First order phase transitions are rounded in solids of finite size. It is shown here that the above rounding is monitored by the correlation length ξL of the finite system, or equivalently by the so-called mass gap for the quantum Hamiltonian version of the model. Scaling with size is studied as a function of variable boundary conditions for the cylinder geometry (infinite strips with finite width), and a striking crossover is found in the mass gap behavior when the coupling g along the boundary becomes anti-periodic. For g>0, the rounding is exponential with size and an accurate determination of the spontaneous magnetization (order parameter) of the infinite system is obtained from numerical extrapolations.
Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from
+Msp to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from
0+ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature
Tc and at
Tc. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.
The finite-size rounding of a first-order transition is studied in systems representable as n-vector ferromagnets, so that O(n) symmetry (n\ensuremath{\ge}2) is broken at the bulk transition point. Both ``block,'' V=
Ld, and ``cylinder,'' ${L}^{d\mathrm{\ensuremath{-}}1}$\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/${L}^{d\mathrm{\ensuremath{-}}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel's phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ``degeneracy kernel'' to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ``spin-wave stiffness'' or ``superfluid density'') and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, that is needed for d>
d>=4. All the results found can be checked in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} against exact results for spherical models (described elsewhere).
The finite-size scaling hypothesis of Privman and Fisher (Phys. Rev. B, 30, 322 (1984)) is applied to systems with O(n) symmetry [Formula: see text], confined to geometry Ld − d′ × ∞d′ (where d and d′ are continuous variables such that 2 < d′ < d < 4) and subjected to periodic boundary conditions. Predictions, involving amplitudes as well as exponents, are made on the singular part of the specific heat c(s), spontaneous magnetization m0, magnetic susceptibility χ0, and correlation length ξ0 in the region of the second-order phase transition [Formula: see text]. Analytical verification of these predictions is carried out in the case of the spherical model of ferromagnetism (n = ∞), which includes evaluation of the shift in the critical temperature of the system.
Finite-size effects on the first-order phase transition in a three-dimensional classical Heisenberg model with nearest-neighbor ferromagnetic interactions and periodic boundary conditions are studied theoretically and numerically in this paper. We find that the current Fisher-Privman theory fails at temperatures close to criticality and/or for small systems. We rederive explicit scaling functions for the longitudinal magnetization and susceptibility under another phenomenological assumption. The results are in very good agreement with Monte Carlo simulations.
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.