Notes on the δ-expansion approach to the 2D Ising susceptibility scaling

We study the scaling of the magnetic susceptibility in the square Ising model based upon the delta-expansion in the high temperature phase. The susceptibility chi is expressed in terms of the mass M and expanded in powers of 1/M. The dilation around M=0 by the delta expansion and the parametric extension of the ratio of derivatives of chi, chi^{(ell+1)}/chi^{(ell)} is used as a test function for the estimation of the critical exponent gamma with no bias from information of the critical temperature. Estimation is done with the help of the principle of minimum sensitivity and detailed analysis revealed that ell=0,1 cases provide us accurate estimation results. Critical exponent of the sub-leading scaling term is also estimated.We study the scaling of the magnetic susceptibility in the square Ising model based upon the delta-expansion in the high temperature phase. The susceptibility chi is expressed in terms of the mass M and expanded in powers of 1/M. The dilation around M=0 by the delta expansion and the parametric extension of the ratio of derivatives of chi, chi^{(ell+1)}/chi^{(ell)} is used as a test function for the estimation of the critical exponent gamma with no bias from information of the critical temperature. Estimation is done with the help of the principle of minimum sensitivity and detailed analysis revealed that ell=0,1 cases provide us accurate estimation results. Critical exponent of the sub-leading scaling term is also estimated.

arXiv:1311.4741Nov 19, 20130 citations

Delta expansion at low temperatures

Yamada, Hirofumi

In the low temperature phase of the square Ising model, we describe the inverse temperature beta as the function of a squared mass M and study the critical behavior of beta(M) via the large M expansion. Using the delta-expansion by which the large mass expansion is transformed into a series exhibiting expected scaling behavior, we perform the estimation of the critical inverse temperature beta_{c} with the help of linear differential equation to be satisfied by ansatz of beta(M) near the critical point M=0. To improve the estimation, the leading correction exponent nu is independently estimated from beta^{(2)}/beta^{(1)} and is used in the estimation of beta_{c}, giving rise to remarkable accuracy improvement.In the low temperature phase of the square Ising model, we describe the inverse temperature beta as the function of a squared mass M and study the critical behavior of beta(M) via the large M expansion. Using the delta-expansion by which the large mass expansion is transformed into a series exhibiting expected scaling behavior, we perform the estimation of the critical inverse temperature beta_{c} with the help of linear differential equation to be satisfied by ansatz of beta(M) near the critical point M=0. To improve the estimation, the leading correction exponent nu is independently estimated from beta^{(2)}/beta^{(1)} and is used in the estimation of beta_{c}, giving rise to remarkable accuracy improvement.

arXiv:1410.5751Oct 21, 20140 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

Series expansions for the zero-temperature transverse Ising model

L.G. Marland

Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .

Journal of Physics A Mathematical and GeneralAug 1, 198150 citations

Microcanonical high-temperature expansion in the theory of ferromagnetism

E. Praveczki

To avoid the difficulties with the usual type of (canonical) high-temperature series which arise when T<Tc and H=O, it is suggested that a high-temperature expansion be accomplished using the microcanonical ensembles of systems defined as involving only those states having values of the order parameter equal to the temperature-averaged one. The method is applied for the Ising and Heisenberg model utilising the molecular-field picture, and for the Heisenberg model using the spin-wave picture. In the latter case, the spin-wave energy is expanded according to the high-temperature parameter. It is shown that (m,m+1)-type Pade approximants are good at all temperatures. It is found that the longitudinal susceptibility chi L, diverges like H-1/2 at T<Tc; consequently, the critical exponent gamma ' no longer holds. A new critical exponent, gamma is defined as scaling the reduced susceptibility, chi 0=H1/2chi L. Deriving the scaling-law relating gamma with beta and delta , gamma is estimated as 0.18.To avoid the difficulties with the usual type of (canonical) high-temperature series which arise when T<Tc and H=O, it is suggested that a high-temperature expansion be accomplished using the microcanonical ensembles of systems defined as involving only those states having values of the order parameter equal to the temperature-averaged one. The method is applied for the Ising and Heisenberg model utilising the molecular-field picture, and for the Heisenberg model using the spin-wave picture. In the latter case, the spin-wave energy is expanded according to the high-temperature parameter. It is shown that (m,m+1)-type Pade approximants are good at all temperatures. It is found that the longitudinal susceptibility chi L, diverges like H-1/2 at T<Tc; consequently, the critical exponent gamma ' no longer holds. A new critical exponent, gamma is defined as scaling the reduced susceptibility, chi 0=H1/2chi L. Deriving the scaling-law relating gamma with beta and delta , gamma is estimated as 0.18.

Journal of Physics C Solid State PhysicsOct 14, 19771 citation

Validity of hyperscaling for thed=3Ising model

Anthony John Guttmann

The availability of extended series expansions for ${\ensuremath{\chi}}_{0}^{(2)}$=${\ensuremath{\partial}}^{2}$\ensuremath{\chi}/\ensuremath{\partial} ${H}^{2}$ , accurate Monte Carlo estimates of the critical temperature for the sc spin-(1/2) Ising model, and refined methods of series analysis allow the hyperscaling relation d\ensuremath{\nu}=2\ensuremath{\Delta}-\ensuremath{\gamma} to be severely tested. From analysis of ${\ensuremath{\chi}}_{0}^{(2)}$ and ${\ensuremath{\chi}}_{0}^{(2)}$/${\ensuremath{\chi}}_{0}^{2}$ we estimate \ensuremath{\gamma}=1.240\ifmmode\pm\else\textpm\fi{}0.006 and \ensuremath{\Delta}=1.565\ifmmode\pm\else\textpm\fi{}0.004. The estimate of \ensuremath{\Delta} is in complete agreement with the prediction of hyperscaling using both the best extant series and renormalization-group estimates of the exponents \ensuremath{\gamma} and \ensuremath{\nu}. With the assumption of hyperscaling, our analysis gives \ensuremath{\nu}=0.630\ifmmode\pm\else\textpm\fi{}0.002.The availability of extended series expansions for ${\ensuremath{\chi}}_{0}^{(2)}$=${\ensuremath{\partial}}^{2}$\ensuremath{\chi}/\ensuremath{\partial}${H}^{2}$, accurate Monte Carlo estimates of the critical temperature for the sc spin-(1/2) Ising model, and refined methods of series analysis allow the hyperscaling relation d\ensuremath{\nu}=2\ensuremath{\Delta}-\ensuremath{\gamma} to be severely tested. From analysis of ${\ensuremath{\chi}}_{0}^{(2)}$ and ${\ensuremath{\chi}}_{0}^{(2)}$/${\ensuremath{\chi}}_{0}^{2}$ we estimate \ensuremath{\gamma}=1.240\ifmmode\pm\else\textpm\fi{}0.006 and \ensuremath{\Delta}=1.565\ifmmode\pm\else\textpm\fi{}0.004. The estimate of \ensuremath{\Delta} is in complete agreement with the prediction of hyperscaling using both the best extant series and renormalization-group estimates of the exponents \ensuremath{\gamma} and \ensuremath{\nu}. With the assumption of hyperscaling, our analysis gives \ensuremath{\nu}=0.630\ifmmode\pm\else\textpm\fi{}0.002.

Physical review. B, Condensed matterApr 1, 198622 citations

The availability of extended series expansions for ${\ensuremath{\chi}}_{0}^{(2)}$=${\ensuremath{\partial}}^{2}$\ensuremath{\chi}/\ensuremath{\partial}

{H}^{2}

, accurate Monte Carlo estimates of the critical temperature for the sc spin-(1/2) Ising model, and refined methods of series analysis allow the hyperscaling relation d\ensuremath{\nu}=2\ensuremath{\Delta}-\ensuremath{\gamma} to be severely tested. From analysis of ${\ensuremath{\chi}}_{0}^{(2)}$ and ${\ensuremath{\chi}}_{0}^{(2)}$/${\ensuremath{\chi}}_{0}^{2}$ we estimate \ensuremath{\gamma}=1.240\ifmmode\pm\else\textpm\fi{}0.006 and \ensuremath{\Delta}=1.565\ifmmode\pm\else\textpm\fi{}0.004. The estimate of \ensuremath{\Delta} is in complete agreement with the prediction of hyperscaling using both the best extant series and renormalization-group estimates of the exponents \ensuremath{\gamma} and \ensuremath{\nu}. With the assumption of hyperscaling, our analysis gives \ensuremath{\nu}=0.630\ifmmode\pm\else\textpm\fi{}0.002.