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Journal of the Physical Society of Japan

Low-Temperature Behavior of Antiferromagnetic Ising Model on Triangular Lattice

Tsuyoshi Horiguchi, Kazuyuki Tanaka, Tohru Morita

Published Jan 15, 1992• 12 citations
Publisher

Abstract

We derive low-temperature expansions of the internal energy, the specific heat and so on for the antiferromagnetic Ising model on a triangular lattice by using the expression obtained by Stephenson. We propose to use Δ T = exp(-2β J ) as the reduced temperature variable near the critical temperature. We then conclude the values of the critical exponents α, ν, γ, δ and Δ as α=0, ν=1, γ=3/2, δ=7 and Δ=7/4.

DOI
10.1143/jpsj.61.64

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We derive low-temperature expansions of the internal energy, the specific heat and so on for the antiferromagnetic Ising model on a triangular lattice by using the expression obtained by Stephenson. We propose to use Δ T = exp(-2β J ) as the reduced temperature variable near the critical temperature. We then conclude the values of the critical exponents α, ν, γ, δ and Δ as α=0, ν=1, γ=3/2, δ=7 and Δ=7/4.
Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .
Low temperature series expansions for the spin S = 1 Ising model on various two and three dimensional lattices have been generated. By extrapolation of these series, it is found that the critical exponents β, γ' and δ appear to be the same as for the spin S = 1/2 Ising model. For the spin S = 1/2 Ising model, we have found that the class of configurations represented by Cayley trees make a significant contribution to the low temperature series coefficients. In particular, we find that the distribution of singularities within the disk | u | ≤ |uc |(uc = exp[- 4 J/kTc], Tc is the critical temperature) is well represented by including only the Cayley trees in the partition function.
The phase transitions (PTs) and critical properties of the antiferromagnetic Ising model on a layered (stacked) triangular lattice have been studied by the Monte Carlo method using a replica algorithm with allowance for the next-nearest-neighbor interactions. The character of PTs is analyzed using the histogram technique and the method of Binder cumulants. It is established that the transition from the disordered to paramagnetic phase in the adopted model is a second-order PT. Static critical exponents of the heat capacity (α), susceptibility (γ), order parameter (β), and correlation radius (ν) and the Fischer exponent η are calculated using the finite-size scaling theory. It is shown that (i) the antiferromagnetic Ising model on a layered triangular lattice belongs to the XY universality class of critical behavior and (ii) allowance for the intralayer interactions of next-nearest neighbors in the adopted model leads to a change in the universality class of critical behavior.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.
Low temperature Ising model series in the usual variable z = exp(−4 J/kT ) are analysed. The positions of the singularities in the disk|z| <or=|z c | and the values of the corresponding critical exponents are determined for the simple cubic, body-centred cubic and face-centred cubic lattices. We obtain the following estimates of the physical critical exponents: β = 0·312 ± 0·002 for the simple cubic and face-centred cubic lattices, and β = 0·312 ± 0·004 for the body-centred cubic lattice. γprime = 1·31 ± 0·02 for the simple cubic lattice, γprime = 1·28 ± 0·04 for the body-centred cubic lattice, and γprime = 1·3 ± 0·1 for the face-centred cubic lattice. For the low temperature specific heat exponent we obtain fraction one-sixteens <or= αprime <or= ¼.