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arXiv:1011.0302 · Physical Review E

Crossover phenomena involving the dense O( n ) phase

Wenan Guo, Henk W. J. Blöte

Published Feb 28, 2011• 7 citations
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Abstract

We explore the properties of the low-temperature phase of the O(n) loop model in two dimensions by means of transfer-matrix calculations and finite-size scaling. We determine the stability of this phase with respect to several kinds of perturbations, including cubic anisotropy, attraction between loop segments, double bonds, and crossing bonds. In line with Coulomb gas predictions, cubic anisotropy and crossing bonds are found to be relevant and introduce crossover to different types of behavior. Whereas perturbations in the form of loop-loop attractions and double bonds are irrelevant, sufficiently strong perturbations of these types induce a phase transition of the Ising type, at least in the cases investigated. This Ising transition leaves the underlying universal low-temperature O(n) behavior unaffected.

DOI
10.1103/physreve.83.021115

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We explore the properties of the low-temperature phase of the O(n) loop model in two dimensions by means of transfer-matrix calculations and finite-size scaling. We determine the stability of this phase with respect to several kinds of perturbations, including cubic anisotropy, attraction between loop segments, double bonds, and crossing bonds. In line with Coulomb gas predictions, cubic anisotropy and crossing bonds are found to be relevant and introduce crossover to different types of behavior. Whereas perturbations in the form of loop-loop attractions and double bonds are irrelevant, sufficiently strong perturbations of these types induce a phase transition of the Ising type, at least in the cases investigated. This Ising transition leaves the underlying universal low-temperature O(n) behavior unaffected.
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.
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 study the O(2) model with Z_{4}-symmetric perturbations within the framework of the nonperturbative renormalization group (RG) for spatial dimensionality d=2 and 3. In a unified framework we resolve the relatively complex crossover behavior emergent due to the presence of multiple RG fixed points. In d=3 the system is controlled by the XY, Ising, and low-T fixed points in the presence of a dangerously irrelevant anisotropy coupling λ. In d=2 the anisotropy coupling is marginal and the physical picture is governed by the interplay between two distinct lines of RG fixed points, giving rise to nonuniversal critical behavior, and an isolated Ising fixed point. In addition to inducing crossover behavior in universal properties, the presence of the Ising fixed point yields a generic, abrupt change of critical temperature at a specific value of the anisotropy field.
We use Monte Carlo, transfer-matrix and finite-size scaling methods to investigate two-dimensional O (n) models with n > 2, in particular the case n = 3 which includes the classical Heisenberg model. Depending on the type of interaction and the lattice structure, two different types of phase transitions are present. One type resembles the hard-hexagon transition and occurs in the loop representation of the honeycomb O (n) model. The other type is a first-order transition which occurs for spin-spin interactions that are strongly nonlinear in the neighbor-spin products. When the nonlinearity is decreased, the first-order line ends in a critical point. The existence of the first-order line is in agreement with mean-field theory as well as with high- and low-temperature approximations.
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.