SURFACE CRITICAL BEHAVIOUR OF SOME TWO-DIMENSIONAL LATTICE MODELS

1990 ◽  
Vol 04 (09) ◽  
pp. 1437-1464 ◽  
Author(s):  
A.L. STELLA ◽  
C. VANDERZANDE
Keyword(s):  
Potts Model ◽  
Conformal Invariance ◽  
Finite Size ◽  
Lattice Models ◽  
Universality Class ◽  
Two Dimensions ◽  
Point Of View ◽  
Critical Behaviour ◽  
Logarithmic Scaling ◽  
Theta Point

A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.

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

1999 ◽  
Vol 10 (01) ◽  
pp. 291-300 ◽  
Author(s):  
WENAN GUO ◽  
HENK W. J. BLÖTE ◽  
BERNARD NIENHUIS
Keyword(s):  
Tricritical Point ◽  
Finite Size ◽  
Universality Class ◽  
Two Dimensions ◽  
Honeycomb Lattice ◽  
Matrix Methods ◽  
First Order ◽  
Universality Classes ◽  
Theta Point ◽  
Positive Integers

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.

SOFTENING OF PHASE TRANSITION BY RANDOM DISTRIBUTION OF BIMODAL-BONDS IN THE 2D 8-STATE POTTS MODEL

1999 ◽  
Vol 10 (05) ◽  
pp. 961-966
Author(s):  
FATIH YAŞAR ◽  
YIĞIT GÜNDÜÇ ◽  
TARIK ÇELIK
Keyword(s):  
Phase Transition ◽  
Potts Model ◽  
Random Distribution ◽  
Finite Size ◽  
Threshold Value ◽  
Two Dimensions ◽  
First Order ◽  
Size Dependent ◽  
First Order Phase Transition ◽  
First Order Phase

We investigated the influence of the distribution of bimodal bonds on phase transition in 8-state Potts model in two dimensions. We show that there is a finite size dependent threshold value of the introduced quenched randomness in the bond distribution for rounding the first-order phase transition.

From the natural continuum to the elastic spring cell in three and two dimensions -- a flexibility approach

2019 ◽  
Vol 36 (5) ◽  
pp. 1676-1698 ◽  
Author(s):  
I St Doltsinis
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Lattice Models ◽  
Anisotropic Elasticity ◽  
Two Dimensions ◽  
Point Of View ◽  
Space Structures ◽  
Content Type ◽  
Novel Approach ◽  
Definition Of

Purpose The employment of spring cell substitutes for the numerical analysis of solids and structures in place of finite elements has occasioned research on the subject with regard to both, the applicability of existing approaches and the advancement of concepts. This paper aims to explore in the context of linear elasticity the substitution of the simplex tetrahedral element in space and the triangle in the plane by corresponding spring cells deduced on a flexibility basis using the natural formalism. Design/methodology/approach The natural formalism is characterized by the homogeneous definition of strain and stress along the lines connecting nodes of the simplex tetrahedron and the triangle. The elastic compliance involves quantities along the prospective spring directions and offers itself for the transition to the spring cell. The diagonal entities are interpreted immediately as spring flexibilities, the off-diagonal terms account for the completeness of the substitution. In addition to the isotropic elastic material, the concept is discussed for anisotropic elasticity in the plane. Findings The natural point of view establishes the spring cell as part of the continuum element. The simplest configuration of pin-joined bars discards all geometrical and physical cross effects. The approach is attracting by its transparent simplicity, revealing deficiencies of the spring cell and identifying directly conditions for the complete substitution of the finite element. Research limitations/implications The spring cell counterparts of the tetrahedral- and the triangular finite elements allow employment in problems in three and two dimensions. However, the deficient nature of the approximation requires attention in the design of the discretization lattice such that the conditions of complete finite element substitution are approached as close as possible. Practical implications Apart from plane geometries, triangular spring cells have been assembled to lattice models of space structures such as membrane shells and similar. Tetrahedral cells have been used, in modelling plates and shell structures exhibiting bending stiffness. Originality/value The natural formalism of simplex finite elements in three and two dimensions is used for defining spring cells on a flexibility basis and exploring their properties. This is a novel approach to spring cells and an original employment of the natural concept in isotropic and anisotropic elasticity.

scholarly journals DYNAMIC CRITICAL EXPONENTS OF THE ISING MODEL WITH MULTISPIN INTERACTIONS

2001 ◽  
Vol 15 (15) ◽  
pp. 487-496 ◽  
Author(s):  
C. S. SIMÕES ◽  
J. R. DRUGOWICH DE FELÍCIO
Keyword(s):  
Ising Model ◽  
Potts Model ◽  
Critical Exponents ◽  
Finite Size ◽  
Two Dimensions ◽  
Time Dynamics ◽  
Spin Interactions ◽  
Dynamic Exponent ◽  
Short Time ◽  
Multispin Interactions

We revisit the short-time dynamics of 2D Ising model with three spin interactions in one direction and estimate the critical exponents z, θ, β and ν. Taking properly into account the symmetry of the Hamiltonian, we obtain results completely different from those obtained by Wang et al.10 For the dynamic exponent z our result coincides with that of the 4-state Potts model in two dimensions. In addition, results for the static exponents ν and β agree with previous estimates obtained from finite size scaling combined with conformal invariance. Finally, for the new dynamic exponent θ we find a negative and close to zero value, a result also expected for the 4-state Potts model according to Okano et al.

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