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arXiv:1301.6828 · Physical Review E

Ising-like transitions in the O(n) loop model on the square lattice

Zhe Fu, Wenan Guo, Henk W. J. Blöte

Published May 14, 2013• 9 citations
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Abstract

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.

DOI
10.1103/physreve.87.052118

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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.
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 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 present a transfer-matrix scaling study of a hard-square lattice-gas model with an exclusion shell extending up to second-nearest neighbors, and with a uniaxial finite repulsion active at third-nearest-neighbor separations. The phase diagram, as a function of temperature and chemical potential, shows three distinct regions bounded by two merging critical lines. At low temperature the system undergoes a transition to an ordered phase whose c(4\ifmmode\times\else\texttimes\fi{}2) structure is characterized---within this calculation---by the asymptotic degeneracy of the four largest eigenvalues of the transfer matrix. At high temperature this phase is reached from the disordered state via an intermediate, degenerate phase which, on average, exhibits long-range order in one dimension only. These two transition regimes are found to be separated by a multicritical point which is likely to arise from the competition between interaction anisotropy and thermal-disordering effects. Evidence of anisotropic critical behavior is found in the correlation lengths exponent ratios, ${\ensuremath{\nu}}_{?}$/${\ensuremath{\nu}}_{\ensuremath{\perp}}$, which appear to vary continuously along both transition lines. However, the accuracy of the predictions we make for the thermodynamics of the model is still limited by sensitive finite-size effects which are presumably emphasized by the relatively extended range of the interactions.
We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is known to undergo a phase transition to one out of four partially ordered phases. We find that this transition displays strong finite-size corrections to scaling and that the correlation functions deviate from isotropy to rather large distances. In contrast with an earlier study, we find that the critical temperature exponent of the transition is not Ising-like.
We present an investigation of the completely packed O(n) loop model on the square lattice by means of the transfer-matrix method and finite-size scaling. We investigate the model for a number of n values covering a wide range. This model is known to be equivalent with the q-state Potts model with q = n 2 , but here we also investigate the range n 2, the system is in an ordered phase, where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence. Furthermore, we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six-vertex model. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu’s 80th birthday.
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.