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Physical Review Letters

Fully packed loop model on the honeycomb lattice

Henk W. J. Blöte, Bernard Nienhuis

Published Feb 28, 1994• 93 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.72.1372

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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.
Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derived for the honeycomb lattice. Finite-size convergence is poor for most of the third branch, so that in this case the classification in terms of known models is uncertain. The authors' data show that the fourth branch can be interpreted in terms of a combination of a low-temperature O(n) and a critical Ising model. Finally, the numerical results for the O(n) model on the remaining branch agree accurately with the known results for the low-temperature O(n+1) model. They prove this equivalence by means of an exact mapping.
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 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.
We explore the phase diagram of an O (n) model on the honeycomb lattice with vacancies, using finite-size scaling and transfer-matrix methods. We make use of the loop representation of the O (n) model, so that n is not restricted to positive integers. For low activities of the vacancies, we observe critical points of the known universality class. At high activities the transition becomes first order. For n=0 the model includes an exactly known theta point, used to describe a collapsing polymer in two dimensions. When we vary n from 0 to 1, we observe a tricritical point which interpolates between the universality classes of the theta point and the Ising tricritical point.
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.