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arXiv:cond-mat/9403095 · Journal of Physics A Mathematical and General

Exact theory of polymer adsorption in analogy with the Kondo problem

Paul Fendley, Hubert Saleur

Published Nov 7, 1994• 13 citations
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Abstract

We conjecture the exact scaling theory for the adsorption of two-dimensional polymers by using boundary S matrices. We compute the boundary free energy (the 'g-function'), study the flow from the adsorbed to the desorbed phase and derive the crossover exponent and all the geometrical exponents at the transition. More generally, we solve the special transition in the O(n) model, the polymer case corresponding to n=0. The n=2 limit appears identical to the ordinary Kondo problem.

DOI
10.1088/0305-4470/27/21/001

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We derive the exact critical couplings (x∗,ya∗)({x}^{*}{,y}_{a}^{*})(x∗,ya∗​), where ya∗/x∗=1+2=1.533…{y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}ya∗​/x∗=1+2​​=1.533… , for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the O(n)O\left(n\right)O(n) loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by ${h}_{\ensuremath{\ell}+1,3}$ in the Kac formula.
The possibility of an exact analytical description of first-order and second-order transitions is demonstrated using a specific microscopic model. Predictions using the exactly calculated partition function are compared with those based on the Landau and Yang–Lee approaches. The model employed is an adsorbed polymer chain with an arbitrary number of links and an external force applied to its end, for which the variation of the partition function with the adsorption interaction parameter and the magnitude of the applied force is calculated. In the thermodynamic limit, the system has one isotropic and two anisotropic, ordered phases, each of which is characterized by two order parameters and between which first-order and second-order transitions occur and a bicritical point exists. The Landau free energy is found exactly as a function of each order parameter separately and, near the bicritical point, as a function of both of them simultaneously. An exact analytical formula is found for the distribution of the complex zeros of the partition function in first-order and second-order phase transitions. Hypotheses concerning the way in which the free energy and the positions of the complex zeros scale with the number of particles N in the system are verified.
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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.