Tuning phase transition between quantum spin Hall and ordinary insulating phases

Abstract

An effective theory is constructed for analyzing a generic phase transition between the quantum spin Hall and the insulator phases. Occurrence of degeneracies due to closing of the gap at the transition are carefully elucidated. For systems without inversion symmetry the gap closing occurs at $\ifmmode\pm\else\textpm\fi{}{\stackrel{P\vec}{k}}_{0}(\ensuremath{\ne}\stackrel{P\vec}{G}∕2)$ while for systems with inversion symmetry, the gap can close only at wave numbers $\stackrel{P\vec}{k}=\stackrel{P\vec}{G}∕2$, where $\stackrel{P\vec}{G}$ is a reciprocal lattice vector. In both cases, following a unitary transformation which mixes spins, the system is represented by two decoupled effective theories of massive two-component fermions having masses of opposite signs. Existence of gapless helical modes at a domain wall between the two phases directly follows from this formalism. This theory provides an elementary and comprehensive phenomenology of the quantum spin Hall system.