Physical review. B, Solid state

Spin- S Ising susceptibility on the triangular lattice

John P. VAN DYKE, William J. Camp

Published Aug 1, 1975• 2 citations

Abstract

The high-temperature susceptibility of the spin-$S$ Ising model on the triangular net is analyzed as a function of $S$ using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.

DOI: 10.1103/physrevb.12.1084

Related papers

High-temperature series for the susceptibility of the spin- s Ising model: Analysis of confluent singularities

William J. Camp, John P. VAN DYKE

Physical review. B, Solid stateApr 1, 1975148 citations

Complex-temperature singularities of the susceptibility in the D=2 Ising model. I. Square lattice

V Matveev, Robert E. Shrock

arXiv:hep-lat/9408020Journal of Physics A Mathematical and GeneralMar 23, 199563 citations

High-temperature series for the susceptibility of the spin- S Ising model

John P. VAN DYKE, William J. Camp

Physical review. B, Solid stateApr 1, 197442 citations

Complex-temperature singularities in thed= 2 Ising model: triangular and honeycomb lattices

Victor Matveev, Robert E. Shrock

arXiv:hep-lat/9411023Journal of Physics A Mathematical and GeneralFeb 21, 199641 citations

Ising model susceptibility amplitudes. I. Two-dimensional lattices

Anthony John Guttmann

Journal of Physics A Mathematical and GeneralAug 1, 197528 citations

Renormalization-group study of the ferromagnetic Ising model on the triangular lattice

Chris Unger

Physical review. B, Condensed matterAug 1, 19840 citations

Physical review. B, Solid state

Spin- S Ising susceptibility on the triangular lattice

John P. VAN DYKE, William J. Camp

Published Aug 1, 1975• 2 citations

Publisher

Abstract

The high-temperature susceptibility of the spin-

S

Ising model on the triangular net is analyzed as a function of

S

using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value

\ensuremath{\gamma}=\frac{7}{4}

for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.

DOI: 10.1103/physrevb.12.1084

Related papers

High-temperature series for the susceptibility of the spin- s Ising model: Analysis of confluent singularities

William J. Camp, John P. VAN DYKE

We have extended the series for the zero-field susceptibility of the spin- $s$ Ising model through tenth order in the reduced inverse temperature $K$ on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients ${h}_{n}(s)$ are expressed as simple polynomials in $X=s(s+1)$ . Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of $s$ . The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For $s=\frac{1}{2}$ only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of $s$ , with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.We have extended the series for the zero-field susceptibility of the spin-$s$ Ising model through tenth order in the reduced inverse temperature $K$ on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients ${h}_{n}(s)$ are expressed as simple polynomials in $X=s(s+1)$. Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of $s$. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For $s=\frac{1}{2}$ only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of $s$, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.

Physical review. B, Solid stateApr 1, 1975148 citations

Complex-temperature singularities of the susceptibility in the D=2 Ising model. I. Square lattice

V Matveev, Robert E. Shrock

We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.

arXiv:hep-lat/9408020Journal of Physics A Mathematical and GeneralMar 23, 199563 citations

High-temperature series for the susceptibility of the spin- S Ising model

John P. VAN DYKE, William J. Camp

We have extended the series for the zero-field susceptibility of the spin- $S$ Ising model to eighth order in the reduced temperature $K$ , on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series ${h}_{n}(S)$ are expressed as simple polynomials in $X=S(S+1)$ . For the face-centered-cubic lattice, an accurate polynomial fit to the critical point ${K}_{c}(S)$ is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all $S$ . However, the large- $S$ series seems to exhibit more rapid apparent convergence.We have extended the series for the zero-field susceptibility of the spin-$S$ Ising model to eighth order in the reduced temperature $K$, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series ${h}_{n}(S)$ are expressed as simple polynomials in $X=S(S+1)$. For the face-centered-cubic lattice, an accurate polynomial fit to the critical point ${K}_{c}(S)$ is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all $S$. However, the large-$S$ series seems to exhibit more rapid apparent convergence.

Physical review. B, Solid stateApr 1, 197442 citations

Complex-temperature singularities in thed= 2 Ising model: triangular and honeycomb lattices

Victor Matveev, Robert E. Shrock

We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point (where ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities and both have divergent singularities at , and our numerical values for the exponents are consistent with the hypothesis that the exact values are . The critical amplitudes at this singularity were calculated. Using our exact results for and together with numerical values for from series analyses, we find that the exponent relation is violated at z = -1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point (where ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities and both have divergent singularities at , and our numerical values for the exponents are consistent with the hypothesis that the exact values are . The critical amplitudes at this singularity were calculated. Using our exact results for and together with numerical values for from series analyses, we find that the exponent relation is violated at z = -1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.

arXiv:hep-lat/9411023Journal of Physics A Mathematical and GeneralFeb 21, 199641 citations

Ising model susceptibility amplitudes. I. Two-dimensional lattices

Anthony John Guttmann

By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.

Journal of Physics A Mathematical and GeneralAug 1, 197528 citations

Renormalization-group study of the ferromagnetic Ising model on the triangular lattice

Chris Unger

The dynamic real-space renormalization group of Mazenko and Valls is applied to the zero-field ferromagnetic Ising model on the triangular lattice. Renormalization equations valid for all temperatures above the critical temperature ${T}_{c}$ are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for $T<{T}_{c}$ . The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.The dynamic real-space renormalization group of Mazenko and Valls is applied to the zero-field ferromagnetic Ising model on the triangular lattice. Renormalization equations valid for all temperatures above the critical temperature ${T}_{c}$ are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for $T<{T}_{c}$. The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.

Physical review. B, Condensed matterAug 1, 19840 citations

The high-temperature susceptibility of the spin-

S

Ising model on the triangular net is analyzed as a function of

S

\ensuremath{\gamma}=\frac{7}{4}

We have extended the series for the zero-field susceptibility of the spin-

s

Ising model through tenth order in the reduced inverse temperature

K

on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients

{h}_{n}(s)

are expressed as simple polynomials in

X=s(s+1)

. Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of

s

. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For

s=\frac{1}{2}

only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of

s

, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.

We have extended the series for the zero-field susceptibility of the spin-

S

Ising model to eighth order in the reduced temperature

K

, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series

{h}_{n}(S)

are expressed as simple polynomials in

X=S(S+1)

. For the face-centered-cubic lattice, an accurate polynomial fit to the critical point

{K}_{c}(S)

is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all

S

. However, the large-

S

series seems to exhibit more rapid apparent convergence.

{T}_{c}

are derived for the susceptibility, specific heat, structure factor, and correlation length. The magnetization is found for

T<{T}_{c}

. The critical exponents and amplitudes for these quantities are calculated. The agreement between the known static properties and the renormalization-group results is good to excellent, and shows that this renormalization-group method can accurately calculate nonuniversal, as well as universal, quantities on different lattices. The computed dynamic structure factor, however, exhibits nonmonotonic behavior as a function of temperature. This nonmonotonic behavior is conjectured to be due to approximations in determining the expansion parameters.