Soft Matter at the Thermal Scale, Ch Frank elasticity, defects and smectic layers
Problem
Someone argues: "Consider a point in a two-dimensional nematic film around which the director rotates by 180∘ (half of a full turn) on a loop enclosing the point. The director must return to its starting orientation after one full circuit, and a rotation of 180∘ does not return an arrow to where it started, so such a defect cannot exist; only whole-turn rotations, s an integer, are allowed." Find the error in this argument, and state which values of the disclination strength s are actually allowed and why.