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Journal of Applied Physics

Systematics of d-dimensional hypercubic spin-1/2 Ising ferromagnets: Critical temperature and critical energy

K. Nath, B. Frank

Published Nov 1, 1982• 8 citations
Publisher

Abstract

The critical temperatures Kc and the nearest-neighbour correlation functions Uc have been calculated for the d-dimensional hypercubic Ising ferromagnet for d=4 to d=30, by generalizing a recent theory reported for d=3. The results are of relevance for Monte Carlo Renormalization Group work involving critical properties in random fields, which calls for a knowledge of Kc in the limit of zero field. As d becomes large, Kc and Uc are found to approach the spherical model values. Polynomial expressions in 1/d are fitted to the calculated values, using the analogous spherical model expressions as guide. Comparison is made with the corresponding expressions of Fisher and Gaunt. Uc/(1−Kc), independent of d for the spherical model, is investigated; it is seen to be approximately independent of lattice structure for d=2 and for d=3.

DOI
10.1063/1.330245

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The critical temperatures Kc and the nearest-neighbour correlation functions Uc have been calculated for the d-dimensional hypercubic Ising ferromagnet for d=4 to d=30, by generalizing a recent theory reported for d=3. The results are of relevance for Monte Carlo Renormalization Group work involving critical properties in random fields, which calls for a knowledge of Kc in the limit of zero field. As d becomes large, Kc and Uc are found to approach the spherical model values. Polynomial expressions in 1/d are fitted to the calculated values, using the analogous spherical model expressions as guide. Comparison is made with the corresponding expressions of Fisher and Gaunt. Uc/(1−Kc), independent of d for the spherical model, is investigated; it is seen to be approximately independent of lattice structure for d=2 and for d=3.
THe authors investigate the spin- 1 / 2 Ising model with nearest-neighbour interactions on the four-dimensional simple hypercubic lattice. High-temperature series expansions are studied for the zero-field susceptibility chi 0 and the fourth-field derivative of the free energy Xi 0 (2) up to order nu 17 . The series are analysed for singularities of the form t -1 mod 1nt mod p where t is the reduced temperature. For chi 0 it is found that p=0.33+or-0.07 when q=1, in good agreement with the prediction p= 1 / 3 , q=1 of renormalisation group theory. The critical temperature is estimated to be nu c -1 =6.7315+or-0.0015. Results for chi 0 (2) are more slowly convergent but are not inconsistent with the renormalisation group prediction p= 1 / 3 , q=4.
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.
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.
The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic ddd-dimensional lattice for all integral values of z=2dz={2}^{d}z=2d between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.
The critical properties of the spin- 1 / 2 Ising ferromagnet on a diamond lattice are calculated using the Monte Carlo technique. The heat capacity, the susceptibility, the magnetisation, and various pair and multispin correlation functions are calculated as functions of the temperature. The critical temperature, and the critical parameters pertaining to the magnetisation, are determined. The results are consistent with series analyses. Using the critical values of the correlation functions, it is shown that the basic assumption in the theory of Frank and Mitran (1978) is not exact for the diamond lattice.