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 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 develop cluster algorithms for a broad class of loop models on two-dimensional lattices, including several standard O (n) loop models at n≥1. We show that our algorithm has little or no critical slowingdown when 1≤n≤2. We use this algorithm to investigate the honeycomb-lattice O(n) loop model, for which we determine several new critical exponents, and a square-lattice O(n) loop model, for which we obtain new information on the phase diagram.

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.

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.