Stability of the quantum spin Hall effect: Effects of interactions, disorder, and ${\mathbb{Z}}_2$ topology

The stability to interactions and disorder of the quantum spin Hall effect (QSHE) proposed for time-reversal-invariant two-dimensional systems is discussed. The QSHE requires an energy gap in the bulk and gapless edge modes that conduct spin-up and spin-down excitations in opposite directions. When the number of Kramers pairs of edge modes is odd, certain one-particle scattering processes are forbidden due to a topological ${\mathbb{Z}}_2$ index. We show that in a many-body description, there are other scattering processes that can localize the edge modes and destroy the QSHE: the region of stability for both classes of models (even or odd number of Kramers pairs) is obtained explicitly in the chiral boson theory. For a single Kramers pair, the QSHE is stable to weak interactions and disorder, while for two Kramers pairs it is not; however, the two-pair case can be stabilized by either finite attractive or repulsive interactions. For the simplest case of a single pair of edge modes, it is shown that changing the screening length in an edge with screened Coulomb interactions can be used to drive a phase transition between the QSHE state and the ordinary insulator.

Figure 1

An allowed two-particle backscattering process: (a) particle at momentum $p_1$ scatters to intermediate state $-p_2$; (b) particles at $\pm p_2$ interact and become intermediate states $\pm p_1$; (c) and (d), particle at intermediate state $p_1$ backscatters to state $-p_2$.

Figure 2

Regions of stability (white) in the $({\tau }_T^1,{\tau }_T^2)$ plane for values (a) ${\tau }_C=-0.5$; (b) ${\tau }_C=-0.15$; (c) ${\tau }_C=-0.075$; (d) ${\tau }_C=0.5$. Note that the region of stability (b) or of instability (c) may be disconnected and that this topological property is coordinate independent.