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arXiv:cond-mat/9903132 · Journal of Physics A Mathematical and General

On the universality of fully packed loop models

Jesper Lykke Jacobsen

Published Jan 1, 1999• 18 citations
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Abstract

Fully packed loop (FPL) models on the square and the honeycomb lattice constitute new classes of critical behaviour, distinct from those of the low-temperature O( n ) model. A simple symmetry argument suggests that such compact phases are only possible when the underlying lattice is bipartite. Motivated by the hope of identifying further compact universality classes we therefore study the FPL model on the square-octagon lattice. Surprisingly, this model is only critical for loop weights n <1.88, and its scaling limit coincides with the dense phase of the O( n ) model. For n = 2 it is exactly equivalent to the self-dual 9-state Potts model. These analytical predictions are confirmed by numerical transfer matrix results. Our conclusions extend to a large class of bipartite decorated lattices.

DOI
10.1088/0305-4470/32/29/305

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Fully packed loop (FPL) models on the square and the honeycomb lattice constitute new classes of critical behaviour, distinct from those of the low-temperature O( n ) model. A simple symmetry argument suggests that such compact phases are only possible when the underlying lattice is bipartite. Motivated by the hope of identifying further compact universality classes we therefore study the FPL model on the square-octagon lattice. Surprisingly, this model is only critical for loop weights n <1.88, and its scaling limit coincides with the dense phase of the O( n ) model. For n = 2 it is exactly equivalent to the self-dual 9-state Potts model. These analytical predictions are confirmed by numerical transfer matrix results. Our conclusions extend to a large class of bipartite decorated lattices.
We investigate the O(n) model on the honeycomb lattice, using its loop representation in the limit of full packing. The universal properties, which we calculate by means of finite-size scaling and transfer-matrix techniques, are different from the branches of O(n) critical behavior known thus far. The conformal anomaly of the model varies between -1 and 2 in the interval 0\ensuremath{\le}n\ensuremath{\le}2. The universality class of the model is characterized as a superposition of a low-temperature O(n) phase, and a solid-on-solid model at a temperature independent of n.
We investigate the critical behaviour of the fully packed O(n) loop model on the square lattice in which each vertex is visited once by a loop. A transfer-matrix analysis shows that this model can be interpreted as a superposition of a low-temperature O(n) model and an solid-on-solid (SOS) model, as for the fully packed model on the honeycomb lattice. However, not all of the critical exponents are the same for both lattices. In contrast, the fully packed model on the triangular lattice appears to behave as a pure low-temperature O(n) model.
A relation between the average length of loops and their free energy is obtained for a variety of O( n )-type models on two-dimensional lattices, by extending to finite temperatures a calculation due to Kast. We show that the (number) averaged loop length stays finite for all non-zero fugacities n , and in particular it does not diverge upon entering the critical regime ( n 2 + ). Fully packed loop (FPL) models with n = 2 seem to obey the simple relation = 3 L min , where L min is the smallest loop length allowed by the underlying lattice. We demonstrate this analytically for the FPL model on the honeycomb lattice and for the 4-state Potts model on the square lattice, and based on numerical estimates obtained from a transfer matrix method we conjecture that this is also true for the two-flavour FPL model on the square lattice. We present, in addition, numerical results for the average loop length on the three critical branches (compact, dense and dilute) of the O( n ) model on the honeycomb lattice, and discuss the limit n 0. Contact is made with the predictions for the distribution of loop lengths obtained by conformal invariance methods.
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.
A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome lattice to the exactly solvable critical q-state Potts model on the honeycomb lattice with q=(n+1)2. The intermediate steps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loop weight n'=sqrt(q) and a dilute loop model with loop weight n , both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O(n) model, as well as some of its critical properties. These properties differ from those of the generic O(n) critical points. For n=0, our model reproduces the known universal properties of the theta point describing the collapse of a polymer. For n not equal 0 it displays a line of multicritical points, with the same universal behavior as a branch of critical points that was found earlier in a dilute O(n) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.
Starting from the Bethe ansatz solution we derive a set of coupled non-linear integral equations for the fully packed double loop model (FPL^2) on the square lattice. As an application we find exact expressions for the central charge and for the scaling dimension corresponding to the simplest charge excitation. We study numerically the low-lying excitations corresponding to more general perturbations of the ground state and discover that the corresponding scaling dimensions are well described by the Cartan matrix of sl_4.