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

We derive the nested Bethe ansatz solution of the fully packed $\mathrm{O}\mathrm{}\left(n\right)\mathrm{}$ 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 $n=0\mathrm{}$ limit we obtain the exact compact exponents $\gamma =1\mathrm{}$ and $\nu =\frac{1}{2}$ for Hamiltonian walks, along with the exact value ${\kappa }^2=3\mathrm{}\sqrt{\frac{3}{4}}$ 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.