The phase diagram of the O (n) model, in particular the special case n=0, is studied by means of transfer-matrix calculations on the loop representation of the O (n) model. The model is defined on the square lattice; the loops are allowed to collide at the lattice vertices, but not to intersect. The loop model contains three variable parameters that determine the loop density or temperature, the energy of a bend in a loop, and the interaction energy of colliding loop segments. A finite-size analysis of the transfer-matrix results yields the phase diagram in a special plane of the parameter space. These results confirm the existence of a multicritical point and an Ising-like critical line in the low-temperature O (n) phase.
The phase diagram of the O (n) model, in particular the special case n=0, is studied by means of transfer-matrix calculations on the loop representation of the O (n) model. The model is defined on the square lattice; the loops are allowed to collide at the lattice vertices, but not to intersect. The loop model contains three variable parameters that determine the loop density or temperature, the energy of a bend in a loop, and the interaction energy of colliding loop segments. A finite-size analysis of the transfer-matrix results yields the phase diagram in a special plane of the parameter space. These results confirm the existence of a multicritical point and an Ising-like critical line in the low-temperature O (n) phase.
We explore the phase diagram of the O(n) loop model on the square lattice in the (x,n) plane, where x is the weight of a lattice edge covered by a loop. These results are based on transfer-matrix calculations and finite-size scaling. We express the correlation length associated with the staggered loop density in the transfer-matrix eigenvalues. The finite-size data for this correlation length, combined with the scaling formula, reveal the location of critical lines in the diagram. For n>>2 we find Ising-like phase transitions associated with the onset of a checkerboardlike ordering of the elementary loops, i.e., the smallest possible loops, with the size of an elementary face, which cover precisely one-half of the faces of the square lattice at the maximum loop density. In this respect, the ordered state resembles that of the hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of n represents a softening of its particle-particle potentials. We also determine critical points in the range -2≤n≤2. It is found that the topology of the phase diagram depends on the set of allowed vertices of the loop model. Depending on the choice of this set, the n>2 transition may continue into the dense phase of the n≤2 loop model, or continue as a line of n≤2 O(n) multicritical points.
We investigate the O(n) nonintersecting loop model on the square lattice under the constraint that the loops consist of 90-deg bends only. The model is governed by the loop weight n, a weight x for each vertex of the lattice visited once by a loop, and a weight z for each vertex visited twice by a loop. We explore the (x,z) phase diagram for some values of n. For 0<n<1, the diagram has the same topology as the generic O(n) phase diagram with n<2, with a first-order line when z starts to dominate and an O(n)-like transition when x starts to dominate. Both lines meet in an exactly solved higher critical point. For n>1, the O(n)-like transition line appears to be absent. Thus, for z=0, the (n,x) phase diagram displays a line of phase transitions for n≤1. The line ends at n=1 in an infinite-order transition. We determine the conformal anomaly and the critical exponents along this line. These results agree accurately with a recent proposal for the universal classification of this type of model, at least in most of the range -1≤n≤1. We also determine the exponent describing crossover to the generic O(n) universality class, by introducing topological defects associated with the introduction of "straight" vertices violating the 90-deg-bend rule. These results are obtained by means of transfer-matrix calculations and finite-size scaling.
We investigate the completely packed O(n) loop model on the square lattice, and its generalization to an Eulerian graph model, which follows by including cubic vertices which connect the four incoming loop segments. This model includes crossing bonds as well. Our study was inspired by existing exact solutions of the so-called coloring model due to Schultz and Perk [Phys. Rev. Lett. 46, 629 (1981)], which is shown to be equivalent with our generalized loop model. We explore the physical properties and the phase diagram of this model by means of transfer-matrix calculations and finite-size scaling. The exact results, which include seven one-dimensional branches in the parameter space of our generalized loop model, are compared to our numerical results. The results for the phase behavior also extend to parts of the parameter space beyond the exactly solved subspaces. One of the exactly solved branches describes the case of nonintersecting loops and was already known to correspond with the ordering transition of the Potts model. Another exactly solved branch, describing a model with nonintersecting loops and cubic vertices, corresponds with a first-order, Ising-like phase transition for n>2. For 1<n<2, this branch is interpreted in terms of a low-temperature O(n) phase with corner-cubic anisotropy. For n>2 this branch is the locus of a first-order phase boundary between a phase with a hard-square, lattice-gas-like ordering and a phase dominated by cubic vertices. A mean-field argument explains the first-order nature of this transition.
We investigate tricritical behavior of the O(n) model in two dimensions by means of transfer-matrix and finite-size scaling methods. For this purpose we consider an O(n) symmetric spin model on the honeycomb lattice with 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 degrees of freedom model on a non-intersecting-loop model, in which the number n of spin components assumes the role of a continuously 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 of the 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.
Abstract We study a model of dilute oriented loops on the square lattice, where each loop is compatible with a fixed, alternating orientation of the lattice edges. This implies that loop strands are not allowed to go straight at vertices, and results in an enhancement of the usual symmetry to . The corresponding transfer matrix acts on a number of representations (standard modules) that grows exponentially with the system size. We derive their dimension and those of the centralizer by both combinatorial and algebraic techniques. A mapping onto a field theory permits us to identify the conformal field theory governing the critical range, . We establish the phase diagram and the critical exponents of low-energy excitations. For generic n , there is a critical line in the universality class of the dilute model, terminating in an point. The case n = 1 maps onto the critical line of the six-vertex model, along which exponents vary continuously.
We obtain phase diagrams of a spin-3/2 Ising model on a square lattice by using a corner transfer matrix renormalization group method. We find that three phases appear in the phase diagrams; they are the ferromagnetic(FM) phase, the paramagnetic(PM) phase and a ferrimagnetic(FI) phase. It turns out that a tricritical point appears in a region of space spanned by parameters of coupling constants and the temperature. We estimate values of critical exponent ν and α at a transition temperature between the FM phase and the PM phase, at that between the FI phase and the FM phase, and also at that between the FI phase and the PM phase, by using the finite size scaling analysis.