Polymer winding numbers and quantum mechanics

  • David R. Nelson
  • Ady Stern
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 492)


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 ħ 2/mk B T, 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 \(\left\langle {W^2 } \right\rangle = \frac{{n_s \hbar ^2 }}{{2\pi mk_B T}}\ln (R/a)\). 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.


Periodic Boundary Condition Vortex Line Flux Line Imaginary Time Single Polymer 
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Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • David R. Nelson
    • 1
  • Ady Stern
    • 2
  1. 1.Lyman Laboratory of PhysicsHarvard UniversityCambridge
  2. 2.Department of Condensed Matter PhysicsWeizmann Institute of SciencesRehovotIsrael

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