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arXiv:1407.2100 · Physical Review E

Large-order aspects of the δ expansion in low-dimensional Ising models

Hirofumi Yamada

Published Sep 29, 2014• 2 citations
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Abstract

We investigate the large order aspects of the δ expansion under the estimation procession of the critical quantities. As illustrative examples, we revisit the one-dimensional Ising model for the analytic study and the two-dimensional square Ising model in the high-temperature phase for the numerical experiment to large orders. In both models, the proposed fundamental base on which the estimation protocol should be constructed is investigated in detail and confirmed to be valid. In the square lattice model, we present a new protocol for the estimation of critical exponents and temperature.

DOI
10.1103/physreve.90.032139

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We investigate the large order aspects of the δ expansion under the estimation procession of the critical quantities. As illustrative examples, we revisit the one-dimensional Ising model for the analytic study and the two-dimensional square Ising model in the high-temperature phase for the numerical experiment to large orders. In both models, the proposed fundamental base on which the estimation protocol should be constructed is investigated in detail and confirmed to be valid. In the square lattice model, we present a new protocol for the estimation of critical exponents and temperature.
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.
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 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.
For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann, and Jensen. We have computed various critical parameters and improved the estimates of others. Moreover, the properties of hyperscaling and of universality (spin-S independence) of exponents and of various dimensionless amplitude combinations have been verified accurately. Assuming the validity of the lattice-lattice scaling, from our estimates of critical amplitudes for the square lattice we have also obtained estimates of the corresponding amplitudes for the spin-S Ising model on the triangular, honeycomb, and kagom\'e lattices.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.