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Journal of Physics A Mathematical Nuclear and General

Critical temperature of finite systems in d dimensions

Cyril Domb

Published Sep 1, 1973• 53 citations
Publisher

Abstract

The critical temperature of an Ising system consisting of n(d-1) dimensional layers in d dimensions is estimated for large n from the properties of random and self-avoiding walks (SAW) in the finite system. Denoting the deviation of T c (n) from T c ( infinity ) by 1/n lambda it is found that there are two contributions, a finite size effect which applies both to a torus and a strip and for which lambda =d-2+ eta , and a free boundary effect which applies to a strip alone and for which lambda =1/ nu d . ( nu d , eta are exponents associated with the spin pair correlation function using the standard notation of Fisher). The results are based on a perturbation expansion for SAW which should be reliable for d>or=4, but which needs further examination when d=3. Some, but not all, of the above features are in agreement with the results of Fisher and Barber for the spherical model.

DOI
10.1088/0305-4470/6/9/005

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The critical temperature of an Ising system consisting of n(d-1) dimensional layers in d dimensions is estimated for large n from the properties of random and self-avoiding walks (SAW) in the finite system. Denoting the deviation of T c (n) from T c ( infinity ) by 1/n lambda it is found that there are two contributions, a finite size effect which applies both to a torus and a strip and for which lambda =d-2+ eta , and a free boundary effect which applies to a strip alone and for which lambda =1/ nu d . ( nu d , eta are exponents associated with the spin pair correlation function using the standard notation of Fisher). The results are based on a perturbation expansion for SAW which should be reliable for d>or=4, but which needs further examination when d=3. Some, but not all, of the above features are in agreement with the results of Fisher and Barber for the spherical model.
The critical temperature T c (b) of a d-dimensional system with finite thickness b in one of its dimensions approaches its bulk limit as b to infinity according to T c ( infinity )-T c (b) varies as b - lambda . To first order in epsilon =4-d the shift exponent lambda is found to be given by lambda =2-((n+2)/(n+8)) epsilon +O( epsilon 2 ), and thus satisfies the scaling relation lambda = nu d -1 , where nu d is the correlation length exponent of the bulk system and n is the order parameter dimensionality.
Using recent results by Cardy (1984) based on conformal invariance of critical correlation functions the authors calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular, elliptical and rectangular shapes and free boundary conditions. Results show the effects of critical order on these quantities. For a circle, S(k) decays as 1/k 2- eta for a large range of intermediate k values with an 'apparent' exponent eta =0.09. The probable influence of end, edge and domain wall effects in the rectangular geometry is discussed. Application of the results to experimental systems and other theoretical models is discussed.
Fisher's rigorous bound chi < chi saw , where chi saw is the susceptibility calculated from the self-avoiding walk (SAW) generating function, is applied to (i) the ferromagnetic Ising alloy, (ii) the dilute Ising ferromagnet, (iii) the layer susceptibility of a free surface, (iv) the Ising ferromagnet in a transverse field, and (v) the dilute Ising ferromagnet in a transverse field. The author obtains an upper bound T UB for the temperature T 0 below which a spontaneous magnetization occurs. The use of a microscopically varying molecule field is analysed. A heuristic argument is given to show that, for the dilute Ising ferromagnet in a transverse field, the critical transverse field at zero temperature jumps from zero to a value at least equal to that of the one-dimensional ground state as x passes through the critical percolation concentration.
A classical n-vector model is considered which is infinitely extended in d-1 dimensions and periodic of period b in the dth dimension. Asymptotically for large values of b the critical temperature is expected to obey a scaling law T c (b)-T c ( infinity ) approximately Ab - lambda . The authors have calculated the shift exponent lambda to first order in a 1/n expansion for dimensions between three and four. To this order the result is consistent with the assertion lambda = nu d -1 where nu d is the correlation length exponent of the infinitely extended d-dimensional system. From the equivalence between this model and (d-1)-dimensional quantum mechanical systems the behaviour of critical lines near the displacive limit (T c =0) can be derived.
We present a unified view of finite-size scaling (FSS) in dimension d above the upper critical dimension, for both free and periodic boundary conditions. We find that the modified FSS proposed some time ago to allow for violation of hyperscaling due to a dangerous irrelevant variable applies only to k=0 fluctuations, and "standard" FSS applies to k≠0 fluctuations. Hence the exponent η describing power-law decay of correlations at criticality is unambiguously η=0. With free boundary conditions, the finite-size "shift" is greater than the rounding. Nonetheless, using T-T(L), where T(L) is the finite-size pseudocritical temperature, rather than T-T(c), as the scaling variable, the data do collapse onto a scaling form that includes the behavior both at T(L), where the susceptibility χ diverges like L(d/2), and at the bulk T(c), where it diverges like L(2). These claims are supported by large-scale simulations on the five-dimensional Ising model.