A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employa number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome latticetothe exactly solvable critical q-state Potts model on the honeycomb lattice with q= (n+1) 2. The intermediatesteps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loopweight n'=q and a dilute loop model with loop weight n, both on the kagome lattice. This mapping enablesthe determination ofa branch of critical points of the dilute O（n）model, as well as some of its criticalproperties. These properties differ from those of the generic O（n）critical points. For n=0, our model reproducesthe known universal properties of the point describing the collapse of a polymer. For n≠0 it displaysa line of multicritical points, with the same universal behavior as a branch of critical points that was foundearlier in a dilute O（n）model on the square lattice. These findings are supported by a finite-size-scalinganalysis in combination with transfer-matrix calculations.

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 consider a special case of the n-component cubic model on the square lattice, for which an expansion exists in Ising-type graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determinethe critical points for several values of n. Furthermore we determine several universal quantities, includingthree critical exponents. For n﹤2, these results agree well with the theoretical predictions for the criticalO (n) branch. This model is also a special case of the (Nα, Nβ) model of Domany and Riedel. It appears that theself-dual plane of the latter model contains the exactly known critical points of the n=1 and 2 cubic models.For this reason we have checked whether this is also the case for 1﹤n﹤2. However, this possibility isexcluded by our numerical results.

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 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 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.