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arXiv:cond-mat/9408083 · Physical Review Letters

Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice

Murray T. Batchelor, Junji Suzuki, C M Yung

Published Nov 14, 1994• 53 citations

Abstract

We derive the nested Bethe ansatz solution of the fully packed $\mathrm{O}(n)$ 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 $n=0$ 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.

DOI: 10.1103/physrevlett.73.2646

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We derive the nested Bethe ansatz solution of the fully packed

\mathrm{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=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.

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.

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.

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.

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.

We derive the exact critical couplings

({x}^{*}{,y}_{a}^{*})

, where

{y}_{a}^{*}{/x}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\sqrt{1+\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.533\dots{}

, for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe ansatz solution of the

O\left(n\right)

loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by ${h}_{\ensuremath{\ell}+1,3}$ in the Kac formula.