Universality of surface critical behavior with respect to surface enhancement is studied for O(n) models with n=1 (Ising), n=2 (planar rotor), and n=3 (Heisenberg) on simple-cubic lattices. Finite-size methods are employed to estimate surface critical exponents for ordinary surface criticality. In addition, it is shown that universal scaling functions, independent of surface enhancement, can be constructed with all nonuniversal features of the finite-size scaling function of the spin-spin surface correlation functions incorporated in (1) a metric factor and (2) an irrelevant scaling field associated with the surface coupling strength.
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
Kc(n=2)=0.4541659 (10) and
Kc(n=3)=0.693003 (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be
yh1(o)=0.7374(15), 0.781 (2), and 0.813 (2) for
n=1, 2, and 3, respectively. Then we vary the surface coupling
K1 and locate the so-called special transition at ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (8) and ${\ensuremath{\kappa}}_{\mathrm{c}}(n=2)=0.6222$ (3), where $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1$. The corresponding surface thermal and magnetic exponents are
yt1(s)=0.715 (1) and
yh1(s)=1.636 (1) for the Ising model, and
yt1(s)=0.608 (4) and
yh1(s)=1.675 (1) for the
XY model. Finite-size corrections with an exponent close to $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.
Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability
Ep, the percolation probability
P, and the probability
Wn for the appearance of
n percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions
L. Using the HMCSM, we calculate the existence probability
Ep and the percolation probability
P for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of
Ep and
P in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that
Ep and
P have universal finite-size scaling functions. This implies that the critical
Ep is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that
Wn for site and bond percolation on sc lattices have universal finite-size scaling functions.
We study the critical behavior at the ordinary surface universality class of the three-dimensional
O(N) model, bounded by a two-dimensional surface. Using high-precision Monte Carlo simulations of an improved lattice model, where the leading bulk scaling correction is suppressed, and finite-size scaling analysis of the fourth cumulant of the surface magnetization, we obtain precise estimates of the scaling dimension of the surface field operator for
N=2,3,4. We also determine the fixed-point values of two renormalization-group invariant observables, which characterize the finite-size scaling behavior at the ordinary transition.
We study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find
yt1=0.718(2) and
yh1=1.6465(6) for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.
The surface scaling behavior of classical isotropic Heisenberg magnets is investigated by Monte Carlo methods in
d=3 dimensions for various values of the surface-to-bulk coupling ratio
J1/J. For
J1/J< 1.0, critical behavior according to the ordinary surface universality class is found. Estimates for magnetic surface exponents are presented and compared to older estimates and their theoretical counterparts. For
J1/J> 2.0, scaling is still valid with effective exponents which depend on
J1/J. The surface structure factor
S1(p,L) is investigated at bulk criticality as a function of the momentum transfer p parallel to the surface and the system size L. For
J1/J< 1.0 and
J1/J> 2.0 the full p dependence of
S1(p,L) can be captured by generalized shape functions to a remarkable accuracy. Profiles of the magnetization and the energy density also confirm scaling, where for
J1/J< 1.0 the ordinary surface universality class is recovered and for
J1/J> 2.0, scaling with
J1/J dependent exponents is found. For
J1/J=1.5 the system displays a striking crossover behavior from spurious long-range surface order to the ordinary surface universality class. For
J1/J> 2.0 the effective scaling laws must be interpreted as nonasymptotic and the value
J1/J=1.5 marks a crossover regime, in which the crossover from the nonasymptotic to the asymptotic (ordinary) surface scaling behavior can be resolved within numerically attainable system sizes.