Spin-Charge Separated Solitons in a Topological Band Insulator

In this Letter we construct a simple, controllable, two-dimensional model based on a topological band insulator. It has many attractive properties. (1) We obtain spin-charge separated solitons that are associated with dynamic $\pi$ fluxes. (2) These solitons obey Bose statistics and their condensation triggers a phase transition from a spin Hall insulator to an easy-plane ferromagnet. (3) It suggests an alternative way to classify the $Z_2$ topological band insulator without resorting to the sample boundary.

Figure 1

Four different types of fluxons. The notation, e.g., $f_{+(1/2)\uparrow ,-(1/2)\downarrow }$, means the up-spin zero mode is filled while down-spin zero mode is empty.

Figure 2

The charge (a) and spin (b) densities of a pair of fluxons on a $24\times 24$ lattice with periodic boundary condition. The fluxon at coordinate $(6,6)$ is $f_{+(1/2)\uparrow ,+(1/2)\downarrow }$. It has charge 1 and $S_z=0$. The fluxon at $(18,18)$ is $f_{+(1/2)\uparrow ,-(1/2)\downarrow }$, it has charge 0 and $S_z=1/2$. Note that the average charge density of the ground state is 2. The spin-charge separation in real space is manifest.

Figure 3

The result for $\theta (f_{1\uparrow },f_{2\uparrow })$ on a $4d\times 4d$ square lattice with periodic boundary condition. We fix the position of $f_{2\uparrow }$ and let $f_{1\uparrow }$ go around a loop of a $2d\times 2d$ square path. The fermionic band parameters used in the computation are $\mu =0.3$, $\Delta =0.5$, $t=1$. (a) Identical fluxons ${\theta }_{id}\equiv \theta (f_{(1/2)\uparrow },f_{(1/2)\uparrow })=\theta (f_{-(1/2)\uparrow },f_{-(1/2)\uparrow })$, and (b) distinguishable fluxons ${\theta }_{op}\equiv \theta (f_{-(1/2)\uparrow },f_{(1/2)\uparrow })=\theta (f_{(1/2)\uparrow },f_{-(1/2)\uparrow })$. Note that due to the particle-hole symmetry $\theta (f_{(1/2)\uparrow },f_{(1/2)\uparrow })=\theta (f_{-(1/2)\uparrow },f_{-(1/2)\uparrow })$ and $\theta (f_{-(1/2)\uparrow },f_{(1/2)\uparrow })=\theta (f_{(1/2)\uparrow },f_{-(1/2)\uparrow })$. The extrapolation to $d\to \infty$ gives $-{\theta }_{id}={\theta }_{op}=\pi /4$.