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arXiv:cond-mat/9410082 · Physical Review Letters

Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions

Murray T. Batchelor, C M Yung

Published Mar 1, 1995• 79 citations
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Abstract

We derive the exact critical couplings $({x}^{*}{,y}_{a}^{*})$, where ${y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}$ , for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the $O\left(n\right)$ 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.

DOI
10.1103/physrevlett.74.2026

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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 \ensuremathφ=12\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}\ensuremathφ=21​, 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\ensuremathℓ+1,3{h}_{\ensuremath{\ell}+1,3}h\ensuremathℓ+1,3​ in the Kac formula.
The O(n) loop model on the honeycomb lattice with mixed ordinary and special boundary conditions is solved exactly by means of the Bethe ansatz. The calculation of the dominant finite-size corrections to the eigenspectrum yields the mixed boundary scaling index and the geometric scaling dimensions describing the universal surface critical behaviour. Exact results follow in the limit n=0 for the polymer adsorption transition with a mixed adsorbing and free boundary. These include the new configurational exponent gamma l =85/64.
We derive the nested Bethe ansatz solution of the fully packed O(n)\mathrm{O}(n)O(n) loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behavior. In the n=0n=0n=0 limit we obtain the exact compact exponents $\ensuremath{\gamma}=1$ and $\ensuremath{\nu}=\frac{1}{2}$ for Hamiltonian walks, along with the exact value ${\ensuremath{\kappa}}^{2}=3\sqrt{\frac{3}{4}}$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.
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.
A mapping is developed between the quantum Hall plateau transition and two-dimensional self-interacting lattice polymers. This mapping is exact in the classical percolation limit of the plateau transition, and diffusive behavior at the critical energy is shown to be related to the critical exponents of a class of chiral polymers at the \ensuremath{\theta} point. The exact critical exponents of the chiral polymer model on the honeycomb lattice are found, verifying that this model is in the same universality class as a previously solved model of polymers on the Manhattan lattice. The mapping is obtained by averaging analytically over the local random potentials in a previously studied lattice model for the classical plateau transition. This average generates a weight on chiral polymers associated with the classical localization length exponent $\ensuremath{\nu}=\frac{4}{3}.$ We discuss the differences between the classical and quantum transitions in the context of polymer models and use numerical results on higher-moment scaling laws at the quantum transition to constrain possible polymer descriptions. Some properties of the polymer models are verified by transfer matrix and Monte Carlo studies.