Entangled polymer complex as Higgs phenomena

We derive an effective Maxwell-London equation for entangled polymer complex under the topological constraint, borrowing the theoretical framework from the topological field theory. We find that the transverse current flux of the test polymer chain, surrounded with entangled chains, decays exponentially from its average position with finite penetration depth, which is analogous to the magnetic-field decay in a superconductor (SC). Like as the mass acquirement of photons in SC is the origin of the magnetic-field decay, the polymer earns uncrossible intersections along the chain due to the preserved linking number, which restricts the deviation of the transverse polymer current in the normal direction. Interestingly, this picture is well incorporated with the most successful phenomenological theory of the so called tube model, of which researchers have long pursued its microscopic origin. The correspondence of our equation of motion to the tube model claims that the confining tube potential is a consequence of the topological constraint (linking number). The tube radius is attributed to the decay length, and increasing the retracting force at intersections or increasing the number of intersections (linking number), the tube becomes narrow and tighter. It further shows that the probability of the tube leakage decays exponentially with the decay length of tube radius.