David R. Nelson (Harvard University): Polymer Winding Numbers and Quantum Mechanics [WPI-SKCM² Colloquia]

The winding of a single polymer in thermal equilibrium interacting with a repulsive cylindrical obstacle is perhaps the simplest example of statistical mechanics in a multiply connected geometry. As shown by S.F. Edwards, this problem is closely related to the quantum mechanics of a charged particle interacting with an Aharonov-Bohm flux. In another development, Pollock and Ceperley have shown that boson world lines in 2 + 1 dimensions with periodic boundary conditions, regarded as ring polymers on a torus, have a mean square winding number given by the superfluid density. Here, we review the mapping of the statistical mechanics of directed polymers onto quantum mechanics, using as an example vortex lines in high temperature superconductors in the presence of columnar defects. An external current passing through the a rod of columnar defects controls the braiding of flux lines in this case.