Physical review. B, Solid state

Ising Model with First-, Second-, and Third-Neighbor Interactions

Joel Philhours

Published Aug 1, 1971• 5 citations

Abstract

The high-temperature expansion of the partition function of an Ising model in zero magnetic field is calculated through terms of ${T}^{\ensuremath{-}9}$ for a fcc lattice and variable first-, second-, and third-neighbor interactions (${J}_{1}$, ${J}_{2}$, and ${J}_{3}$). Values of the transition temperature ${T}_{c}$ and of the first-, second-, and third-neighbor correlation coefficients at $T={T}_{c}$ are given for $\frac{{J}_{2}}{{J}_{1}}$ and $\frac{{J}_{3}}{{J}_{1}}$ each varying between 0 and 2.

DOI: 10.1103/physrevb.4.929

Related papers

Critical Point Behavior of the Ising Model with Higher-Neighbor Interactions Present

N W Dalton, David W. Wood

Journal of Mathematical PhysicsJul 1, 1969128 citations

LOW-TEMPERATURE SERIES EXPANSIONS FOR SQUARE-LATTICE ISING MODEL WITH FIRST AND SECOND NEIGHBOUR INTERACTIONS

Yee-Mou Kao, MALL CHEN, Keh Ying Lin

International Journal of Modern Physics BDec 20, 20020 citations

First-order phase transitions in the Ising square lattice with first- and second-neighbor interactions

José Luis Morán-López, F. Aguilera‐Granja, J. M. Sánchez

Physical review. B, Condensed matterAug 1, 199378 citations

Ising models with n>1 : A series-expansion approach

Michael Plischke, J. Oitmaa

Physical review. B, Condensed matterJan 1, 197916 citations

Extension of the High Temperature Susceptibility Expansion for General Ising Systems

Agoes Soehianie, J. Oitmaa

We have obtained the graphs and weights for the twelfth and thirteenth terms of the high temperature multigraph expansion for the susceptibility of general Ising systems. As applications we have computed the coefficients through thirteenth order for the square lattice with first and second neighbor interactions and first and third neighbor interactions.

Modern Physics Letters BJun 20, 19974 citations

Partial Test of the Universality Hypothesis: The Case of Next-Nearest-Neighbor Interactions

Gerald Paul, Eugene Stanley

Physical review. B, Solid stateMay 1, 197230 citations

Physical review. B, Solid state

Ising Model with First-, Second-, and Third-Neighbor Interactions

Joel Philhours

Published Aug 1, 1971• 5 citations

Publisher

Abstract

The high-temperature expansion of the partition function of an Ising model in zero magnetic field is calculated through terms of

{T}^{\ensuremath{-}9}

for a fcc lattice and variable first-, second-, and third-neighbor interactions (

{J}_{1}

{J}_{2}

, and

{J}_{3}

). Values of the transition temperature

{T}_{c}

and of the first-, second-, and third-neighbor correlation coefficients at

T={T}_{c}

are given for

\frac{{J}_{2}}{{J}_{1}}

and

\frac{{J}_{3}}{{J}_{1}}

each varying between 0 and 2.

DOI: 10.1103/physrevb.4.929

Related papers

Critical Point Behavior of the Ising Model with Higher-Neighbor Interactions Present

N W Dalton, David W. Wood

The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.The method of developing exact power-series expansions for the partition function ZN and related thermodynamic functions for the Ising model valid below the critical point is generalized to include exchange interactions between first-, second-, and third-neighbor pairs. Expansions of the spontaneous magnetization M0(T) and zero field susceptibility χ0(T) are derived through to sixth order of perturbation for the s.q. lattice, and through to fifth order of perturbation for the Δ′r, b.c.c., s.c., and f.c.c. lattices, when interactions J1SizSjz and J2SkzSlz are present between first- and second-neighbor spins, respectively (second-neighbor model). These expansions have also been obtained for the case where interactions of equal magnitude (J1 = J2 = J3) are present between first-, second-, and third-neighbor pairs (third-equivalent-neighbor model); here expansions through to fifth order of perturbation are obtained for the s.q., Δ′r, b.c.c., and s.c. lattices and through to fourth order for the f.c.c. lattice. The Padé approximant·procedure is employed to discuss the effects of an extended but finite range of interaction on the behavior of M0(T) and χ0(T) for T→Tc− as characterized by the critical exponents β and γ′, respectively. For the second-equivalent-neighbor model lattices, it is found that 0.122 ≤ β ≤ 0.134 in two dimensions, and that 0.308 ≤ β ≤ 0.328 in three dimensions; from which it is concluded that β remains unchanged from its value in the nearest-neighbor model. The corresponding limits for γ′ in three dimensions are 1.18 ≤ γ′ ≤ 1.28; from this and the results for the b.c.c. lattice in particular, it is concluded that γ′ is probably 54 and hence the transition in χ0 is symmetrical about Tc (γ′ = γ). A repetition of this analysis for the third-equivalent-neighbor model three-dimensional lattices shows a marked shift in the estimated range of β and γ′; the results are 0.345 ≤ β ≤ 0.365, and 1.01 ≤ γ′ ≤ 1.14. In each of the above cases, the corresponding high-temperature (T > Tc) expansions of χ0(T) obtained previously have been analyzed to yield estimates of the critical exponent γ. The over-all results and in particular the estimates of γ for the s.q. and b.c.c. lattices suggest that this index is unaffected by extending the range of interaction, and that if γ is a rational fraction then it is the same fraction for the n.n. model and second- and third-equivalent neighbor models. Finally the high-temperature expansions of ZN in zero field, and of χ0(T) for the second-neighbor model are used to examine the dependence of the critical temperature Tc, the critical energy (E∞ − Ec)/kTc, and the critical entropy (S∞ − Sc)/k on the relative strengths of J1 and J2 for values of J2/J1 in the range 0 to 1. It is found that the variation of the critical point is well represented by Tc(α)=Tc(0)[1+m1α],where α = J2/J1 and lies in the range 0 ≤ α ≤ 1; and Tc(0) is the critical temperature of the nearest-neighbor model. The values of m1 are 0.61, 2.47, 0.84, 1.45, and 1.35 for the f.c.c., s.c., b.c.c., s.q., and Δ′r lattices, respectively. All these calculations are compared with the corresponding results for the Heisenberg model.

Journal of Mathematical PhysicsJul 1, 1969128 citations

LOW-TEMPERATURE SERIES EXPANSIONS FOR SQUARE-LATTICE ISING MODEL WITH FIRST AND SECOND NEIGHBOUR INTERACTIONS

Yee-Mou Kao, MALL CHEN, Keh Ying Lin

We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.We have calculated the low-temperature series expansions of the spontaneous magnetization and the zero-field susceptibility of the square-lattice ferromagnetic Ising model with first-neighbour interaction J1 and second-neighbour interaction J2 to the 30th and 26th order respectively by computer. Our results extend the previous calculations by Lee and Lin to six more orders. We use the Padé approximants to estimate the critical exponents and the critical temperature for different ratios of R = J2/J1. The estimated critical temperature as a function of R agrees with the estimation by Oitmaa from high-temperature series expansions.

International Journal of Modern Physics BDec 20, 20020 citations

First-order phase transitions in the Ising square lattice with first- and second-neighbor interactions

José Luis Morán-López, F. Aguilera‐Granja, J. M. Sánchez

The phase diagram of the Ising square lattice with first- and second-nearest-neighbor antiferromagnetic interactions ( ${\mathit{J}}_{1}$ and ${\mathit{J}}_{2}$ ) is calculated within the framework of the cluster-variation method. The Hamiltonian is solved within the nine-point-cluster approximation. In contrast to previous studies we find that the phase transition from the superantiferromagnetic to the paramagnetic state is of first order in the range 0.5R= ${\mathit{J}}_{1}$ /${\mathit{J}}_{2}$1.144. The non-Ising critical behavior observed in that range of interactions might be due to that phenomenon.The phase diagram of the Ising square lattice with first- and second-nearest-neighbor antiferromagnetic interactions (${\mathit{J}}_{1}$ and ${\mathit{J}}_{2}$) is calculated within the framework of the cluster-variation method. The Hamiltonian is solved within the nine-point-cluster approximation. In contrast to previous studies we find that the phase transition from the superantiferromagnetic to the paramagnetic state is of first order in the range 0.5R=${\mathit{J}}_{1}$/${\mathit{J}}_{2}$1.144. The non-Ising critical behavior observed in that range of interactions might be due to that phenomenon.

Physical review. B, Condensed matterAug 1, 199378 citations

Ising models with n>1 : A series-expansion approach

Michael Plischke, J. Oitmaa

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.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 matterJan 1, 197916 citations

Extension of the High Temperature Susceptibility Expansion for General Ising Systems

Agoes Soehianie, J. Oitmaa

Modern Physics Letters BJun 20, 19974 citations

Partial Test of the Universality Hypothesis: The Case of Next-Nearest-Neighbor Interactions

Gerald Paul, Eugene Stanley

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 $D$ -dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in $\frac{1}{{k}_{B}T}$ for the Ising ( $D=1$ ), classical-planar ( $D=2$ ), and classical-Heisenberg ( $D=3$ ) 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.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 $D$-dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in $\frac{1}{{k}_{B}T}$ for the Ising ($D=1$), classical-planar ($D=2$), and classical-Heisenberg ($D=3$) 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.

Physical review. B, Solid stateMay 1, 197230 citations

The high-temperature expansion of the partition function of an Ising model in zero magnetic field is calculated through terms of

{T}^{\ensuremath{-}9}

for a fcc lattice and variable first-, second-, and third-neighbor interactions (

{J}_{1}

{J}_{2}

, and

{J}_{3}

). Values of the transition temperature

{T}_{c}

and of the first-, second-, and third-neighbor correlation coefficients at

T={T}_{c}

are given for

\frac{{J}_{2}}{{J}_{1}}

and

\frac{{J}_{3}}{{J}_{1}}

each varying between 0 and 2.

The phase diagram of the Ising square lattice with first- and second-nearest-neighbor antiferromagnetic interactions (

{\mathit{J}}_{1}

and

{\mathit{J}}_{2}

) is calculated within the framework of the cluster-variation method. The Hamiltonian is solved within the nine-point-cluster approximation. In contrast to previous studies we find that the phase transition from the superantiferromagnetic to the paramagnetic state is of first order in the range 0.5R=

{\mathit{J}}_{1}

/${\mathit{J}}_{2}$1.144. The non-Ising critical behavior observed in that range of interactions might be due to that phenomenon.

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.

D

-dimensional unit vectors. The two-spin correlation function, ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is calculated to tenth, ninth, and eighth order in

\frac{1}{{k}_{B}T}

for the Ising (

D=1

), classical-planar (

D=2

), and classical-Heisenberg (

D=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.