We investigate tricritical behavior of the O(n) model in two dimensions by means of transfer-matrix andfinite-size scaling methods. For this purpose we consider an O(n) symmetric spin model on the honeycomb latticewith vacancies; the tricritical behavior is associated with the percolation threshold of the vacancies. The vacancies are represented by face variables on the elementary hexagons of the lattice. We apply a mapping of the spin degreesof freedom model on a non-intersecting-loop model, in which the number n of spin components assumes the role of acontinuously variable parameter. This loop model serves as a suitable basis for the construction of the transfer matrix.Our results reveal the existence of a tricritical line, parametrized by n, which connects the known universality classes ofthe tricritical Ising model and the theta point describing the collapse of a polymer. On the other side of the Ising point,the tricritical line extends to the n = 2 point describing a tricritical O(2) model.

We determine the phase diagram of the O(n) loop model on the honeycomb lattice, in particular, in therange n＞2, by means of a transfer-matrix method. We find that, contrary to the prevailing expectation,there is a line of critical points in the range between n＝2 and ∞. This phase transition, which belongsto the three-state Potts universality class, is unphysical in terms of the O (n) spin model, but falls insidethe physical region of the n-component corner-cubic model. It can also be interpreted in terms of theordering of a system of soft particles with hexagonal symmetry.

We investigate the two-dimensional classical Heisenberg model with a nonlinear nearest-neighbor interaction V(s,s')＝2K [(1＋s s')2]p. The analogous nonlinear interaction for the XY model was introduced by Domany, Schick, and Swendsen, who find that for large p the Kosterlitz-Thouless transition is preempted by a first-order transition. Here we show that, whereas the standard (p＝ 1) Heisenberg model has no phase transition, for large enough pa first-order transition appears. Both phases have only short-range order, but with a correlation length that jumps at the transition.

We test the performance of the Monte Carlo renormalization method in the context of the Ising model on atriangular lattice. We apply a block-spin transformation which allows for an adjustable parameter so that the transformation can be optimized. This optimization purportedly brings the fixed point of the transformation to a location where the corrections to scaling vanish. To this purpose we determine corrections to scaling of the triangular Ising model with nearest- and next-nearest-neighbor interactions by means of transfer-matrix calculationsand finite-size scaling. We find that the leading correction to scaling just vanishes for the nearestneighbormodel. However, the fixed point of the commonly used majority-rule block-spin transformation appears to lie well away from the nearest-neighbor critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because the standard assumptions of real-space renormalizationimply that corrections to scaling vanish at the fixed point. We avoid this inconsistency by means of theoptimized transformation which shifts the fixed point back to the vicinity of the nearest-neighbor criticalHamiltonian. The results of the optimized transformation in terms of the Ising critical exponents are moreaccurate than those obtained with the majority rule.

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 thevacancies, we observe critical points of the known universality class.At high activities the transition becomes first order. For n=0 themodel includes an exactly known theta point, used to describe a collapsingpolymer in two dimensions. When we vary n from 0 to 1, weobserve a tricritical point which interpolates between the universalityclasses of the theta point and the Ising tricritical point.