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Journal of Physics A Mathematical and General

Surface critical behaviour of an O(n) loop model related to two Manhattan lattice walk problems

Murray T. Batchelor, Aleksander L. Owczarek, Katherine Seaton, C M Yung

Published Feb 21, 1995• 9 citations
Publisher

Abstract

We find and discuss the scaling dimensions of the branch 0 manifold of the Nienhuis O(n) loop model on the square lattice, concentrating on the surface dimensions. The results are extracted from a Bethe ansatz calculation of the finite-size corrections to the eigenspectrum of the six-vertex model with free boundary conditions. These results are especially interesting for polymer physics at two values of the crossing parameter lambda . Interacting self-avoiding walks on the Manhattan lattice at the collapse temperature ( lambda = pi /3) and Hamiltonian walks on the Manhattan lattice ( lambda = pi /2) are discussed in detail. Our calculations illustrate the importance of examining both odd and even strip widths when performing finite-size correction calculations to obtain scaling dimensions.

DOI
10.1088/0305-4470/28/4/011

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We find and discuss the scaling dimensions of the branch 0 manifold of the Nienhuis O(n) loop model on the square lattice, concentrating on the surface dimensions. The results are extracted from a Bethe ansatz calculation of the finite-size corrections to the eigenspectrum of the six-vertex model with free boundary conditions. These results are especially interesting for polymer physics at two values of the crossing parameter lambda . Interacting self-avoiding walks on the Manhattan lattice at the collapse temperature ( lambda = pi /3) and Hamiltonian walks on the Manhattan lattice ( lambda = pi /2) are discussed in detail. Our calculations illustrate the importance of examining both odd and even strip widths when performing finite-size correction calculations to obtain scaling dimensions.
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.
The phase diagram and surface critical behaviour of the vertex-interacting self-avoiding walk are examined using transfer matrix methods extended using DMRG and coupled with finite-size scaling. Particular attention is paid to the critical exponents at the ordinary and special points along the collapse transition line. The question of the bulk exponents (ννν and γγγ) is addressed, and the results found are at variance with previously conjectured exact values.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.
The Bethe ansatz equations are derived for the six-vertex model with general boundary weights on a lattice in a diagonal orientation. These are solved in the thermodynamic limit. Finite-size corrections to the free energy (for a restricted class of boundary weights, including those corresponding to a Potts model with free boundaries) are calculated in the critical region. The first-order term gives the surface free energy of the model. The second-order term is found to be - pi okT tan( pi nu /2 mu )c/48N' 2 where c=(1-6 mu 2 /( pi 2 - pi mu )) is the conformal anomaly. This can be compared to - pi kT sin( pi nu / mu )c/6N' 2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions. This difference can be explained geometrically using conformal invariance.
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.
Exact results are obtained for a square lattice fifteen-vertex model and related O(n) model recently proposed by Nienhuis (1990). The partition function per site and finite-size corrections are obtained, along with the critical exponents via the Temperley-Lieb equivalent six-vertex model. The exact exponents nu = 4 / 7 and gamma = 6 / 7 follow in the limit n=0 for even system sizes. Odd system sizes are also discussed. The fifteen-vertex model is also the vertex formulation of the anisotropic, or q-deformed, spin-1 biquadratic model.