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

The susceptibility of the square lattice Ising model: New developments

William P. Orrick, Bernie G. Nickel, Anthony John Guttmann, Jacques H. H. Perk

Published Nov 15, 2012• 69 citations
OpenAlex

Abstract

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T −Tc | 9/4, though high-low temperature symmetry is still preserved. At terms of order |T −Tc | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − Tc) p (log |T − Tc|) q with p ≥ q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.

Related papers

Critical Behavior of the Two-Dimensional Ising Susceptibility

William P. Orrick, Bernie G. Nickel, Anthony John Guttmann +1 more

We report computations of the short- and long-distance (scaling) contributions to the square-lattice Ising susceptibility. Both computations rely on summation of correlation functions, obtained using nonlinear partial difference equations. In terms of a temperature variable $\ensuremath{\tau}$, linear in ${T/T}_{c}\ensuremath{-}1$, the short-distance terms have the form ${\ensuremath{\tau}}^{p}(\mathrm{ln}|\ensuremath{\tau}|{)}^{q}$ with $p\ensuremath{\ge}{q}^{2}$. A high- and low-temperature series of $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}323$ terms, generated using an algorithm of complexity $\mathrm{O}({N}^{6})$, are analyzed to obtain the scaling part, which when divided by the leading $|\ensuremath{\tau}{|}^{\ensuremath{-}7/4}$ singularity contains only integer powers of $\ensuremath{\tau}$. Contributions of distinct irrelevant variables are identified and quantified at leading orders $|\ensuremath{\tau}{|}^{9/4}$ and $|\ensuremath{\tau}{|}^{17/4}$.

arXiv:cond-mat/0009059•Physical Review Letters•Apr 30, 2001•52 citations

Higher-order corrections for the quadratic Ising lattice susceptibility

S. Gartenhaus, W. Scott McCullough

Four terms in the expansion for the zero-field susceptibility of the quadratic Ising lattice at criticality are known exactly. We have computed three additional terms in this expansion by analyzing Nickel's high-temperature series for this lattice with second-order homogeneous differential approximants and Pad\'e techniques. These three terms are found to vary as \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$,t, and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ with t=1-${T}_{c}$/T, and their respective coefficients are obtained to within 0.01% accuracy. It is shown that, if terms of the form \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} were present, their amplitudes would be less than ${10}^{\mathrm{\ensuremath{-}}5}$ of the amplitudes of the \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ terms, respectively. The results are in accord with the predictions of the renormalization group if irrelevant variables are neglected and indicate that, if singularities due to irrelevant variables are present, their associated exponents must exceed 2. The results are also used to predict the coefficients of the low-temperature series for this system. Agreement to as much as one part in ${10}^{5}$ with the known first eleven coefficients of this series is obtained.

Physical review. B, Condensed matter•Dec 1, 1988•42 citations

Zero-field susceptibility of the antiferromagnetic square Ising lattice

Stuart S. C. Burnett, S. Gartenhaus

The first two terms in the expansion near ${\mathit{T}}_{\mathit{C}}$ of the zero-field susceptibility of the antiferromagnetic square Ising lattice are known exactly. We have isolated the next four terms in this expansion by carrying out a differential-approximant analysis of Nickel's 55-term high-temperature ferromagnetic series with the variable v=tanhJ/${\mathit{k}}_{\mathit{B}}$T replaced by -v. The four new terms are found to vary as t, ${\mathit{t}}^{2}$ln\ensuremath{\Vert}t\ensuremath{\Vert}, ${\mathit{t}}^{2}$, and ${\mathit{t}}^{3}$ln\ensuremath{\Vert}t\ensuremath{\Vert} where t=1-${\mathit{T}}_{\mathit{C}}$/T and values for their respective coefficients are obtained. It is established that to order ${\mathit{t}}^{3}$ln\ensuremath{\Vert}t\ensuremath{\Vert}, there are no confluent singularities associated with irrelevant variables for this system. A direct extrapolation of Nickel's series coefficients is also carried out and found to be consistent with the present results.

Physical review. B, Condensed matter•Apr 1, 1993•6 citations

Antiferromagnetic Susceptibility and the Short-Range Correlation Function in the Ising Square Lattice

K. Yamada, Toshimichi Suzuki

The zero·field susceptibility x ( T) of the antiferromagnet in the Ising square lattice with the symmetric interaction was evaluated by carrying out the lattice sum of the short · range correlation function X ( m, n) (O~m, n~4), which is exactly and explicitly calculated on the basis of a "generalized Wronskian " form with dimension m. It is naturally derived from the singularity of X(m, n) that the highest singularity of x ( T) (per site) near the critical temperature Tc is logarithmic as kB Tz ( T) =A + B(l- T/Tc)lnI1- T/Tcl, with A =0.1596264029 and B=0.2009359779 in the best approximation of the present calculation. Both A and B are completely equal below and above the critical temperature.

Progress of Theoretical Physics•Sep 1, 1986•3 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

Low-temperature series expansion of the square-lattice Ising model: A study based on Fisher zeros

De-Zhang Li, Xin Wang, Xiao‐Bao Yang

Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.

arXiv:2412.07328•Physical review. B./Physical review. B•Apr 21, 2025•3 citations
We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T −Tc | 9/4, though high-low temperature symmetry is still preserved. At terms of order |T −Tc | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − Tc) p (log |T − Tc|) q with p ≥ q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
We report computations of the short- and long-distance (scaling) contributions to the square-lattice Ising susceptibility. Both computations rely on summation of correlation functions, obtained using nonlinear partial difference equations. In terms of a temperature variable $\ensuremath{\tau}$, linear in ${T/T}_{c}\ensuremath{-}1$, the short-distance terms have the form ${\ensuremath{\tau}}^{p}(\mathrm{ln}|\ensuremath{\tau}|{)}^{q}$ with $p\ensuremath{\ge}{q}^{2}$. A high- and low-temperature series of N=323N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}323N=323 terms, generated using an algorithm of complexity O(N6)\mathrm{O}({N}^{6})O(N6), are analyzed to obtain the scaling part, which when divided by the leading $|\ensuremath{\tau}{|}^{\ensuremath{-}7/4}$ singularity contains only integer powers of $\ensuremath{\tau}$. Contributions of distinct irrelevant variables are identified and quantified at leading orders $|\ensuremath{\tau}{|}^{9/4}$ and $|\ensuremath{\tau}{|}^{17/4}$.
Four terms in the expansion for the zero-field susceptibility of the quadratic Ising lattice at criticality are known exactly. We have computed three additional terms in this expansion by analyzing Nickel's high-temperature series for this lattice with second-order homogeneous differential approximants and Pad\'e techniques. These three terms are found to vary as \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$,t, and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ with t=1-Tc{T}_{c}Tc​/T, and their respective coefficients are obtained to within 0.01% accuracy. It is shown that, if terms of the form \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ln\ensuremath{\Vert}t\ensuremath{\Vert} were present, their amplitudes would be less than ${10}^{\mathrm{\ensuremath{-}}5}$ of the amplitudes of the \ensuremath{\Vert}t${\ensuremath{\Vert}}^{1/4}$ and \ensuremath{\Vert}t${\ensuremath{\Vert}}^{5/4}$ terms, respectively. The results are in accord with the predictions of the renormalization group if irrelevant variables are neglected and indicate that, if singularities due to irrelevant variables are present, their associated exponents must exceed 2. The results are also used to predict the coefficients of the low-temperature series for this system. Agreement to as much as one part in 105{10}^{5}105 with the known first eleven coefficients of this series is obtained.
The first two terms in the expansion near TC{\mathit{T}}_{\mathit{C}}TC​ of the zero-field susceptibility of the antiferromagnetic square Ising lattice are known exactly. We have isolated the next four terms in this expansion by carrying out a differential-approximant analysis of Nickel's 55-term high-temperature ferromagnetic series with the variable v=tanhJ/kB{\mathit{k}}_{\mathit{B}}kB​T replaced by -v. The four new terms are found to vary as t, t2{\mathit{t}}^{2}t2ln\ensuremath{\Vert}t\ensuremath{\Vert}, t2{\mathit{t}}^{2}t2, and t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert} where t=1-TC{\mathit{T}}_{\mathit{C}}TC​/T and values for their respective coefficients are obtained. It is established that to order t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert}, there are no confluent singularities associated with irrelevant variables for this system. A direct extrapolation of Nickel's series coefficients is also carried out and found to be consistent with the present results.
The zero·field susceptibility x ( T) of the antiferromagnet in the Ising square lattice with the symmetric interaction was evaluated by carrying out the lattice sum of the short · range correlation function X ( m, n) (O~m, n~4), which is exactly and explicitly calculated on the basis of a "generalized Wronskian " form with dimension m. It is naturally derived from the singularity of X(m, n) that the highest singularity of x ( T) (per site) near the critical temperature Tc is logarithmic as kB Tz ( T) =A + B(l- T/Tc)lnI1- T/Tcl, with A =0.1596264029 and B=0.2009359779 in the best approximation of the present calculation. Both A and B are completely equal below and above the critical temperature.
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.
Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.