Geometrical engineering of a two-band Chern insulator in two dimensions with arbitrary topological index

Two-dimensional 2-band insulators breaking time-reversal symmetry can present topological phases indexed by a topological invariant called the Chern number. We propose an efficient procedure to determine this topological index, which makes it possible to conceive 2-band, tight-binding Hamiltonians with arbitrary Chern numbers. The technique is illustrated by a step-by-step construction of a model exhibiting five topological phases indexed by Chern numbers $\{0,\pm 1\pm 2\}$. On a finite cylindrical geometry, this insulator possesses up to two edge states which are characterized analytically. The model can be combined with its time-reversal copy to form a quantum spin Hall insulator. It is shown that edge states in the latter can be destroyed by a time-reversal-invariant one-particle perturbation if the Chern number equals $\pm 2$.

Figure 1

(a) Domains associated with the mass term $\cos (k_x+k_y)$. Phases with Chern number $\pm 2$ are realized. (b) Domains associated with the mass term $\sin \left(k_x\right)+\sin \left(k_y\right)$. Phases with Chern number $\pm 1$ are possible after the remaining Dirac points are identically gapped. In both figures, dashed lines denote the locus of points of zero mass term, while ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ denote regions with opposite values for the mass term. The solid circles represent Dirac points of the initial gapless system; the corresponding chiralities are marked by the signs within.

Figure 2

Phase diagram of the system for $t_2=1$. Each region is denoted by the corresponding Chern number and represents an insulating topological phase. The boundaries of the regions represent topological transitions where the system becomes gapless.

Figure 3

Direct-space realization of the model (20); see Eq. (22). $(i,j)$ denotes lattice sites. ${\mathbf{a}}_{1/2}$ are Bravais lattice vectors. The filled and open circles represent orbitals with energy $m$ and $-m$, respectively. The vertical axis represents the on-site energy difference between the two inequivalent orbitals. Black lines represent $t_1$ hoppings, blue lines, $t_2$ hoppings, and red lines, $t_3$ hoppings. An arrow on a link indicates that an electron hopping in the corresponding direction gains a $\pi /2$ phase. Similarly a double line indicates a $\pi$ phase gain.

Figure 4

Energy dispersions for (a) $t_3=0.4$ ($\text{Ch}=-2$), (b) $t_3=0.85$ (topological phase transition at closing bulk gap), (c) $t_3=1.6$ ($\text{Ch}=-1$). The other parameters are $t_1=1$, $t_2=1$, $m=-1.4$. The simulation is done when the system is on a cylinder with a height of 40 sites and a circumference of 180 sites. The number of edge states is $2\times \left|\text{Ch}\right|$ because there are two edges. (d) Representation of chosen points on the phase diagram.

Figure 5

Energy spectrum as a function of momentum $k_x$ on a cylindrical geometry (height 40 sites and circumference 180 sites). Two edge states located near $j=1$ are represented in blue, and two at $j=40$ in green. The position and chirality of the edges is schematically represented in the inset.