Universal phase diagrams for the quantum spin Hall systems

We describe how the three-dimensional (3D) quantum spin Hall phase arises from the insulator phase by changing an external parameter. In 3D systems without inversion symmetry, a gapless phase should appear between the two phases with a bulk gap. The gapless points are monopoles and antimonopoles (in $\mathbf{k}$ space), whose topological nature is the source of this gapless phase. In general, when the external parameter is changed from the ordinary insulator phase, two monopole-antimonopole pairs are created and the system becomes gapless. The gap-closing points (monopoles and antimonopoles) then move in the $\mathbf{k}$ space as the parameter is changed further. They eventually annihilate in pairs, with changing partners from the pair creations, and the system opens a gap again entering into the quantum spin Hall phase.

Figure 1

(Color online) Phase diagram for the QSH and insulator (I) phases in (a) 3D and (b) 2D. $m$ is a parameter driving the phase transition and $\delta$ represents breaking of inversion symmetry. $\delta =0$ corresponds to the inversion-symmetric system.

Figure 2

(Color online) Trajectory for the gapless points in the $\mathbf{k}$ space. At $m=m_0$ a monopole-antimonopole pair is created and they run to the opposite directions when $m$ is changed. We assume ${\left(Q^{-1}\mathbf{N}\right)}_1<0$, in which gapless points exist only for $\Delta m\ge 0$.

Figure 3

(Color online) Trajectory of the gapless points for (a) inversion-asymmetric and (b) symmetric systems. For (b) inversion-symmetric systems, the gapless point is located at $\mathbf{k}={\Gamma }_i$ and is an isolated point in the $m\text{−}\mathbf{k}$ space. Only at $m=m_0$ the system is gapless. For (a) inversion-asymmetric systems, on the other hand, the gapless points are created in monopole-antimonopole pairs at $m=m_1$ and move in $\mathbf{k}$ space as $m$ is varied. The system opens a gap only by pair annihilation of these gapless points at $m=m_2$.

Figure 4

(Color online) Trajectory of the gapless points for inversion-asymmetric systems, but without the phase transition.

Figure 5

(Color online) Phase diagrams for the Fu-Kane-Mele model with $\delta t_3=0$ and $\delta t_4=0$. $t_1$ and $t_2$ are the bonds along the [111] and [$1\overline{1}\overline{1}$] directions. We put ${\lambda }_{\text{SO}}=0.1t$. The axes are in the unit of $t$. (a) The phase diagram in $\delta t_1-\delta t_2$ plane obtained in Ref. 12. ${\lambda }_v$ is set as zero. Each phase is indexed by cubic Miller indices following Ref. 12. (b) The phase diagram in the $\delta t_{+}-{\lambda }_v$ plane. ${\lambda }_v$ is newly introduced into the Fu-Kane-Mele model. Here $\delta t_{+}=\delta t_1+\delta t_2$, while we fix $\delta t_{-}=\delta t_1-\delta t_2=0.1t$. The arrows (red) in (a) and (b) correspond to the identical change in parameters.

Figure 6

(Color online) (a) Trajectory of the gapless points in $\mathbf{k}$ space as we change the parameter $\delta t_{+}$ with ${\lambda }_v$ fixed. The wave number $\mathbf{k}$ is shown in the unit of $\left(2\pi /a\right)$. The solid and broken curves are the trajectories for the monopoles and antimonopoles, respectively. This corresponds to the arrow in the phase diagram in (b).

Figure 7

(Color online) (a) Dispersion around the gapless points when the monopole-antimonopole pairs annihilate for the Fu-Kane-Mele model. The direction $\mathbf{L}$ is along the trajectory of the monopole and the antimonopole, while ${\mathbf{N}}_1$ and ${\mathbf{N}}_2$ are perpendicular to $\mathbf{L}$. These directions are shown in (b) in the trajectory for the gapless point in the $\mathbf{k}$ space. Energy is shown in the unit $t$ and the wave number is in the unit $2\pi /a$.

Figure 8

The string which is a set of gapless points in the $\left(m,\mathbf{k}\right)$ space. The arrow describes the direction of the flow vector $\rho$.