Fisher's rigorous bound chi < chi saw , where chi saw is the susceptibility calculated from the self-avoiding walk (SAW) generating function, is applied to (i) the ferromagnetic Ising alloy, (ii) the dilute Ising ferromagnet, (iii) the layer susceptibility of a free surface, (iv) the Ising ferromagnet in a transverse field, and (v) the dilute Ising ferromagnet in a transverse field. The author obtains an upper bound T UB for the temperature T 0 below which a spontaneous magnetization occurs. The use of a microscopically varying molecule field is analysed. A heuristic argument is given to show that, for the dilute Ising ferromagnet in a transverse field, the critical transverse field at zero temperature jumps from zero to a value at least equal to that of the one-dimensional ground state as x passes through the critical percolation concentration.
The critical temperature of an Ising system consisting of n(d-1) dimensional layers in d dimensions is estimated for large n from the properties of random and self-avoiding walks (SAW) in the finite system. Denoting the deviation of T c (n) from T c ( infinity ) by 1/n lambda it is found that there are two contributions, a finite size effect which applies both to a torus and a strip and for which lambda =d-2+ eta , and a free boundary effect which applies to a strip alone and for which lambda =1/ nu d . ( nu d , eta are exponents associated with the spin pair correlation function using the standard notation of Fisher). The results are based on a perturbation expansion for SAW which should be reliable for d>or=4, but which needs further examination when d=3. Some, but not all, of the above features are in agreement with the results of Fisher and Barber for the spherical model.
For spin 1 / 2 Ising models with ferromagnetic pair interactions J ij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q ) is less than or equal to the value ( sigma p sigma q ) J for the corresponding non-random system in which each J ij is replaced by its average value J ij . The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi <or= chi j . With additional assumptions about the existence and nature of the phase transitions for the systems, one concludes that the critical temperature tau c for the averaged system is not greater than the corresponding critical temperature tau (J) for the nonrandom system. In the process of obtaining the results it is noticed that Griffiths' inequalities ( sigma p sigma q )>or=0 and delta ( sigma p sigma q )/ delta J rs >or=0 imply that delta 2 ( sigma p sigma q )/ delta J rs 2 <or=0.
The effect of random bond (site) dilution on the phase transition of a three-dimensional transverse Ising model is studied by a cluster variation, along with perturbation expansion method. Complete phase diagrams in the temperature-concentration-transverse-field space are constructed for both the cases. The value of critical transverse field as a function of concentration, at zero temperature, exhibits a discontinuity at the critical value of concentration for the existence of phase transition, as conjectured by Harris. As is expected for the existence of a second-order phase transition, the static susceptibility diverges on the critical surface.
Within the framework of the effective-field theory, we examine the phase transitions of a transverse spin-
21 Ising film. The critical temperatures of the film as a function of the interactions, transverse fields, and film thickness are studied. It is found that for the ratio of the surface interactions to the bulk ones
R=Js/J less than a critical value
Rc, the critical temperature
Tc/J of the film is smaller than the bulk critical temperature
TcB/J and as the film thickness L is increased further,
Tc/J increases and approaches asymptotically
TcB/J for large values of L. However, for
R>Rc,Tc/J is larger both than the bulk
TcB/J and the surface
TcS/J critical temperatures of the corresponding semi-infinite system and as the film thickness L is increased further,
Tc/J decreases and approaches asymptotically, for large values of L, the surface magnetic transition
TcS/J observed in the corresponding semi-infinite system. We calculate also some magnetic properties of the film such as the layer magnetizations, their averages and their profiles and the longitudinal susceptibility of the film. The film longitudinal susceptibility still diverges at the film critical temperature as does the bulk longitudinal susceptibility, but its magnitude is reduced. Also there is a rounded peak at the bulk critical temperature when
R>Rc. The bulk related character of the susceptibility is more pronounced and the surface related character is less pronounced when the film thickness is large.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.