First order transitions are rounded in finite cells or in infinite strips of finite width (cylinders). In this work we have considered the rounding of the jump of the magnetization of an Ising system at a fixed temperature T below Tc when the external field h goes from a small positive, to a small negative value. This rounding is well-known in a finite cell (block geometry) but the situation is more complicated, and more interesting, in a cylinder. Indeed it is known from the work of Privman and Fisher that these rounding effects are related to the behaviour of the correlation length along the axis of the cylinder which increases exponentially with the cross-sectional area A of the strip. It then follows that if one studies, for a given magnetization M, the external field h, this field approaches exponentially with A its bulk limit, but the coefficient inside the exponential is different if M is smaller, larger than or equal to the bulk spontaneous magnetization M0. We present numerical data for the two-dimensional case (or equivalently for the one-dimensional quantum model in a transverse field) which are in satisfactory agreement with the theory.
A scaling hypothesis is set up for the correlation function G(R,T;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (2d4, d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in the correlation length \ensuremath{\xi}(T;L) are predicted in the region of first-order phase transition (T
Tc) as well as the region of second-order phase transition (T\ensuremath{\simeq}
Tc); for the correlation function, the predictions relate mostly to the regime T
Tc. To test these predictions, a detailed analytical study is carried out in the case of the spherical model and all predictions are seen to be fully borne out. In the process we find that, though a single correlation length \ensuremath{\xi} governs the scaling of the variable R in all directions, the functional dependence of the correlation function on the scaled parameter R/\ensuremath{\xi} is highly anisotropic, in that it depends very differently on the components, ${\mathrm{R}}_{\ensuremath{\perp}}$ and
R?, of R pertaining to the finite and the infinite dimensions, respectively. Most importantly, while long-range order at temperatures below
Tc prevails all the way in the direction of ${\mathrm{R}}_{\ensuremath{\perp}}$ it is severely limited in the direction of
R?. In fact, for
R?\ensuremath{\gg}L, the qualitative features of the correlation function become characteristic of a d'-dimensional bulk system, though \ensuremath{\xi} still pertains to the actual, finite-sized system; the net result is that long-range order in the direction of
R? persists only to distances small in comparison with \ensuremath{\xi} which, for d'2, is found to be O(L(L/a${)}^{(d\mathrm{\ensuremath{-}}2)/(2\mathrm{\ensuremath{-}}d\mathcal{'})}$), a being the lattice constant.
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.
A scaling hypothesis is set up for the magnetization m(T,H;L), susceptibility \ensuremath{\chi}(T,H;L), and correlation length \ensuremath{\xi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2) and long-range interactions decaying as 1/${r}^{d+\ensuremath{\sigma}}$ (0<\ensuremath{\sigma}<2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (\ensuremath{\sigma}<d<2\ensuremath{\sigma},d'\ensuremath{\le}\ensuremath{\sigma}) and subjected to periodic boundary conditions. Finite-size effects are predicted, in the region of a first-order phase transition (T
Tc) as well as in the region of a second-order phase transition (T\ensuremath{\simeq}
Tc), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out. In situations where L\ensuremath{\gg}\ensuremath{\xi}, finite-size corrections to standard bulk values of the various physical quantities pertaining to these models are found to vary as (\ensuremath{\xi}/L${)}^{d+\ensuremath{\sigma}}$, rather than as ${e}^{\mathrm{\ensuremath{-}}L/\ensuremath{\xi}}$, which is characteristic of models with short-range interactions.
We propose a finite-size scaling hypothesis for the ``singular'' part of the free-energy density,
f(s)(T,H;L), and the correlation function, G(R,T,H;L), of a system with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${L}^{d\mathrm{\ensuremath{-}}d\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\infty}}^{d\mathcal{'}}$ (d>4, d'\ensuremath{\le}2) and subjected to periodic boundary conditions. On the basis of this hypothesis, we predict finite-size effects in the zero-field susceptibility, \ensuremath{\chi}(T;L), the correlation length, \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat,
c(s)(T;L), in the region of first-order phase transition (T
Tc) as well as the region of second-order phase transition (T\ensuremath{\simeq}
Tc). It turns out that in both these regions the system in zero field can be described by a single, unifying variable v\ensuremath{\sim}${L}^{2(d\mathrm{\ensuremath{-}}d\mathcal{'})/(4\mathrm{\ensuremath{-}}d\mathcal{'})}$t\ifmmode \tilde{}\else \~{}\fi{}, where t\ifmmode \tilde{}\else \~{}\fi{} is an appropriate temperature parameter of the system. To test the various predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions are seen to be fully borne out. For general n, we compare our findings with those of the previous authors, wherever possible, and in each case the comparison turns out to be perfect.
A scaling hypothesis is set up for the magnetization m(T,H;L) and susceptibility \ensuremath{\chi}(T,H;L) of a finite-sized system, with O(n) symmetry (n\ensuremath{\ge}2), confined to geometry ${\mathit{L}}^{\mathit{d}\mathrm{\ensuremath{-}}\mathit{d}\mathcal{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathit{d}\mathcal{'}}$(2d4,d'\ensuremath{\le}2) and subjected to periodic boundary conditions. Finite-size effects in both m and \ensuremath{\chi} are predicted, in the region of the first-order phase transition (T
Tc) as well as in the region of the second-order phase transition (T\ensuremath{\simeq}
Tc), for different regimes of the parameters H and L. To test these predictions, a detailed analytical study is carried out in the case of the spherical model of ferromagnetism (n=\ensuremath{\infty}), and all predictions based on the scaling hypothesis are seen to be fully borne out.