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.
This thesis is concerned with several aspects of the Ising and tri-critical Ising models in two-dimensions. These are much-studied models relevant in both condensed matter physics as descriptions of the critical phenomena of two- dimensional systems and in String theory as building blocks of the string world sheet theory. The first part of the thesis is concerned with the derivation of differential equations for the critical four-point function in the Ising model. We present a method which provides the well-known standard solutions by a new and efficient route. The second part of the thesis is concerned with off-critical behaviour, and in particular the numerical study of perturbations of conformal field theory through the truncated conformal space approach. We show that the coupling constant undergoes significant renormalization in this scheme, and in particular the Ising model can be found as a fixed point for a finite value of the bare coupling constant. The renormalization group equations we find are of general use in the TCSA approach. The final part of the thesis considers off-critical boundary conditions in the tri-critical Ising model. We study them using a variant of the mean-field method and find a qualitative description of the space of boundary conditions that is in accord with the exact conformal field theory description. This is both a test of the method and its applicability in new domains, and also shows that previously published results are in error.
We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large
m,
m being the distance from the edge. The case
A<0 of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when
A>0. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.
We explore the phase diagram of Ising spins on one-dimensional chains that criss-cross in two perpendicular directions and that are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian that has been proposed as a model of magnetic behavior in Nb_{12}O_{29} and also, conceptually, as a geometry that is intermediate between one and two dimensions. Using mean-field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit, we demonstrate nontrivial critical behavior, including tricritical points that separate first- and second-order phase transitions. Finally, we present evidence that this model belongs to the two-dimensional Ising universality class.
Using Monte Carlo techniques and finite-size analysis, we investigate several two-dimensional lattice models with open edges, including the Blume-Capel model and the q=1 and 3 Potts models with vacancies. At bulk tricriticality, we find that the open edges are dominated by the vacancies when the surface coupling K(s) and the chemical potential D(s) of the vacancies assume the bulk values. When K(s) and/or D(s) is sufficiently enhanced, an edge phase transition takes place, beyond which spontaneous one-dimensional order occurs on the edges. Edge phase transitions can also be induced by a surface magnetic field H(s) . We numerically determine a number of edge critical exponents and derive phase diagrams in terms of K(s) , D(s) , and H(s) . In the low-temperature region, we observe first-order transitions when K(s) and D(s) are varied; the associated hysteresis loops of surface quantities are remarkably asymmetric. Some further insight into these edge transitions is provided by the exact equivalence of the tricritical q=1 Potts model and the Ising model.
We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field
H. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of
H and proportional to the total number of lattice sites, while the other depends on
H and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature
Tc. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For
T=Tc, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small
H. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at
T=Tc, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for
T<Tc, may be analytically continued beyond the point
H=0. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.