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The effect of finite size of the system on correlation length behaviour at the presence of external field

Mykhailo P. Kozlovskii

Published Jan 1, 2007• 1 citation
Publisher

Abstract

The explicit analytical expression for the effective correlation length ξ as a function of reduced temperature , external field h and system size L for the Ising-like system near the phase transition point Tc is obtained. The role of these quantities in the formation of the correlation length value is ascertained. It is shown that the irregular increase of the correlation length exists only in the case of τ → 0, h → 0 and L → ∞. With deviation from these values the essential slowing down exists for the increasing . The criterium of the permissible range of temperature values (field values), where the correlation length behaviour is defined by temperature (or field) variable, is established for the fixed size of the system. Beyond this range τ < τc, h < hcr the system size becomes crucial for forming the correlation length.

DOI
10.5488/cmp.10.2.173

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The explicit analytical expression for the effective correlation length ξ as a function of reduced temperature , external field h and system size L for the Ising-like system near the phase transition point Tc is obtained. The role of these quantities in the formation of the correlation length value is ascertained. It is shown that the irregular increase of the correlation length exists only in the case of τ → 0, h → 0 and L → ∞. With deviation from these values the essential slowing down exists for the increasing . The criterium of the permissible range of temperature values (field values), where the correlation length behaviour is defined by temperature (or field) variable, is established for the fixed size of the system. Beyond this range τ < τc, h < hcr the system size becomes crucial for forming the correlation length.
The explicit analytical expression for the effective correlation length ξ as a function of reduced temperature τ , external field h and system size L for the Ising-like system near the phase transition point Tc is obtained.The role of these quantities in the formation of the correlation length value is ascertained.It is shown that the irregular increase of the correlation length exists only in the case of τ → 0, h → 0 and L → ∞.With deviation from these values the essential slowing down exists for the increasing ξ.The criterium of the permissible range of temperature values (field values), where the correlation length behaviour is defined by temperature (or field) variable, is established for the fixed size of the system.Beyond this range τ < τc, h < hcr the system size becomes crucial for forming the correlation length.
The explicit dependence of correlation length as function of temperature and external field is found for the Ising model on a simple cubic lattice near the phase transition point. , where and h c are the renormalized values of the external field h and reduced temperature τ. The critical amplitude ξ0 is proportional to the lattice constant through the coefficient, which is determined by the parameter of the interaction potential. The size of the critical region is established by the temperature and field. It is shown that the essential increasing of the system correlation length takes place at () with h→0 and at h′<h* (, h′=βh) with τ→0.
Abstract For systems above the marginal dimension d*, where mean field theory starts to become valid, such as Ising models in d = 5 for which d* = 4, hyperscaling is invalid and hence it was suggested that finite size scaling is not ruled by the correlation length ξ (∝ |t| −1/2 in Landau theory, t being the distance from the critical point) but by a “thermodynamic length” l (∝ |t| −2/d). Early simulation work by Binder et al. using nearest neighbor hypercubic L5 lattices with L ⩽ 7 yielded some evidence for this prediction, but the renormalized coupling constant gL = −3 + 〈M4〉/〈M2〉2 at Tc was gL ≈ −1.0 instead of the prediction of Brézin and Zinn‐Justin, gL(Tc) = −3 + Γ4(1/4)/(8 π2) ≈ −0.812. In the present work, we try to shed light on this controversy obtaining much more precise Monte Carlo data using multihistogram techniques and lattices as large as L = 17 (i.e., 1419857 Ising spins). While our value of Tc agrees nicely with recent high temperature series work (KBTc/J ≈ 8.7774 ± 0.0035), the coupling at Tc {gL(Tc) ≈ −0.958 ± 0.050} confirms the work for small lattices and disagrees with the analytical prediction. Hence the data are better consistent with a shift of the effective critical temperature Tc(L) ‐ Tc(∝) ∝ L−d/2 rather than with L−(d−2) according to the analytical theory. If the latter behavior is correct, it can hence be seen for extremely large systems only.
The validity of finite-size scaling in the presence of an inhomogeneous external field vanishing in the thermodynamic-limit is studied using a fully finite three-dimensional mean spherical model. The external field is chosen to change sign stepwise in one space dimension and to be translationally invariant in the other two dimensions, in which the lattice is assumed periodic. The boundary conditions in the direction of broken translational invariance are (i) periodic, and (ii) free, and (iii) fixed. Exact expressions for the magnetization profile are derived and studied. An extended, coordinate-dependent finite-size scaling is found to hold near the shifted critical temperature. Different scaling forms hold near the bulk critical temperature: in case (ii) the distance from the boundary scales with the finite-size correlation length, and in case (iii) with the linear size of the system.