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Journal of Physics A Mathematical and General

The critical isotherm of the four-dimensional Ising model

Sati McKenzie, M F Sykes, David S. Gaunt

Published May 1, 1979• 10 citations
Publisher

Abstract

The critical isotherm of the Ising model for the four-dimensional simple hypercubic lattice is studied using a high-field expansion through mu 15 . The series for the magnetisation is analysed for singularities of the asymptotic form E(1- mu ) 1/3 mod ln(1- mu ) mod p , predicted by renormalisation group theory. Good convergence is obtained for values of p around 1 / 3 (the renormalisation group prediction) and we estimate p=0.30+or-0.05. Assuming p= 1 / 3 , the critical amplitude E is estimated to be 0.896+or-0.02.

DOI
10.1088/0305-4470/12/5/020

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The critical isotherm of the Ising model for the four-dimensional simple hypercubic lattice is studied using a high-field expansion through mu 15 . The series for the magnetisation is analysed for singularities of the asymptotic form E(1- mu ) 1/3 mod ln(1- mu ) mod p , predicted by renormalisation group theory. Good convergence is obtained for values of p around 1 / 3 (the renormalisation group prediction) and we estimate p=0.30+or-0.05. Assuming p= 1 / 3 , the critical amplitude E is estimated to be 0.896+or-0.02.
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.
Using extended high-field series expansions and more precise estimates of the critical temperature the critical isotherm of the spin 1 / 2 Ising ferromagnet is re-analysed. The conjectures delta =15 (d=2) and delta =5 (d=3) are confirmed to within 1 / 2 % and 1%, respectively. Good agreement is found, particularly in two dimensions, between series estimates of the critical amplitudes and the predictions of the lattice-lattice scaling hypothesis. Some tentative conclusions regarding the coincident and nonphysical singularities are drawn.
The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K 9 for the hyper face centred (HFC) lattice, to order K 11 for the hyper body centred cubic (HBCC) lattice and to order K 12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t -1 mod In t mod p , predicted by the renormalisation group theory (t=1-K c /K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K9 for the hyper face centred (HFC) lattice, to order K11 for the hyper body centred cubic (HBCC) lattice and to order K12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t-1 mod In t mod p, predicted by the renormalisation group theory (t=1-Kc/K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.
The basic low temperature series expansion data for the spin one-half Ising model on the hydrogen peroxide lattice are used to obtain series in z=exp(-2J/k B T) along the coexistence curve for the specific heat C H , the magnetization M, and its first five derivatives delta 1 M/ delta mu 1 , where mu =exp(-2mH/k B T). The same data are used to derive series in mu along the critical isotherm for M and its first five derivatives delta l M/ delta z l . Ratio and Pade approximant analysis yield estimates of the critical exponents and critical amplitudes. On the whole the estimates of the critical exponents support scaling theory although a few of the exponent estimates are not in good agreement with scaling.