Ihnsouk Guim

Conformal theory of energy correlations in the semi-infinite two-dimensional O(N) model

Abstract With conformal-invariance methods we calculate the energy-energy correlation function of the semi-infinite two-dimensional O(N) model with −2 ⩽ N ⩽ 2. Boundary conditions corresponding to ordinary, extraordinary, and special surface critical behaviour are considered. The crossover exponent φ of the special transition has the value φ = 1 2 . Particular attention is given to the limit N → 0, corresponding to the self-avoiding walk or polymer.

Universality class of trails in two dimensions

A trail is a walk on a lattice that may visit a site more than once but a bond at most once. We have carried out transfer-matrix studies of trails on the square lattice and of hybrid walks that interpolate between self-avoiding walks and trails. The results are in agreement with the same universal exponents as self-avoiding walks. However, the finite-size corrections are much larger than for self-avoiding walks. An explanation in terms of an irrelevant variable with scaling index is given.

Finite-size scaling of the quantum Ising chain with periodic, free, and antiperiodic boundary conditions

The authors give exact results for the energy spectrum of a chain of N Ising spins in a transverse field with periodic, free, and antiperiodic boundary conditions. The dependence of the energy gaps on boundary conditions is compatible with predictions of conformal invariance for correlation lengths in two-dimensional strips. The feasibility of calculating surface critical indices using phenomenological renormalisation with free boundary conditions and the convergence for large N are discussed.

Self-avoiding walks that cross a square

The authors consider self-avoiding walks that traverse an L*L square lattice. Whittington and Guttmann (1990) have proved the existence of a phase transition in the infinite-L limit at a critical value of the step fugacity. They make several finite-size scaling predictions for the critical region, using the relation between self-avoiding walks and the N-vector model of magnetism. Adsorbing as well as nonadsorbing boundaries are considered. The predictions are in good agreement with numerical data for L<or=9.

Correlation length in Ising strips with free and fixed boundary conditions

The correlation length of boundary spins in the Ising model, defined on strips of triangular lattice with free boundary conditions, is determined with an efficient numerical procedure based on the star-triangle transformation. In the case of isotropic critical interactions, the extrapolated amplitude of the correlation length is in excellent agreement with the value 2/( pi eta /sub ///) predicted by conformal invariance. An analytical formula for the amplitude in strips with anisotropic interactions is proposed. Fixing the spins on one edge reduces the amplitude of the correlation length on the other edge by a factor 1/2. The convergence of phenomenological renormalisation with free boundary conditions is studied.

Non-universal and anomalous surface critical behaviour in an inhomogeneous semi-infinite Gaussian model

Considers a semi-infinite Gaussian model with spatially inhomogeneous short-range couplings that depend on the distance z from the surface. Far from the surface the coupling constants vary as K(z)=KB-Az-y. For y=>2 the pair correlation function of the surface spins decays as a power law with a universal exponent eta /sub ///=2 at the bulk critical temperature. For y=2, eta /sub /// is non-universal, and for y<2 there is an anomalous exponential decay.

Conformal theory of spin correlations in the semi-infinite 3-state Potts and self-dual ZN models

In the conformal theory the magnetization operator of the critical 3-state Potts or self-dual Z3 model is degenerate at level 6. Solving a sixth-order differential equation, the authors calculate the bulk 4-spin correlation function and the spin-spin correlation function in the half space. The spin-spin correlation function of the semi-infinite self-dual ZN model is also obtained for arbitrary N. Both free- and fixed-spin boundary conditions are considered.

Star–triangle approach to boundary behavior in the two-dimensional Ising model

Hilhorst and van Leeuwen showed how to calculate boundary properties of the Ising model on the triangular lattice by iterating a mapping based on the star–triangle transformation. We apply this approach to the Ising model with homogeneous initial couplings in both the semi-infinite and strip geometries. Several exact results for the boundary correlation length and the magnetization are reproduced. The correlation-dimensionality transition for enhanced edge couplings (dual of Abraham’s interface-unbinding transition) is also considered.

Real-space renormalisation of the smoothly inhomogeneous semi-infinite Ising model

Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am-y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v-1, a surface magnetic phase at the bulk critical temperature for y 0, and a non-universal A-dependent surface magnetic index in the marginal case y=v-1.