arXiv:cond-mat/9812270 · International Journal of Modern Physics C

FIRST AND SECOND ORDER TRANSITIONS IN DILUTE O (n) MODELS

Wenan Guo, Henk W. J. Blöte, Bernard Nienhuis

Published Feb 1, 1999• 5 citations

Abstract

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 the vacancies, we observe critical points of the known universality class. At high activities the transition becomes first order. For n=0 the model includes an exactly known theta point, used to describe a collapsing polymer in two dimensions. When we vary n from 0 to 1, we observe a tricritical point which interpolates between the universality classes of the theta point and the Ising tricritical point.

DOI: 10.1142/s012918319900019x

Related papers

Phase Transitions of a Dilute O(n) Model

Wenan Guo, Henk W. J. Blöte, Yuanyuan Liu

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.

Communications in Theoretical PhysicsJun 15, 20044 citations

PHASE TRANSITIONS IN THE TWO-DIMENSIONAL O(3) MODEL

Henk W. J. Blöte, Wenan Guo

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.

International Journal of Modern Physics BJun 20, 20023 citations

Critical properties of a dilute O(n) model on the kagome lattice

Biao Li, Wenan Guo, Henk W. J. Blöte

A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome lattice to the exactly solvable critical q-state Potts model on the honeycomb lattice with q=(n+1)2. The intermediate steps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loop weight n'=sqrt(q) and a dilute loop model with loop weight n , both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O(n) model, as well as some of its critical properties. These properties differ from those of the generic O(n) critical points. For n=0, our model reproduces the known universal properties of the theta point describing the collapse of a polymer. For n not equal 0 it displays a line of multicritical points, with the same universal behavior as a branch of critical points that was found earlier in a dilute O(n) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.

arXiv:0805.2678Physical Review EAug 21, 20083 citations

Tricritical Points with Different Universality Classes for Polymers in Two Dimensions

Pablo Serra, Jürgen F. Stilck

We report a numerical investigation of the tricritical behaviour of the dilute equilibrium polymerization model with annealed dilution on the square lattice using transfer matrix techniques. In general, the tricritical exponent ν t is found to be compatible with the value for the tricritical n = 1 model. In the limit where the dilute model is equivalent to the interacting polymer model (whose tricritical point is known as the Θ-point), we observe a crossover of ν t to the tricritical n → 0 model value.

Europhysics Letters (EPL)Feb 1, 19929 citations

Fully packed loop model on the honeycomb lattice

Henk W. J. Blöte, Bernard Nienhuis

We investigate the O(n) model on the honeycomb lattice, using its loop representation in the limit of full packing. The universal properties, which we calculate by means of finite-size scaling and transfer-matrix techniques, are different from the branches of O(n) critical behavior known thus far. The conformal anomaly of the model varies between -1 and 2 in the interval 0\ensuremath{\le}n\ensuremath{\le}2. The universality class of the model is characterized as a superposition of a low-temperature O(n) phase, and a solid-on-solid model at a temperature independent of n.

Physical Review LettersFeb 28, 199493 citations

Frustration and disorder in discrete lattice models

Jesper Lykke Jacobsen

PART I: DISORDER The effect of quenched impurities on systems which undergo first-order phase transitions is studied within the framework of the q-state Potts model. For large q a mapping to the random field Ising model is introduced which provides a simple physical explanation of the absence of any latent heat in two dimensions, and suggests that in higher dimensions such systems should exhibit a tricritical point with a correlation length exponent related to the exponents of the random field model by \nu = \nu_RF/(2-\alpha_RF-\beta_RF). A phase diagram unifying pure, percolative and non-trivial random behaviour is proposed. In two dimensions we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a magnetic exponent \beta/\nu which varies continuously with q, and a weakly varying correlation length exponent \nu ~ 1. For q>4 the first-order transitions of the pure model are softened due to the impurities, and the resulting universality class is different from that of the pure Ising model. We find strong evidence for multiscaling of the correlation functions, as expected for such random systems. PART II: FRUSTRATION Exact results for conformational statistics of compact polymers are derived from the two-flavour fully packed loop model on the square lattice. This loop model exhibits a two-dimensional manifold of critical fixed points each one characterised by an infinite set of geometrical scaling dimensions. We calculate these dimensions exactly by mapping the loop model to an interface model whose scaling limit is described by a Liouville field theory. The formulae for the central charge and the first few scaling dimensions are compared to numerical transfer matrix results and excellent agreement is found. Compact polymers are identified with a particular point in the phase diagram of the loop model, and the non-mean field value of the conformational exponent \gamma = 117/112 is calculated for the first time. We also present the very precise numerical estimate \kappa = 1.472801(10) for the connective constant. Interacting compact polymers are described by a line of fixed points along which \gamma and \kappa vary continuously. Our identification of compact polymers with a critical model has bearings on the cooperativity of protein folding thermodynamics. Namely, in the large chain limit, the absence of an energy gap separating the denatured states from the native (compact) ones implies that homopolymer collapse is a one-state process. We comment on the prospects of modeling specific sequence (amino acid) information through the imposition of quenched disorder.

Sep 11, 19980 citations