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

Abstract

We derive the exact critical couplings (x * , y * a ), where y * a /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) loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent φ = 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 aregiven by h ℓ+1,3 in the Kac formula.61. 41.+e, 64.60.Cn, 64.60.Kw, 64.60.Fr Typeset using REVT E X 1 A long flexible polymer in a good solvent with an attractive short-range force between the polymer and the container wall is known to undergo an adsorption transition [1][2][3][4].A standard model for this phenomenon is a self-avoiding walk (SAW) on a d-dimensional lattice interacting with a (d − 1)-dimensional substrate. In the lattice model, the SAW has a Boltzmann weight x per monomer (in the bulk), with weight y per adsorbed monomer (on the substrate). At the adsorption transition, y * a , the number of adsorbed monomers scales with the total length L as L a ∼ L φ , where φ is a crossover exponent. The polymer is in the adsorbed phase for y > y * a and the desorbed phase for y < y * a , where the surface attractions are not effective. In the language of surface critical phenomena, the adsorption transition is a special transition [5]. where the sum is over all configurations of closed and nonintersecting loops. Here P is the total number of closed loops of fugacity n in a given configuration. In the limit n → 0 this reduces to the required SAW generating function, with x the fugacity of a step in the bulk and y the fugacity of a step along the surface. Here L is the length of a walk in the bulk and L s is the length of a walk along the surface of the strip.The partition function can be conveniently rewritten in terms of the Boltzmann weights of the empty vertices. To do this we need to distinguish between three classes of vertices: − and − which appear (i) in the bulk and (ii) on the surface, and (iii) and on the surface.For each class we define the weights t b , tb and t s , respectively. We then considerwhere N b , Nb and N s are the total numbers of vertices (either full or empty) of class (i),(ii) and (iii). Apart from harmless normalisation factors, the two partition functions are equivalent if t b = tb, along with the identificationwhere In particular, we found that the known diagonal reflection matrices for the Izergin-Korepin model [24] lead to two integrable sets of boundary weights which preserve the O(n) symmetry [22,25]. For each case t b = tb = 2 cos λ. Thus from (3) the critical bulk fugacity iswhich is the well known bulk critical value [16]. The SAW point occurs at λ = π/8, whereThe two integrable sets of boundary weights are [22] (A) t s = sin 2λ sin λ and (B) t s = cos 2λ cos λ .Thus from (3) case (A) gives the critical surface fugacity y * o = x...

Microcanonical simulations of adsorbing self-avoiding walks

“…This result is consistent with φ = 1 2 [2,4,9], and compares well with other estimates in the literature (for example, φ = 0.501 ± 0.014 in reference [23]). …”

“…Scaling of the generating and partition functions are found to be consistent with the exact values of critical exponents determined elsewhere in two dimensions [2,9], and with the value φ = 1 2 in three dimensions [19,23,31]. Similary, the data are also consistent with estimates for the surface exponent γ s ; this is seen particularly in the scaling of the partition function (see equation (16)).…”

“…However, it was not clear how to generalise the algorithm to sample states in models of interacting walks. In this paper our purpose is (1) to generalise the algorithm to a model of adsorbing walks, (2) to examine the behaviour of the algorithm by computing critical points and exponents of the walk, and to compare this to results found elsewhere, and (3) to use our data to examine scaling in adsorbing walks by computing critical exponents and examining the scaling of thermodynamic functions.…”

“…Physically, E(a) is the density of visits per unit length, and C(a) is the rate of change in E(a) as a function of changes in log a (it has a maximum at a + c ). For adsorbing walks it is thought that φ = 1 2 in all dimensions d ≥ 2 [2,9], and numerical evidence supporting this in dimensions lower than d = 4 (the upper critical dimension) are available in references [4,23,27,29,31]. If φ = 1 2 , then α = 0, so, for example, the specific heat has scaling C n (a) = h c (n φ (a−a + c )), and plotting measurements of C n (a) against the rescaled variable τ = n φ (a−a + c ) for small values of τ should collapse the curves to a limiting curve (with some finite size corrections to scaling), exposing the scaling function h c .…”

Exact solution of the Izergin-Korepin model with general non-diagonal boundary terms

“…The R-matrix of the model corresponds to the simplest twisted affine algebra A (2) 2 . The IK model with open boundary condition is related to the loop models [60] and self-avoiding walks at a boundary [61]. The IK model with U(1)-symmetry, i.e.…”