We propose exact expressions for the conformal anomaly and for three critical exponents of thetricritical O(n)loop model as a function of n in the range-2 ≤n≤3/2. These findings are based on ananalogy with known relations between Potts and O (n)models and on an exact solution of a "ritricritical" Potts model described in the literature. We verify the exact expressions for the tricritical O(n)model by means of a finite-size scaling analysis based on numerical transfer-matrix calculations.

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.

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 investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-sizescalinganalysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponentsis consistent with that of the exactly solved Ising model, and that the critical exponents and correlationlengthamplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectraare exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and thetemperature and magnetic exponents with numerical margins of 10211 or less. These results are in a perfectagreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degreeof precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic.The latter resource provided a substantial improvement of the analysis, despite the fact that it restrictedthe transfer-matrix calculations to finite sizes of at most 34 lattice units.