In this study, we explored the magnetic coupling in a multilayer system consisting of thin layers separated by a distance
d. We have employed Monte Carlo (MC) simulations to calculate the thermodynamic quantities such as the magnetization per spin
mLμ, magnetic susceptibility
χLμ, and the reduced fourth-order Binder cumulant
ULμ as a function of temperature
T and for several values of lattice size
L. These quantities were obtained for each layer (
μ=l), for the bulk system (
μ=b), as a function of interlayer distance
d and the parameter
α that defines the scale length of the exponential decay of the interaction. Furthermore, we applied the finite-size scaling theory to calculate the critical exponents. Our results reveal that the system exhibits 2D Ising exponents when the layers are sufficiently separated. On the other hand, in the compact limit, where
d equals the distance between two adjacent sites within the same layer, our bulk results show that the system exhibits 3D Ising critical exponents, provided that the number of layers
Lz increases proportionally to the layer sizes. Even with the separation increasing up to
d=2.0, the layers are so correlated that the set of critical exponents retains the values of 3D critical exponents. However, when the number of layers
Lz remains fixed, even in the compact limit with periodic boundary conditions, only the exponent
β aligns closely with the predicted literature values, on the other hand, the other exponents show significant deviations.
The authors present a simple argument showing that the surface energy density of a semi-infinite d-dimensional spin system has, in general, a leading thermal singularity of the same form, mod T-T c mod 2- alpha , as the bulk free energy. At bulk criticality energy-energy correlations decay as r -2d parallel to the surface. These results hold for both free and fixed boundary spins, i.e. the 'ordinary' and 'extraordinary' transitions. The extraordinary critical behaviour of the magnetisation is the same as that of the energy density. They also confirm and generalise these results in two dimensions with an independent approach based on conformal invariance.
In order to investigate the effects of connectivity and proximity in the specific heat, a special class of exactly solvable planar layered Ising models has been studied in the thermodynamic limit. The Ising models consist of repeated uniform horizontal strips of width
m connected by sequences of vertical strings of length
n mutually separated by distance
N, with
N=1,2 and
3. We find that the critical temperature
Tc(N,m,n), arising from the collective effects, decreases as
n and
N increase, and increases as
m increases, as it should be. The amplitude
A(N,m,n) of the logarithmic divergence at the bulk critical temperature
Tc(N,m,n) becomes smaller as
n and
m increase. A rounded peak, with size of order
lnm and signifying the one-dimensional behavior of strips of finite width
m, appears when
n is large enough. The appearance of these rounded peaks does not depend on
m as much, but depends rather more on
N and
n, which is rather perplexing. Moreover, for fixed
m and
n, the specific heats are not much different for different
N. This is a most surprising result. For
N=1, the spin-spin correlation in the center row of each strip can be written as a Toeplitz determinant with a generating function which is much more complicated than in Onsager's Ising model. The spontaneous magnetization in that row can be calculated numerically and the spin-spin correlation is shown to have two-dimensional Ising behavior.
We studied the ferromagnetic Ising model on two-dimensional finite systems with non-integer characteristic lengths. First we considered very long strips with a finite number I of complete rows and one partially filled row with probability x , which is the two-dimensional version of a layer-by-layer growth. At fixed temperature, when the characteristic length L=I+x increases, the free energy per spin, the specific heat and the magnetic susceptibility oscillate, attaining relative extremes at integer L . The oscillations in the free energy are interpreted as surface corrections related to the oscillations in the mean coordination number. Finite-size scaling relations are not satisfied with continuous L , but still hold for fixed x and variable I , where the differences of mean widths are integers. For fixed x , the fits of the free energy give the conformal anomaly with very good accuracy. We also studied strips with discretized Gaussian distributions of widths, with non-integer means L and rms deviation . In these structures, the thermodynamic quantities vary monotonically with continuous L , but some methods for calculating critical exponents do not work properly when generalized to continuous L . We also obtain with good accuracy in these systems. We discuss the possible implications of our results to real systems behaviour.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.
This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!