Topological Insulators Quantum Mechanics

Topological insulators in three dimensions are studied as a problem of supersymmetric quantum mechanics. The spin-orbit coupling is induced as a consequence of the supersymmetrization procedure and we show that it is equivalent to the appearance of a $SU(2)$ connection. The procedure presented in this letter is general and valid for any three-dimensional quantum system. The approach allows – in principle – to study a wide range of topological insulators as standard quantum mechanical problems. As an illustration the three-dimensional harmonic oscillator and the Aharonov-Bohm effect are studied in detail.