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Journal de Physique II

“Fluctuating cluster” equations for a polymer to O(ϵ2). (Toward exact Flory equation?)

Z. Alexandrowicz

Published Mar 1, 1994• 3 citations
Publisher

Abstract

Although Flory's approximation describes very well the radius r of a polymer, it disagrees sharply with an exact ϵ expansion (derived from an analogy to n = 0 spins). Here the polymer is described explicitly as a critical cluster of n = 0 spins. While ordinary clusters resemble trees bifurcating into b branches, a cluster of n = 0 spins constitutes a single line, weighted by $b~phantom$ bifurcations. Analogous description of the polymer leads to a modified Flory-like equation, which describes r and b, jointly, and to two more equations for r and b in separate. These give very accurate exponents ν and γ, correct to O(ϵ2). In addition, the approach can be extended to a many polymer system, and to other critical clusters.

DOI
10.1051/jp2:1994143

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Although Flory's approximation describes very well the radius r of a polymer, it disagrees sharply with an exact ϵ expansion (derived from an analogy to n = 0 spins). Here the polymer is described explicitly as a critical cluster of n = 0 spins. While ordinary clusters resemble trees bifurcating into b branches, a cluster of n = 0 spins constitutes a single line, weighted by b phantomb~phantomb phantom bifurcations. Analogous description of the polymer leads to a modified Flory-like equation, which describes r and b, jointly, and to two more equations for r and b in separate. These give very accurate exponents ν and γ, correct to O(ϵ2). In addition, the approach can be extended to a many polymer system, and to other critical clusters.
A critical cluster at a temperature ${T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T}_{c}(1+\ensuremath{\epsilon})$ is described as a tree consisting of ${b}_{\ensuremath{\epsilon}}\ensuremath{\simeq}{\ensuremath{\epsilon}}^{\ensuremath{-}(1+\ensuremath{\Delta})/2}$ branches of length ${l}_{\ensuremath{\epsilon}}\ensuremath{\simeq}{\ensuremath{\epsilon}}^{\ensuremath{-}(1+\ensuremath{\chi})}$. Exponents \ensuremath{\Delta} and \ensuremath{\chi} represent loop avoidance, and an overlapping correlation in n-vector components. They, and the conventional exponents \ensuremath{\nu} and \ensuremath{\gamma}, are approximately related to n and to the dimension d by $\ensuremath{\Delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremath{\nu}\ensuremath{-}1$, $\ensuremath{\chi}/\ensuremath{\Delta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.5n/(1+0.5n)$, $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\gamma}(\ensuremath{\Delta},\ensuremath{\chi})$, and $2\ensuremath{\nu}\ensuremath{-}1\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(4\ensuremath{-}d)/(2d\ensuremath{-}2\ensuremath{\chi}/\ensuremath{\Delta})$. The results give insight into critical behavior and its relation to polymers.
I consider a connected self-avoiding polymer network made of identical long chains, with fixed topology. Using renormalization theory and conformal invariance, I conjecture in 2D, and give in $d=4\ensuremath{-}\ensuremath{\epsilon}$, to order $O(\ensuremath{\epsilon})$, the exact value of its critical exponent $\ensuremath{\gamma}$ as a function of the topological invariants. In 2D, the exact result fits with recent numerical data for three- and four-leg stars by Lipson et al.
It is well known that the properties of scaling and universality exhibited by the polymer system have a very natural description in terms of the critical behaviour of an analogous magnetic system. This analogy is used to derive renormalisation-group equations directly for the polymer system. Solving these renormalisation-group equations a compact parametric description is obtained of the crossover from the Wilson-Fisher to gaussian fixed point displayed by the mean square size of a polymer (R 2 ) and the second A 2 and third A 3 virial coefficients correct to O( epsilon 2 ) in d=4- epsilon spatial dimensions.
Strong disorder renormalization for the random transverse field Ising model leads to a complicated topology of surviving clusters as soon as d > 1. Even if one starts from a Cayley tree, the network of surviving renormalized clusters will contain loops, so that no analytical solution can be obtained. Here we introduce a modified procedure called 'boundary strong disorder renormalization' that preserves the tree structure, so that one can write simple recursions with respect to the number of generations. We first show that this modified procedure allows one to recover exactly most of the critical exponents for the one-dimensional chain. After this important check, we study the RG equations for the quantum Ising model on a Cayley tree with a uniform ferromagnetic coupling J and random transverse fields with support [ h min , h max ]. We find the following picture: (i) for J > h max , only bonds are decimated, so that the whole tree is a quantum ferromagnetic cluster; (ii) for J < h min , only sites are decimated, so that no quantum ferromagnetic cluster is formed, and the ferromagnetic coupling to the boundary coincides with the partition function of a directed polymer model in a random medium; (iii) for h min < J < h max , both sites and bonds can be decimated, and the quantum ferromagnetic clusters can either remain finite (the physics is then similar to (ii), with a quantitative mapping to a modified directed polymer model) or an infinite quantum ferromagnetic cluster appears. We find that the quantum transition can be of two types: (a) either the quantum transition takes place in the region where quantum ferromagnetic clusters remain finite, and the singularity of the ferromagnetic coupling to the boundary involves the typical correlation length exponent ν typ = 1; (b) or the quantum transition takes place at the point where an extensive quantum ferromagnetic cluster appears, with a correlation length exponent ν ≃ 0.75.