The specific heats of exactly solvable alternating layered planar Ising models with strips of width m_{1} lattice spacings and "strong" couplings J_{1} sandwiched between strips of width m_{2} and "weak" coupling J_{2}, have been studied numerically to investigate the effects of connectivity and proximity. We find that the enhancements of the specific heats of the strong layers and of the overall or "bulk" critical temperature T_{c}(J_{1},J_{2};m_{1},m_{2}) arising from the collective effects reflect the observations of Gasparini and co-workers in experiments on confined superfluid helium. Explicitly, we demonstrate that finite-size scaling holds in the vicinity of the upper limiting critical point T_{1c} (∝J_{1}/k_{B}) and close to the corresponding lower critical limit T_{2c} (∝J_{2}/k_{B}) when m_{1} and m_{2} increase. However, the residual enhancement, defined via appropriate subtractions of leading contributions from the total specific heat, is dominated (away from T_{1c} and T_{2c}) by a decay factor 1/(m_{1}+m_{2}) arising from the seams (or boundaries) separating the strips; close to T_{1c} and T_{2c} the decay is slower by a factor lnm_{1} and lnm_{2}, respectively.
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.
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 specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of
n parallel layers with free boundary conditions and a surface magnetic field
H1=kBTh1 imposed on the last layer, is analyzed for large
n in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions
Wi(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where
a is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant
p is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
A scaling theory and a related conjecture presented recently by de Gennes and Fisher to predict the effects of walls inserted in two near-critical binary fluid mixtures, are checked theoretically by exact analytic calculations for $n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$, two-dimensional Ising model strips with a surface field,
h1, imposed on the first layer; exact calculations with a second surface field,
hn, imposed on the nth layer are also reported. It is verified, in particular, that the effects on properties observed close to one wall of a second wall at distance
D decay, at the critical point, as
Dd1, where
d is the spatial dimensionality. In addition to the scaling limits, the leading corrections are calculated explicitly and presented graphically.
We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat Cv based on the finite-size scaling of its maximal values Cvmax depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of Cv. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of Cvmax in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.