Surface and bulk transitions in three-dimensional O(n) models

Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the O$(n)$ models on the simple-cubic lattice with $n=1$, 2, and 3, i.e. the Ising, XY, and Heisenberg models. For the critical couplings we find $K_{\rm c}(n=2)=0.454 1655 (10)$ and $K_{\rm c}(n=3)= 0.693 002 (2)$. We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be $y_{h1}^{\rm (o)}=0.7374 (15)$, $0.781 (2)$, and $0.813 (2)$ for $n=1$, 2, and 3, respectively. Then we vary the surface coupling $K_1$ and locate the so-called special transition at $\kappa_{\rm c} (n=1)=0.50214 (8)$ and $\kappa_{\rm c} (n=2)=0.6222 (3)$, where $\kappa=K_1/K-1$. The corresponding surface thermal and magnetic exponents are $y_{t1}^{\rm (s)} =0.715 (1)$ and $y_{h1}^{\rm (s)} =1.636 (1)$ for the Ising model, and $y_{t1}^{\rm (s)} =0.608 (4)$ and$y_{h1}^{\rm (s)} =1.675 (1)$ for the XY model. Finite-size corrections with an exponent close to -1/2 occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.Comment: TeX paper and 10 eps figure

Transfer-matrix approach to the three-dimensional bond percolation: An application of Novotny's formalism

A transfer-matrix simulation scheme for the three-dimensional (d=3) bond percolation is presented. Our scheme is based on Novotny's transfer-matrix formalism, which enables us to consider arbitrary (integral) number of sites N constituting a unit of the transfer-matrix slice even for d=3. Such an arbitrariness allows us to perform systematic finite-size-scaling analysis of the criticality at the percolation threshold. Diagonalizing the transfer matrix for N =4,5,...,10, we obtain an estimate for the correlation-length critical exponent nu = 0.81(5)

Critical temperature of a fully anisotropic three-dimensional Ising model

The critical temperature of a three-dimensional Ising model on a simple cubic lattice with different coupling strengths along all three spatial directions is calculated via the transfer matrix method and a finite size scaling for L x L oo clusters (L=2 and 3). The results obtained are compared with available calculations. An exact analytical solution is found for the 2 x 2 oo Ising chain with fully anisotropic interactions (arbitrary J_x, J_y and J_z).Comment: 17 pages in tex using preprint.sty for IOP journals, no figure

Improved Phenomenological Renormalization Schemes

An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the three-dimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent.Comment: 14 pages with 1 figure in late

Finite temperature results on the 2d Ising model with mixed perturbation

A numerical study of finite temperature features of thermodynamical observables is performed for the lattice 2d Ising model. Our results support the conjecture that the Finite Size Scaling analysis employed in the study of integrable perturbation of Conformal Field Theory is still valid in the present case, where a non-integrable perturbation is considered.Comment: 9 pages, Latex, added references and improved introductio

Hyperuniversality of Fully Anisotropic Three-Dimensional Ising Model

For the fully anisotropic simple-cubic Ising lattice, the critical finite-size scaling amplitudes of both the spin-spin and energy-energy inverse correlation lengths and the singular part of the reduced free-energy density are calculated by the transfer-matrix method and a finite-size scaling for cyclic L x L x oo clusters with L=3 and 4. Analysis of the data obtained shows that the ratios and the directional geometric means of above amplitudes are universal.Comment: RevTeX 3.0, 24 pages, 2 figures upon request, accepted for publication in Phys. Rev.

Numerical Studies of the Two Dimensional XY Model with Symmetry Breaking Fields

We present results of numerical studies of the two dimensional XY model with four and eight fold symmetry breaking fields. This model has recently been shown to describe hydrogen induced reconstruction on the W(100) surface. Based on mean-field and renormalization group arguments,we first show how the interplay between the anisotropy fields can give rise to different phase transitions in the model. When the fields are compatible with each other there is a continuous phase transition when the fourth order field is varied from negative to positive values. This transition becomes discontinuous at low temperatures. These two regimes are separated by a multicritical point. In the case of competing four and eight fold fields, the first order transition at low temperatures opens up into two Ising transitions. We then use numerical methods to accurately locate the position of the multicritical point, and to verify the nature of the transitions. The different techniques used include Monte Carlo histogram methods combined with finite size scaling analysis, the real space Monte Carlo Renormalization Group method, and the Monte Carlo Transfer Matrix method. Our numerical results are in good agreement with the theoretical arguments.Comment: 29 pages, HU-TFT-94-36, to appear in Phys. Rev. B, Vol 50, November 1, 1994. A LaTeX file with no figure

Conformal invariance and linear defects in the two-dimensional Ising model

Using conformal invariance, we show that the non-universal exponent eta_0 associated with the decay of correlations along a defect line of modified bonds in the square-lattice Ising model is related to the amplitude A_0=xi_n/n of the correlation length \xi_n(K_c) at the bulk critical coupling K_c, on a strip with width n, periodic boundary conditions and two equidistant defect lines along the strip, through A_0=(\pi\eta_0)^{-1}.Comment: Old paper, for archiving. 5 pages, 4 figures, IOP macro, eps