With simple, exact arguments we show that the surface magnetization
m1 at the extraordinary and normal transitions and the surface energy density ${\ensuremath{\epsilon}}_{1}$ at the ordinary, extraordinary, and normal transitions of semi-infinite
d-dimensional Ising systems have leading thermal singularities $B\ifmmode\pm\else\textpm\fi{}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, with the same critical exponent and amplitude ratio as the bulk free energy
fb(t,0). The derivation is carried out in three steps: (i) By tracing out the surface spins, the semi-infinite Ising model with supercritical surface enhancement
g and vanishing surface magnetic field
h1 is mapped exactly onto a semi-infinite Ising model with subcritical surface enhancement, a nonzero surface field, and irrelevant additional surface interactions. This establishes the equivalence of the extraordinary (
h1=0,g>0) and normal (${h}_{1}\ensuremath{\ne}0, g<0$) transitions. (ii) The magnetization
m1 at the interface of an infinite system with uniform temperature
t and a nonzero magnetic field
h in the half-space
z>0 only is shown to be proportional to ${f}_{b}(t,0)\ensuremath{-}{f}_{b}(t,h)$. (iii) The energy density ${\ensuremath{\epsilon}}_{1}$ at the interface of an infinite system with temperatures
t+ and
t in the half-spaces
z>0 and
z<0 and no magnetic field is shown to be proportional to ${f}_{b}(t,0)\ensuremath{-}{f}_{b}({t}_{+},0)$.
Thermal and magnetic effects in a system consisting of thin layers of coupled Ising spins with S=1/2 and S=1 are considered. The specific heat and the correlation length display maxima at two different temperatures. It is discussed in what sense these maxima can be interpreted as a finite-size rounding of a thermodynamic singularity associated with a phase transition. The connection with ordinary, extraordinary, and special surface phase transitions is made. In two dimensions (2D), the surface critical exponents are calculated from conformal invariance. The bulk and surface finite-size scaling of the order-parameter profiles at the transition points is discussed. In 2D, an exact scaling function for the profiles is suggested through conformal invariance arguments for the (extra)ordinary transition.
The critical behaviour of a semi-infinite system with O(n) spin symmetry is investigated using four techniques: (i) mean-field theory, (ii) exact solution for n= infinity , (iii) the epsilon expansion, and (iv) scaling arguments. The surface equation of state is computed using mean-field theory, and the four phase transitions defined by Lubensky and Rubin, the 'ordinary', 'surface', 'extraordinary', and ' lambda = infinity ' ('special') transitions, are identified. The first three may be observed in the bulk system by adding a suitable 'surface' perturbation which destroys translational invariance. Scaling arguments give their critical exponents exactly in terms of bulk exponents. Critical exponents for the ' lambda = infinity ' transition are determined for n= infinity and also to O( epsilon ). Spin-spin correlation functions at T=T c for the 'ordinary', 'extraordinary' and ' lambda = infinity ' transitions are calculated exactly for n= infinity .
By solving partial differential equations for the correlation functions implied by conformal invariance, we obtain exact results at the critical point for the bulk six-point function 〈\ensuremath{\sigma}(1)...\ensuremath{\sigma}(6)〉 of the two-dimensional Ising model and the three-point functions 〈\ensuremath{\sigma}(1)\ensuremath{\sigma}(2)\ensuremath{\sigma}(3)〉, 〈\ensuremath{\sigma}(1)\ensuremath{\mu}(2)\ensuremath{\mu}(3)〉 of the semi-infinite two-dimensional Ising model with fixed boundary spins. Here \ensuremath{\sigma} and \ensuremath{\mu} denote order (spin) and disorder variables, respectively. We also derive the explicit form of the magnetization profile at the bulk critical temperature in an Ising strip of infinite length and finite width, with spin-up boundary conditions on one edge and spin-down boundary conditions on the other.
We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings
JS and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For
JS\ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$ parallel to the surface. For
JS=1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{r}}^{\mathrm{\ensuremath{-}}3}$ for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.