Real-Space Calculation of the Conductivity Tensor for Disordered Topological Matter

We describe an efficient numerical approach to calculate the longitudinal and transverse Kubo conductivities of large systems using Bastin’s formulation. We expand the Green’s functions in terms of Chebyshev polynomials and compute the conductivity tensor for any temperature and chemical potential in a single step. To illustrate the power and generality of the approach, we calculate the conductivity tensor for the quantum Hall effect in disordered graphene and analyze the effect of the disorder in a Chern insulator in Haldane’s model on a honeycomb lattice.

Figure 1

(a) ${\sigma }_{xx}$ (solid line) and ${\sigma }_{xy}$ (dashed line) for $k_{B}T=0$ (black) and $k_{B}T/t=0.004$ (red/light grey). (b) Electronic density of states (c) ${\sigma }_{xx}$ away from the Dirac point where Shubnikov—de Haas oscillations can be observed for $k_{B}T/t=0.002$ (black), $k_{B}T/t=0.004$ (red/light grey), and $k_{B}T/t=0.008$ (blue/grey). The parameters in panels (a)–(c) are $\varphi /{\varphi }_0\approx 1\times {10}^{-3}$, $SR=200$, $M=6144$, and $N=2\times 128\times 1024$ sites where we use a rectangular geometry to minimize the magnetic flux per unit cell in a system with periodic boundary conditions.

Figure 2

Analysis of the dependency on the number of random vectors and disorder realizations $SR$ and moments $M$: (a) ${\sigma }_{xx}$ and (b) ${\sigma }_{xy}$ for $SR=200$ and $M=2048$ (blue/grey), 4096 (red/light grey), and 6144 (black). (c) ${\sigma }_{xx}$ and (d) ${\sigma }_{xy}$ for $M=6144$ and $SR=25$ (blue/grey), 50 (red/light grey), and 200 (black). In panels (a)–(d) $\varphi /{\varphi }_0=5\times {10}^{-3}$ and $N=2\times 128\times 1024$ sites.

Figure 3

${\rho }_{xx}$ (solid line) and ${\rho }_{xy}$ (dashed line) for HM with (a) ${\mathrm{\Delta }}_T=0.1t$, ${\mathrm{\Delta }}_{AB}=0$, and $\gamma =0.2t$ for $k_{B}T=0$ (black) and $k_{B}T=0.16t$ (red/light grey); (b) ${\mathrm{\Delta }}_T=0.1t$, ${\mathrm{\Delta }}_{AB}=0$, and $k_{B}T=0$ for $\gamma =0.4t$ (black) and $\gamma =1.8t$ (red/light grey); (c) ${\mathrm{\Delta }}_T=0.5t$, ${\mathrm{\Delta }}_{AB}=0.4t$, and $k_{B}T=0$ for $\gamma =0.2t$ (black) and $\gamma =1.8t$ (red/light grey). The green arrow indicates the increase of the topological region with disorder. The system sizes in panels $D=2\times L\times L$ with $L=512$ (a), $L=256$ (b)–(c), and $SR>200$ . Panel (d) illustrates the different gap sizes at $K$ and $K^{\prime }$ for ${\mathrm{\Delta }}_{AB}\ne 0$.