Numerical study of universal conductance fluctuations in three-dimensional topological semimetals

We study the conductance fluctuation in topological semimetals. Through statistical distribution of energy levels of topological semimetals, we determine the dominant parameters of the universal conductance fluctuation (UCF), i.e., the number of uncorrelated bands $k$, the level degeneracy $s$, and the symmetry parameter $\beta$. These parameters allow us to predict the zero-temperature intrinsic UCF of topological semimetals using the Altshuler-Lee-Stone theory. Then, we obtain numerically the conductance fluctuations for topological semimetals of quasi-one-dimensional geometry. We find that for Dirac (Weyl) semimetals, the theoretical prediction coincides with the numerical results. However, a nonuniversal conductance fluctuation behavior is found for topological nodal line semimetals; that is, the conductance fluctuation amplitude increases with the enlargement of spin-orbit-coupling strength. We find that such unexpected parameter-dependent phenomena of conductance fluctuation are related to the Fermi-surface shape of three-dimensional (3D) topological semimetals. These results will help us to understand the existing and future experimental results of UCF in 3D topological semimetals.

Figure 1

$\mathrm{\Delta }G$ versus $\nu =L_z/L_x=L_z/L_y$ with open boundary conditions. $\nu =1/3,2/5,1/2,2/3,1,2,3,5,50$ from left to right.

Figure 2

Nearest-neighbor energy-level distance distribution $P\left(S\right)$ for different symmetry classes of Hamiltonian $H, L_x=L_y=L_z=10, A=0.5$, Fermi energy $E_F=0, W=7$, averaged over 1000 ensembles. (a) $D=0, m_z=0$. (b) $D=1, m_z=0$. (c) $D=0, m_z=0.01$. (d) $D=1, m_z=0.01$. The solid green, red, and orange lines represent the Wigner surmise of orthogonal, unitary, and symplectic ensembles, respectively, and the blue line represents the Poisson distribution.

Figure 3

Evolution of UCF with symmetry parameters $L_x=10,L_y=20,L_z=100, A=1, E_F=0.01$. (a) $\mathrm{\Delta }G$ versus $W$ under the condition of $D=0, \varphi =0\sim 0.1\pi$. (b) $\mathrm{\Delta }G$ versus $W$ under the condition of $D=3, \varphi =0\sim 0.1\pi$. The legend in (a) also applies to (b). (c) UCF for $D=0\sim 4, m_z={10}^{-6}\sim 1$. Horizontal lines in (a), (b), and (c) from top to bottom correspond to $\mathrm{\Delta }G=c_1/\sqrt{2},2,2\sqrt{2}=0.516,0.365,0.258$. (d) Localization length $\xi$ versus disorder strength $W$, calculated with the transfer matrix method with $L_x=10,L_y=20$ and periodic boundary conditions applied along each transverse dimension.

Figure 4

$\mathrm{\Delta }G$ versus $W$ under the conditions of (a) size $L_x=15,L_y=15,L_z=100, B=0, E_F=3$ and (b) size $L_x=10,L_y=10,L_z=67$ ($L_x=15,L_y=15,L_z=100, L_x=20,L_y=20,L_z=133$), $A=0, E_F=0.5$. Larger symbol size corresponds to larger sample size. (c) Nearest-neighbor distribution $P\left(s\right)$ with $E_F=0.5, W=7, L_x=L_y=L_z=10, B=0.5,B=6$ (inset). (d) $\mathrm{\Delta }G$ versus $W$ under the condition of $A=B, E_F=0.5$, and size $L_x=15,L_y=15,L_z=100$. In (a), (b), and (d), horizontal dashed lines correspond to $\mathrm{\Delta }G=0.729/2,2\sqrt{2}=0.365,0.258$, respectively.

Figure 5

(a)–(c) Shapes of the Fermi surface and (d)–(f) the projections onto the $k_y-k_z$ plane for $A=2,3,6, E_F=3,B=0$. Other parameters are the same as those used in Fig. 4.

Figure 6

(a)–(c) Shapes of the Fermi surface and (d)–(f) the projections onto the $k_y-k_z$ plane for $B=2,3,6, E_F=0.5,A=0$. Other parameters are the same as those used in Fig. 4.

Figure 7

(a) Evolution of conductance fluctuation versus disorder strength for different choices of $\mu$; other parameters are $E_F=0,L_x=L_y=L_z=10$. (b) Comparison between UCF amplitude versus Fermi-surface anisotropy $\mu$ and spatial size anisotropy $\nu$. The error bars are estimated from $\mathrm{\Delta }G$ at the plateau region in (a). The horizontal lines in both (a) and (b) are $\mathrm{\Delta }G=0.55,0.365$, corresponding to 3D and quasi-1D conductance fluctuations, respectively.

Figure 8

Evolution of the Fermi surface versus $\mu$, with $E_F=0$ fixed.