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Polymer Winding Numbers and Quantum Mechanics

The winding of a single polymer in thermal equilibrium around 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 a 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 $<W^2> = 2n_s\hbar^2/mk_BT$, where $m$ is the boson mass and $n_s$ is the superfluid number density. Here, we review the mapping of the statistical mechanics of polymers with constraints onto quantum mechanics, and show that there is an interesting generalization of the Pollock-Ceperley result to directed polymer melts interacting with a repulsive rod of radius $a$. When translated into boson language, the mean square winding number around the rod for a system of size $R$ perpendicular to the rod reads $<W^2> = {n_s\hbar^2\over 2πmk_BT}\ln(R/a)$. This result is directly applicable to vortices in Type II superconductors in the presence of columnar defects. An external current passing through the rod couples directly to the winding number in this case.