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Physical review. B, Solid state

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

Fredric Harbus, Eugene Stanley

Published Aug 1, 1973• 41 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.8.1156

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We investigate the Ising model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions on the square and body-centered-cubic lattices. For $0<\frac{{J}_{1}}{{J}_{2}}<\frac{3}{2}$ on the bcc lattice the model has two nontrivially degenerate ground states and has been conjectured to have $\mathrm{XY}$ ($n=2$) critical exponents. For $\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}$ there are three degenerate ground states and Heisenberg critical exponents ($n=3$) are expected. We have obtained nine terms in the high-temperature expansion of the staggered susceptibility. For $0<\frac{{J}_{1}}{{J}_{2}}<~1$ we find $\ensuremath{\gamma}=1.25$ ($n=1$). At $\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}$ the results are not conclusive, but $\ensuremath{\gamma}=1.33$ ($n=2$) is not ruled out by the series analysis whereas $\ensuremath{\gamma}=1.4$ ($n=3$) is not probable. Results for the square lattice are inconclusive.

Physical review. B, Condensed matter•Jan 1, 1979•16 citations
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 \ensuremathχst{\ensuremath{\chi}}_{\mathrm{st}}\ensuremathχst​ appears to diverge along the critical line in the H\ensuremath−TH\ensuremath{-}TH\ensuremath−T plane with a constant exponent \ensuremathγst=54{\ensuremath{\gamma}}_{\mathrm{st}}=\frac{5}{4}\ensuremathγst​=45​, consistent with the universality hypothesis. However, in contrast to the meta model, it is found that the direct susceptibility \ensuremathχ\ensuremath{\chi}\ensuremathχ diverges at the tricritical point with an exponent \ensuremathγ‾\ensuremath≃14\overline{\ensuremath{\gamma}}\ensuremath{\simeq}\frac{1}{4}\ensuremathγ​\ensuremath≃41​. 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.
High-temperature series expansions are used to examine the dependence of critical-point exponents upon the presence of second-neighbor interactions. We consider the Hamiltonian ${\mathcal{H}}_{\mathrm{nnn}}=\ensuremath{-}{J}_{1}\ensuremath{\Sigma}\stackrel{\mathrm{nn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)}\ensuremath{-}{J}_{2}\ensuremath{\Sigma}\stackrel{\mathrm{nnn}}{〈\mathrm{ij}〉}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}^{(D)}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{j}^{(D)},$ where the first and second sums are over pairs of nearest-neighbor (nn) and next-nearest-neighbor (nnn) sites, and where the spins ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}^{(D)}$ are DDD-dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in 1kBT\frac{1}{{k}_{B}T}kB​T1​ for the Ising (D=1D=1D=1), classical-planar (D=2D=2D=2), and classical-Heisenberg (D=3D=3D=3) models, respectively, for various values of the parameter ${R}^{\ensuremath{'}}\ensuremath{\equiv}\frac{{J}_{2}}{{J}_{1}}$ and for various cubic lattices (fcc, bcc, and sample cubic). These represent the first series expansions of the spin correlation function for nnn interactions. From ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ we obtain series for the specific heat, susceptibility, and second moment. Analysis of these series and detailed comparisons with the exactly soluble spherical model ($D=\ensuremath{\infty}$) lead us to conclude that the exponents $\ensuremath{\gamma}$ (susceptibility) and $\ensuremath{\nu}$ (correlation length) may be independent of ${R}^{\ensuremath{'}}$; this suggestion is consistent with the universality hypothesis.
Monte Carlo data for a simple cubic Ising antiferromagnet with nearest- and next-nearest-neighbor interactions reveal asymptotic tricritical behavior of the order parameter which is mean-field-like modified by logarithmic corrections and high-temperature susceptibilities which are mean-field-like without corrections, in agreement with renormalization-group calculations. Crossover between tricritical and critical behavior is observed in the temperature variation of the order parameter and high-temperature susceptibility.
The tricritical behavior of a model studied by Harbus and Stanley is reanalyzed. The Harbus-Stanley analysis of the direct susceptibility suggested anomalous (i.e., non-mean-field-like) tricritical exponents. The tricritical point cannot be unambiguously located by present series data. We point out that, if the true tricritical temperature were actually somewhat lower than the Harbus-Stanley value, then standard ratio analysis would give tricritical exponents consistent with the Gaussian-tricritical-fixed-point analysis of Riedel and Wegner. However, the present situation remains inconclusive.
We investigate the Ising model with nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic interactions on the square and body-centered-cubic lattices. For 0<J1J2<320<\frac{{J}_{1}}{{J}_{2}}<\frac{3}{2}0<J2​J1​​<23​ on the bcc lattice the model has two nontrivially degenerate ground states and has been conjectured to have XY\mathrm{XY}XY (n=2n=2n=2) critical exponents. For J1J2=32\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}J2​J1​​=23​ there are three degenerate ground states and Heisenberg critical exponents (n=3n=3n=3) are expected. We have obtained nine terms in the high-temperature expansion of the staggered susceptibility. For 0<J1J2< 10<\frac{{J}_{1}}{{J}_{2}}<~10<J2​J1​​< 1 we find $\ensuremath{\gamma}=1.25$ (n=1n=1n=1). At J1J2=32\frac{{J}_{1}}{{J}_{2}}=\frac{3}{2}J2​J1​​=23​ the results are not conclusive, but $\ensuremath{\gamma}=1.33$ (n=2n=2n=2) is not ruled out by the series analysis whereas $\ensuremath{\gamma}=1.4$ (n=3n=3n=3) is not probable. Results for the square lattice are inconclusive.