Coexisting Edge States and Gapless Bulk in Topological States of Matter

We consider two-dimensional systems in which edge states coexist with a gapless bulk. Such systems may be constructed, for example, by coupling a gapped two-dimensional state of matter that carries edge states to a gapless two-dimensional system in which the spectrum is composed of a number of Dirac cones. We find that, in the absence of disorder, the edge states could be protected even when the two systems are coupled, due to momentum and energy conservation. We distinguish between weak and strong edge states by the level of their mixing with the bulk. In the presence of disorder, the edge states may be stabilized when the bulk is localized or destabilized when the bulk is metallic. We analyze the conditions under which these two cases occur. Finally, we propose a concrete physical realization for one of our models based on bilayer Hg(Cd)Te quantum wells.

Figure 1

Band structure and edge state wave functions of model I for different edge orientations. Edges along the $y$ direction in (a) and (b): Localized edge states (red, green) coexist with zero modes in the bulk (blue). Edges along the $x$ direction in (c) and (d): Because of energy and momentum overlap, hybridized edge states (red) coexist with the bulk states.

Figure 2

(a) Disorder averaged bulk conductance of model I as a function of disorder strength $\delta$ and the gap-opening parameter $m$. Chern numbers of the two phases are shown. (b) Average conductance $G/G_0$, for $m=0$, as a function of $\delta$, for periodic (red line) and hard wall (blue line) boundary conditions; see Ref. [19] for simulation parameters.

Figure 3

(a) Disorder averaged bulk conductance of model III, as a function of $M_0$ and disorder strength $\delta$. (b) Average bulk conductance for $M_0=-1$ and $\delta =0.95$, as a function of the system size $L$ for different coupling strengths. The conducting region around $\delta =1$ becomes localized as the system size is increased. See Ref. [23] for simulation parameters.

Figure 4

(a) Disorder averaged bulk conductance of two coupled BHZ models, as a function of disorder strength $\delta$ and $M_{0,2}$. (b) Average bulk conductance for $\delta =1$ and $M_{0,2}=-1.47$, as a function of the system size $L$ for different couplings $t$. See Ref. [26] for simulation parameters.