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.
By the use of the Migdal–Kadanoff renormalization group technique and the mean field approximation, we have explored the critical behavior of the semi-infinite mixed spin-7/2 and spin-1/2 Blume–Capel model. As a function of the computation ratios (bulk-surface) R and Y, different phase diagrams in the bulk and on the surface are classified and determined in the (surface anisotropy, temperature) plane. We have found four types of phase diagrams characterized by ordinary, extraordinary, surface, and special phase transitions. The derivative of the free energy and the behavior of the bulk and surface magnetizations are plotted at very low temperatures proving the existence of first-order transitions for both the surface and bulk. We have also presented the related fixed points and the critical exponents manifesting several classes of universality at the surface. Otherwise, a comparison was made between the two methods as well as with previous studies.
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 spin-1/2 Ising model and the Blume-Capel model at various values of the parameter
D on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter
D, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for
D=0.641 and 0.655 on lattices of a linear size up to
L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.
We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to
L=1600. As a check, also simulations of the spin-
21 Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of
N-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where
L is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading
Z2-even scalar $\ensuremath{\epsilon}$.
We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.