Helical Liquid and the Edge of Quantum Spin Hall Systems

The edge states of the recently proposed quantum spin Hall systems constitute a new symmetry class of one-dimensional liquids dubbed the “helical liquid,” where the spin orientation is determined by the direction of electron motion. We prove a no-go theorem which states that a helical liquid with an odd number of components cannot be constructed in a purely 1D lattice system. In a helical liquid with an odd number of components, a uniform gap in the ground state can appear when the time-reversal symmetry is spontaneously broken by interactions. On the other hand, a correlated two-particle backscattering term by an impurity can become relevant while keeping the time-reversal invariance.

Figure 1

The band structure in 1D lattice systems with time-reversal symmetry. Even numbers of Kramers doublets appear at a given energy except at the extremum points.

Figure 2

(a) The two-particle correlated backscattering process at $K<1/4$ divides the line into two half-lines. (b) The instanton process of $O_{1,2}(0,\tau )$ restores the TR symmetry.

Figure 3

(a) The RG flow of the Kondo problem in the spin Hall edge state. The antiferromagnetic FP (dashed line) describes the formation of the Kondo singlet. The ferromagnetic FP (solid dot) shows a nonzero critical value of $J_z$. (b) Kondo screening spin-current vortex around the local moment.