We consider the six-vertex model with anti-periodic boundary conditions across a finite strip. The row-to-row transfer matrix is diagonalised by the 'commuting transfer matrices' method. From the exact solution we obtain an independent derivation of the interfacial tension of the six-vertex model in the anti-ferroelectric phase. The nature of the corresponding integrable boundary condition on the XXZ spin chain is also discussed. Short Title: The anti-periodic six-vertex model ANU preprint MRR 012-95, hep-th/9502040

The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two-and three-state vertex models. These models include the six-vertex model, the 15-vertex A(1) 2 model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices do not coincide with those which lead to quantum group-invariant spin chains. The exact solution of the integrable loop models -including an O(n) model on the square lattice with open boundaries -is of relevance to the surface critical behaviour of two-dimensional polymers.Physics and Astronomy Classifications. 05.50.+q

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...

We derive the nested Bethe Ansatz solution of the fully packed 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 behaviour. In the n = 0 limit we obtain the exact compact exponents γ = 1 and ν = 1/2 for Hamiltonian walks, along with the exact value κ 2 = 3 √ 3/4 for the connective constant (entropy Typeset using REVT E X 1

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