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We derive the spin texture of a weak topological insulator via asupersymmetric approach that includes the roles of the bulk gap edge states andsurface band bending. We find the spin texture can take one of four forms: (i)helical, (ii) hyperbolic, (iii) hedgehog, with spins normal to the Dirac-Weylcone of the surface state, and (iv) hyperbolic hedgehog. Band bendingdetermines the winding number in the case of a helical texture, and for alltextures can be used to tune the spin texture polarization to zero. For theweak topological insulator SnTe, we show that inclusion of band bending iscrucial to obtain the correct texture winding number for the (111) surfacefacet $\Gamma$-point Dirac-Weyl cone. We argue that hedgehogs will be foundonly in low symmetry situations.

Spin texture in weak topological insulators: the role of bulk states and band bending