Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9604080 · Journal of Physics A Mathematical and General

Low-temperature series expansions for the square lattice Ising model with spin

Iwan Jensen, Anthony John Guttmann, Ian Graham Enting

Published Jul 21, 1996• 18 citations
arXivHTMLPDF

Abstract

We derive low-temperature series (in the variable ) for the spontaneous magnetization, susceptibility, and specific heat of the spin-S Ising model on the square lattice for , 2, , and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with S. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of S. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of S, though consistent with universality.

DOI
10.1088/0305-4470/29/14/009

Related papers

Low-temperature series expansions for the spin-1 Ising model

Ian Graham Enting, Anthony John Guttmann, Iwan Jensen

The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta 1 =1.1+or-0.1 is found.

arXiv:hep-lat/9410005•Journal of Physics A Mathematical and General•Nov 7, 1994•49 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.

arXiv:hep-lat/9408020•Journal of Physics A Mathematical and General•Mar 23, 1995•63 citations

Extrapolation procedure for low-temperature series for the square lattice spin-1 Ising model

Iwan Jensen, Anthony John Guttmann

The finite-lattice method of series expansions has been combined with a new extrapolation procedure to extend the low-temperature series for the specific heat, spontaneous magnetization, and susceptibility of the spin-1 Ising model on the square lattice. The extended series were derived by directly calculating the series to order 99 (in the variable and using the new extrapolation procedure to calculate an additional 13 - 14 terms.

arXiv:cond-mat/9604079•Journal of Physics A Mathematical and General•Jul 21, 1996•7 citations

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

V. Matveev, Robert E. Shrock

king model on a square lattice. From an analysis of low-temperature series expansions. we find evidence that, as one approaches the point U = us =- I (where U = e-") from within the complex extensions of the FM or AFM phases. the susceptibility has 3. divergent singularity of the farm,y- A:(l + u)-y: with exponent r: = i. The critical amplitude A: is calculated. Other critical exponents are found to be ci = cs = 0 and ps = $, so thY the scaling relation U: + 26, + y: = 2 is satisfied. However, using exact results for ps on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: ps is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length nnd hence susceptibility are finite GS one approaches the point I ( =-1 from within the symmetric phase. This is m n h e d by an explicit study of high-tempemture series expansions. We investigate the complex-temperature singularities of the susceptibility of the 1.

•Jan 1, 2015•30 citations

Determination of critical behaviour in lattice statistics from series expansions III

Anthony John Guttmann

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.

Journal of Physics C Solid State Physics•Nov 1, 1969•32 citations

Low temperature critical behaviour of the Ising model with spin S>1/2

P F Fox, Anthony John Guttmann

Low temperature series expansions for the Ising model with spin S=1 and S= 3 / 2 are derived for a number of two and three dimensional lattices. For each spin-lattice combination the critical point is estimated, and then the critical exponents beta and gamma ' with their corresponding critical amplitudes are also estimated. The principal conclusion is that there is no compelling evidence to suggest that the above critical exponents differ from their S= 1 / 2 values.

Journal of Physics C Solid State Physics•Mar 14, 1973•62 citations
We derive low-temperature series (in the variable ) for the spontaneous magnetization, susceptibility, and specific heat of the spin-S Ising model on the square lattice for , 2, , and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with S. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of S. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of S, though consistent with universality.
The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta 1 =1.1+or-0.1 is found.
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.
king model on a square lattice. From an analysis of low-temperature series expansions. we find evidence that, as one approaches the point U = us =- I (where U = e-") from within the complex extensions of the FM or AFM phases. the susceptibility has 3. divergent singularity of the farm,y- A:(l + u)-y: with exponent r: = i. The critical amplitude A: is calculated. Other critical exponents are found to be ci = cs = 0 and ps = $, so thY the scaling relation U: + 26, + y: = 2 is satisfied. However, using exact results for ps on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: ps is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length nnd hence susceptibility are finite GS one approaches the point I ( =-1 from within the symmetric phase. This is m n h e d by an explicit study of high-tempemture series expansions. We investigate the complex-temperature singularities of the susceptibility of the 1.
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.