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Series studies of the Potts model: I. the simple cubic Ising model

Anthony John Guttmann, Ian Graham Enting

Published Jan 1, 1993• 31 citations
OpenAlex

Abstract

The finite lattice method of series expansion is generalised to the $q$-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising ($q=2$) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to $u^{26}$ from $u^{20}$. The high-temperature series for the zero-field partition function is extended from $v^{18}$ to $v^{22}$. Subsequent analysis gives critical exponents in agreement with those from field theory.

Related papers

Series studies of the Potts model. I. The simple cubic Ising model

A J Guttmann, I G Enting

The finite-lattice method of series expansion is generalized to the q-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2) case the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.

arXiv:hep-lat/9212032•Journal of Physics A Mathematical and General•Feb 21, 1993•49 citations

Series studies of the Potts model: III. The 3-state model on the simple cubic lattice

Anthony John Guttmann, Ian Graham Enting

For paper II, see ibid, vol. 27, p.1503 (1994). 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 3-state Potts model on the simple cubic lattice to order z 43 and the high-temperature expansion of the partition function to order v 21 . We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.

arXiv:hep-lat/9312083•Journal of Physics A Mathematical and General•Sep 7, 1994•33 citations

Series studies of the Potts model. II. Bulk series for the square lattice

Keith Briggs, Ian Graham Enting, Anthony John Guttmann

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 q-state Potts model to order z 56 (i.e. u 28 ), z 47 , z 43 , z 39 , z 39 , z 39 , z 35 , z 31 and z 31 for q=2, 3, 4, ..., 9 and 10 respectively. These series are used to test techniques designed to distinguish first-order transitions from continuous transitions. New numerical values are also obtained for the q-state Potts model with q>4.

arXiv:hep-lat/9312082•Journal of Physics A Mathematical and General•Mar 7, 1994•36 citations

Analytic Properties of Potts and Ising Model Partition Functions and the Relationship between Analytic Properties and Phase Transitions in Equilibrium Statistical Mechanics

Siti Fatimah Zakaria

The Ising and $Q$-state Potts models are statistical mechanical models of spins interaction on crystal lattices. \nWe study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena $-$ how higher level structure is affected by the detailed activity of a very large number of lower level structures. \nWe investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. \nOur study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. \nHere we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. \nThe partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. \nWe extend the study of $Q$-state Potts models to a more general case called the $Z_Q$-symmetric model. \nWe evidence the existence of multiple phase transitions for this model in case $Q=5,6,$ and discuss the possible connection to different stages of disordered state. \nGiven sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. \nOur analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. \nWe evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent $\\alpha$.

•Nov 1, 2016•3 citations

High-field series expansions and critical properties for the three-state Potts model

S Mivashita, Donald D. Betts, C. J. Elliott

High-field series expansions are presented for the three-state Potts model on the square, simple cubic and body centred cubic lattices. Estimates of the critical exponents for two dimensions are beta =0.1064+or-0.0005, gamma '=1.50+or-0.04 and delta =15.5+or-1.5. It is plausible that the three-dimensional model undergoes a second-order phase transition with exponents beta =0.203+or-0.004, gamma '=1.18+or-0.05 and delta =7.0+or-0.3.

Journal of Physics A Mathematical and General•Sep 1, 1979•29 citations

High-temperature series for the susceptibility of the spin- S Ising model

John P. VAN DYKE, William J. Camp

We have extended the series for the zero-field susceptibility of the spin-$S$ Ising model to eighth order in the reduced temperature $K$, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series ${h}_{n}(S)$ are expressed as simple polynomials in $X=S(S+1)$. For the face-centered-cubic lattice, an accurate polynomial fit to the critical point ${K}_{c}(S)$ is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all $S$. However, the large-$S$ series seems to exhibit more rapid apparent convergence.

Physical review. B, Solid state•Apr 1, 1974•42 citations
The finite lattice method of series expansion is generalised to the qqq-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2q=2q=2) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to u26u^{26}u26 from u20u^{20}u20. The high-temperature series for the zero-field partition function is extended from v18v^{18}v18 to v22v^{22}v22. Subsequent analysis gives critical exponents in agreement with those from field theory.
The finite-lattice method of series expansion is generalized to the q-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising (q=2) case the authors have extended the low-temperature series for the partition functions, magnetization and zero-field susceptibility to u 26 from u 20 . The high-temperature series for the zero-field partition function is extended from nu 18 to nu 22 . Subsequent analysis gives critical exponents in agreement with those from field theory.
For paper II, see ibid, vol. 27, p.1503 (1994). 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 3-state Potts model on the simple cubic lattice to order z 43 and the high-temperature expansion of the partition function to order v 21 . We use the numerical data to show that the transition is first-order, and estimate the latent heat, the discontinuity in the magnetization, and a number of other critical parameters.
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 q-state Potts model to order z 56 (i.e. u 28 ), z 47 , z 43 , z 39 , z 39 , z 39 , z 35 , z 31 and z 31 for q=2, 3, 4, ..., 9 and 10 respectively. These series are used to test techniques designed to distinguish first-order transitions from continuous transitions. New numerical values are also obtained for the q-state Potts model with q>4.
The Ising and QQQ-state Potts models are statistical mechanical models of spins interaction on crystal lattices. \nWe study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena −-− how higher level structure is affected by the detailed activity of a very large number of lower level structures. \nWe investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. \nOur study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. \nHere we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. \nThe partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. \nWe extend the study of QQQ-state Potts models to a more general case called the ZQZ_QZQ​-symmetric model. \nWe evidence the existence of multiple phase transitions for this model in case Q=5,6,Q=5,6,Q=5,6, and discuss the possible connection to different stages of disordered state. \nGiven sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. \nOur analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. \nWe evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent alpha\\alphaalpha.
We have extended the series for the zero-field susceptibility of the spin-SSS Ising model to eighth order in the reduced temperature KKK, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series 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). For the face-centered-cubic lattice, an accurate polynomial fit to the critical point Kc(S){K}_{c}(S)Kc​(S) is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all SSS. However, the large-SSS series seems to exhibit more rapid apparent convergence.