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Journal of Physics A Mathematical and General

Critical behaviour of a model with NNN interaction

Wei Fan, Bo-Wei Xu

Published Sep 7, 1990• 9 citations
Publisher

Abstract

A one-dimensional quantum model with next-nearest-neighbour interaction is studied. Under the long-wavelength approximation, the authors predict that this model belongs to the same universality class as the Ising model. This conclusion is checked with a numerical solution which is consistent with the prediction in both critical point and critical exponent.

DOI
10.1088/0305-4470/23/17/012

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High-Temperature-Series Study of Models Displaying Tricritical Behavior. II. A Nearest-Neighbor Ising Antiferromagnet with Next-Nearest-Neighbor Ferromagnetic Interactions

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The nnn model is an Ising model with nearest-neighbor antiferromagnetic interactions (${J}_{1}<0$) but also next-nearest-neighbor ferromagnetic exchange (${J}_{2}>0$). This model is analyzed in external magnetic field using the same techniques as applied to the meta model of Paper I. Again, the staggered susceptibility ${\ensuremath{\chi}}_{\mathrm{st}}$ appears to diverge along the critical line in the $H\ensuremath{-}T$ plane with a constant exponent ${\ensuremath{\gamma}}_{\mathrm{st}}=\frac{5}{4}$, consistent with the universality hypothesis. However, in contrast to the meta model, it is found that the direct susceptibility $\ensuremath{\chi}$ diverges at the tricritical point with an exponent $\overline{\ensuremath{\gamma}}\ensuremath{\simeq}\frac{1}{4}$. Implications of the scaling hypothesis at the tricritical point are discussed and the results for both the meta and nnn models are utilized to obtain the scaling power ${\overline{a}}_{2}$ corresponding to the "weak" direction (in the sense of Griffiths and Wheeler). Included in this discussion is the double-power-law form, predicted to hold within the crossover region by tricritical-point scaling.

Physical review. B, Solid state•Aug 1, 1973•41 citations
A one-dimensional quantum model with next-nearest-neighbour interaction is studied. Under the long-wavelength approximation, the authors predict that this model belongs to the same universality class as the Ising model. This conclusion is checked with a numerical solution which is consistent with the prediction in both critical point and critical exponent.
The critical behaviour of the Ising model on the square lattice with ferromagnetic nearest-neighbour interactions (J) and anti-ferromagnetic next-nearest neighbour interactions (J') is discussed for small lambda =J/ mod J' mod . The singular part of the free energy is calculated to second order in lambda by perturbation theory. Near the critical temperature of the unperturbed system (J=0), this expansion is found to have a form which may be exponentiated yielding, for non-zero lambda , a non-universal critical line along which the exponents vary continuously with lambda . For small lambda , the specific heat exponent, alpha = alpha 1 lambda 2 +O( lambda 2 ) with alpha 1 approximately=1.5.
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.
The nnn model is an Ising model with nearest-neighbor antiferromagnetic interactions (J1<0{J}_{1}<0J1​<0) but also next-nearest-neighbor ferromagnetic exchange (J2>0{J}_{2}>0J2​>0). This model is analyzed in external magnetic field using the same techniques as applied to the meta model of Paper I. Again, the staggered susceptibility ${\ensuremath{\chi}}_{\mathrm{st}}$ appears to diverge along the critical line in the $H\ensuremath{-}T$ plane with a constant exponent ${\ensuremath{\gamma}}_{\mathrm{st}}=\frac{5}{4}$, consistent with the universality hypothesis. However, in contrast to the meta model, it is found that the direct susceptibility $\ensuremath{\chi}$ diverges at the tricritical point with an exponent $\overline{\ensuremath{\gamma}}\ensuremath{\simeq}\frac{1}{4}$. Implications of the scaling hypothesis at the tricritical point are discussed and the results for both the meta and nnn models are utilized to obtain the scaling power a‾2{\overline{a}}_{2}a2​ corresponding to the "weak" direction (in the sense of Griffiths and Wheeler). Included in this discussion is the double-power-law form, predicted to hold within the crossover region by tricritical-point scaling.