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

Physical review. B, Condensed matter

Zero-field susceptibility of the antiferromagnetic square Ising lattice

Stuart S. C. Burnett, S. Gartenhaus

Published Apr 1, 1993• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.47.7944

Related papers

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

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.

Physical review. B, Solid state•Apr 1, 1975•148 citations

High temperature series for the susceptibility of the Ising model. I. Two dimensional lattices

M F Sykes, David S. Gaunt, P D Roberts +1 more

Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.

Journal of physics. A, Proceedings of the Physical Society. General•May 1, 1972•90 citations

The susceptibility of the square lattice Ising model: New developments

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

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.

•Nov 15, 2012•69 citations

Exact solution and high-temperature series expansion study of the one-fifth-depleted square-lattice Ising model

Simeon Hanks, Trinanjan Datta, J. Oitmaa

The critical behavior of the one-fifth-depleted square-lattice Ising model with nearest-neighbor ferromagnetic interaction has been investigated by means of both an exact solution and a high-temperature series expansion study of the zero-field susceptibility. For the exact solution we employ a decoration transformation followed by a mapping to a staggered eight-vertex model. This yields a quartic equation for the critical coupling giving K(c)(≡βJ(c))=0.695. The series expansion for the susceptibility, to O(K(18)), when analyzed via standard Padé approximant methods gives an estimate of K(c), consistent with the exact solution result to at least four significant figures. The series expansion is also analyzed for the leading amplitude and subdominant terms.

arXiv:1303.3068•Physical Review E•Jun 28, 2013•3 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
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.
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.
We have extended the series for the zero-field susceptibility of the spin-sss Ising model through tenth order in the reduced inverse temperature KKK on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients hn(s){h}_{n}(s)hn​(s) are expressed as simple polynomials in X=s(s+1)X=s(s+1)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 sss. 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=12s=\frac{1}{2}s=21​ 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 sss, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
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.
The critical behavior of the one-fifth-depleted square-lattice Ising model with nearest-neighbor ferromagnetic interaction has been investigated by means of both an exact solution and a high-temperature series expansion study of the zero-field susceptibility. For the exact solution we employ a decoration transformation followed by a mapping to a staggered eight-vertex model. This yields a quartic equation for the critical coupling giving K(c)(≡βJ(c))=0.695. The series expansion for the susceptibility, to O(K(18)), when analyzed via standard Padé approximant methods gives an estimate of K(c), consistent with the exact solution result to at least four significant figures. The series expansion is also analyzed for the leading amplitude and subdominant terms.
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.