Using a confluent singularity analysis based on the generalised recurrence method, the series for the correlation length in terms of a dimensionless coupling constant is analysed, extending an approach of Nickel and Sharpe (1979). The results suggest that the spin- 1 / 2 BCC Ising model satisfies hyperscaling and has the same confluent singularity structure expected for models of the (n=1, D=3) universality class.

Journal of Physics A Mathematical and GeneralJul 1, 197923 citations

Field-theoretical analysis of singularities at critical end points

Hans Werner Diehl, Martin Smock

arXiv:cond-mat/9908311Physica A Statistical Mechanics and its ApplicationsJun 1, 20007 citations

Journal of Mathematical Chemistry

New method for confluent singularity analysis of power series

Gustavo A. Arteca, Francisco Marcelo Fernández, Eduardo Alberto Castro

Published Jul 1, 1988• 0 citations

Publisher

DOI: 10.1007/bf01167203

Related papers

Resolution Dissipation in Ising Susceptibility Series: A Twenty-First Century Reanalysis of Confluent Singularities

Michael George

Jun 30, 20260 citations

The Problem of Confluent Singularities

Bernie G. Nickel

Springer eBooksJan 1, 198211 citations

Confluent singularities in 3D continuum Φ4 theory: resolving critical point discrepancies

Bernie G. Nickel

Physica A Statistical Mechanics and its ApplicationsSep 1, 199142 citations

Hyperscaling in the three-dimensional Ising model

Martin V. Ferer, Michael J. Velgakis

We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin- $s$ Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility $\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})$, where $t=1\ensuremath{-}\frac{{T}_{c}}{T}$. The method determines sequences for these five parameters, e.g., ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}$, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin- $s$ Ising-model series shows that the indices $\ensuremath{\gamma}$ and $2\ensuremath{\nu}$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate $\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)$ and $2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)$, in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin-$s$ Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility $\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})$, where $t=1\ensuremath{-}\frac{{T}_{c}}{T}$. The method determines sequences for these five parameters, e.g., ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}$, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin-$s$ Ising-model series shows that the indices $\ensuremath{\gamma}$ and $2\ensuremath{\nu}$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate $\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)$ and $2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)$, in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.

Physical review. B, Condensed matterMar 1, 198317 citations

Confluent singularities and hyperscaling in the spin-1/2Ising model

John J. Rehr

Journal of Physics A Mathematical and GeneralJul 1, 197923 citations

Field-theoretical analysis of singularities at critical end points

Hans Werner Diehl, Martin Smock

arXiv:cond-mat/9908311Physica A Statistical Mechanics and its ApplicationsJun 1, 20007 citations

We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin-

s

Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility $\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})$, where $t=1\ensuremath{-}\frac{{T}_{c}}{T}$. The method determines sequences for these five parameters, e.g., ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}$, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin-

s

Ising-model series shows that the indices $\ensuremath{\gamma}$ and $2\ensuremath{\nu}$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate $\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)$ and $2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)$, in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.