A new strategy to microscopic modeling of topological entanglement in polymers based on field theory

Abstract

In this work a new strategy is proposed in order to build analytic and microscopic models of fluctuating polymer rings subjected to topological constraints. The topological invariants used to fix these constraints belong to a wide class of the so-called numerical topological invariants. For each invariant it is possible to derive a field theory that describes the statistical behavior of knotted and linked polymer rings following a straightforward algorithm. The treatment is not limited to the partition function of the system, but it allows also to express the expectation values of general observables as field theory amplitudes.Our strategy is illustrated taking as examples the Gauss linking number and a topological invariant belonging to a class of invariants due to Massey. The consistency of the new method developed here is checked by reproducing a previous field theoretical model of two linked polymer rings. After the passage to field theory, the original topological constraints imposed on the fluctuating paths of the polymers become constraints over the configurations of the topological fields that mediate the interactions of topological origin between the monomers. These constraints involve quantities like the crosshelicity which are of interest in other disciplines, like for instance in modeling the solar magnetic field.While the calculation of the expectation values of generic observables remains still challenging due to the complexity of the problem of topological entanglement in polymer systems, we succeed here to reduce the evaluation of the moments of the Gauss linking number for two linked polymer rings to

Size of a 2D ring polymer topologically unentangled with a planar array of obstacles

“…Size of a 2D ring polymer topologically unentangled with a planar array of obstacles Hamiltonian (see, e.g., [11,12,14,17] for related work):…”

“…-We firstly proceed to provide below a concise derivation of the theory to be used. The interested reader is invited to consult [11,12,[14][15][16][17] for more general, background, information.…”

“…where in the second line, and for calculational convenience and tractability, we pass to the continuum approximation (with a short-distance cutoff a corresponding to the lattice spacing of obstacles). The required Boltzmann factor, e −HC , (in the continuum approximation) can be simply unpacked via the introduction of a vector gauge field, A α (x), as follows (for the analogous 3D case see [14][15][16][17]):…”

“…which includes the usual gauge fixing term, controlled typically by the parameter η → 0 [14][15][16][17].…”

“…We are now in a position to form the requisite Green function, G(x, x ; L), as a path integral [14][15][16][17]:…”