C_{+}=0$ (tilted Weyl node with $C_{+}=0.6$) results in null (substantial) Magnus responses. We note that MH and MN conductivities are exactly opposite at two opposite Weyl nodes with $\zeta =+1$ and $-1$ owing to the antisymmetric nature of BC [Eq. (33)].

Figure 10

MH (in the unit of $10e^2/\hslash$) conductivity is shown for different strengths of the tilting parameter ($C_{+}$) for fixed $n=1$. Gradual increase in the response is observed with increasing tilt strength $C_{+}$. All other parameters are kept the same as those mentioned in Fig. 9.