Impurity-induced current in a Chern insulator

Chern insulators arguably provide the simplest examples of topological phases. They are characterized by a topological invariant and can be identified by the presence of protected edge states. In this paper, we show that a local impurity in a Chern insulator induces a twofold response: bound states that carry a chiral current and a net current circulating around the impurity. This is a manifestation of broken time reversal symmetry and persists even for an infinitesimal impurity potential. To illustrate this, we consider a Coulomb impurity in the Haldane model. Working in the low-energy, long-wavelength limit, we show that an infinitesimal impurity strength suffices to create bound states. We find analytic wave functions for the bound states and show that they carry a circulating current. In contrast, in the case of a trivial analog—graphene with a gap induced by a sublattice potential—bound states occur but carry no current. In the many-body problem of the Haldane model at half-filling, we use a linear response approach to demonstrate a circulating current around the impurity. Impurity textures in insulators are generally expected to decay exponentially; in contrast, this current decays polynomially with distance from the impurity. Going beyond the Haldane model, we consider the case of coexisting trivial and nontrivial masses. We find that the impurity induces a local chiral current as long as time-reversal symmetry is broken. However, the decay of this local current bears a signature of the overall topology—the current decays polynomially in a nontrivial system and exponentially in a trivial system. In all cases, our analytic results agree well with numerical tight-binding simulations.

Figure 1

Honeycomb lattice with a Coulomb impurity. The complex nearest-neighbor hopping of the Haldane model is shown in one plaquette.

Figure 2

Bound-state spectrum as a function of impurity strength for (a) trivial and (b) nontrivial gaps. We only show the four highest bound states. The shaded regions represent conduction and valence bands.

Figure 3

Wave function of highest energy bound state for $g=0.2$ [(a),(b)], 0.35 [(c),(d)] and 0.55 [(e),(f)]. The upper panels plot probability density vs distance from impurity at points along the line shown in the inset to (a). The lower panels show current vs distance. Current is evaluated on bonds intersecting the line indicated in the inset to (b). Red lines show analytical results scaled by an overall factor as indicated in each panel. Dots are tight-binding results.

Figure 4

Tight-binding results for current carried by bound states. (a) Current in the highest bound state for $g=0.72$. (b) Current in the second highest bound state for $g=0.3$. The color and length of an arrow show the amplitude of current on the bond; current direction is indicated by the arrow head. The color bar gives the current in units of $t$, the nearest-neighbor hopping.

Figure 5

Tight-binding results for total current around impurity. (a) Currents on bonds near impurity ($g=0.5$); the sites surrounding the impurity organize themselves into loops. (b) Current vs distance from impurity for $g=0.5$. The red line is the best fit (excluding the first two data points) to the form $J=a{R}^{-b}$. We find $b=2.06257\pm 0.02286$, close to the linear response result of $b=2$. (c) Total current vs $g$. The current is linear up to $g\sim 1.15$.

Figure 6

Current vs $R$ for the general Hamiltonian with both mass terms, with $g=0.3$. Keeping $t^{\prime }$ fixed, we increase the sublattice potential $V$. A topological transition occurs at the $V\sim 0.2598$. In each case, we plot the tight-binding results for current with the best fit to an exponential function $J=A{e}^{-BR}$ (black line) and to a polynomial function $J=C{R}^{-D}$ (red line). (a),(b) With both mass terms present, but staying on the topological side, the polynomial function gives a better fit. The exponent is $D\approx 2$. (c),(d) On the trivial side, the exponential function gives a better fit.