Proximity effects and a topological invariant in a Chern insulator connected to leads

Abstract

The observed robustly quantized Hall conductance in quantum Hall systems and Chern insulators (CI) have so far been understood in terms of the topology of isolated systems, which are not coupled to leads. It is assumed that the leads act as inert reservoirs that simply supply/absorb electrons to/from the sample. Within a model of a CI coupled to leads with a cylindrical geometry, we show that this is not true. In the proximity of the CI, the edge current leaks into the leads, with the Hall conductance quantized only if this novel proximity effect is taken into account. For a special choice of leads, we identify the conductance with a topological invariant of the system, in terms of the winding number of the phase of the reflection coefficients of the scattering states.

Figures (4)

• Figure 1: (a) Schematic of a Chern insulator connected to metallic leads in the cylindrical geometry. (b) Equivalent Rice-Mele type one-dimensional setup obtained after Fourier transforming along the $y$-direction. The crossed sites indicate the sites belonging to the leads. The parameters $a_m$ and $\eta_m$ indicate onsite potentials in the CI and lead respectively, $b_m$ and $\eta_x$ are hopping terms in the CI and lead respectively, and $\eta_c$ is the CI-lead coupling.
• Figure 2: Variation of the Hall conductance $G_H(\mu)$ with chemical potential $\mu$ for two different models of leads: (i) 1D case with $\eta_x=1,\eta_y=0$ and (ii) case with $\eta_x=1,~\eta_y=0.25$. For (i), we see a quantized plateau of unit value in the bulk gap of the Chern insulator, while for (ii), we do not see quantization. Other parameters: $\mu_w = 1$, $\eta_c = 1$.
• Figure 3: Contributions to the Hall conductance from the CI and metallic leads are plotted as a function of the chemical potential for 2D leads. The inset shows the equivalence of local Hall conductance computed using the NEGF formalism and scattering approach for the 2D leads case. For both plots, $\mu_w= 1, \eta_x = 1, \eta_y = 0.25, \eta_c = 1$. For the inset $\mu=0.8$.
• Figure A1: Winding of the function $z(k) = e^{i\phi(k)}$ as $k$ varies from $0$ to $2\pi$. (a) and (b) are for topological regimes with $\mu_w=1$ and $\mu_w=-1$, respectively and (c) corresponds to the trivial regime with $\mu_w=2.2$. The arrows labelled from $a$ to $d$ indicate the direction of change of $\phi(k)$ as $k$ increases from $0$ to $2\pi$. The common parameters for the three panels are $\mu=0.1,~\eta_x=1$ and $\eta_c=1$.