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arXiv:1905.06183

An empirical partition function for the simple cubic Ising Model with a zero external magnetic field

Rong Wei

Published May 8, 2019• 0 citations
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Abstract

There is no an accepted exact partition function (PF) for the three dimensional (3D) Ising model to our knowledge. Mainly based on the connection between the lattice Green function (LGF) for the simple cubic lattice and that for the honeycomb lattice, we infer an empirical partition function (EPF) for the simple cubic Ising model in the absence of an external magnetic field. This ${\rm EPF}_{_{\rm 3D}}=\frac{1}{2{π^3}}\int_0^π\int_0^π\int_0^π\log [2(2{{\cosh }^3}2z + 3{{\sinh }^2}2z + 2)^{\frac{1}{2}} -2α\sinh 2z \ (\cosω_1 + \cos ω_2 + \cosω_3)] {\rm{d}}ω_1{\rm{d}}ω_2{\rm{d}}ω_3, α\in[\sqrt{2},\sqrt{3}]$ (where $z=\fracε{kT}$, $ε$ the interaction energy, $T$ the temperature, and $k$ Boltzmann constant). When $α=\sqrt{2}$, this EPF is consistent well numerically with the result from high temperature expansions by Guttmann and Enting (1993). The specific heat from this EPF approaches infinity non-logarithmically at the critical temperature $T_c$. $\fracε{kT_c}=\cosh^{-1}[\frac{1}{4} (17-3\sqrt{17})]/2\approx 0.277212$, which is greater than 0.221654 from the recent Monte Carlo study.

DOI
10.48550/arxiv.1905.06183

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There is no an accepted exact partition function (PF) for the three dimensional (3D) Ising model to our knowledge. Mainly based on the connection between the lattice Green function (LGF) for the simple cubic lattice and that for the honeycomb lattice, we infer an empirical partition function (EPF) for the simple cubic Ising model in the absence of an external magnetic field. This EPF3D=12π3∫0π∫0π∫0πlog⁡[2(2cosh⁡32z+3sinh⁡22z+2)12−2αsinh⁡2z (cos⁡ω1+cos⁡ω2+cos⁡ω3)]dω1dω2dω3,α∈[2,3]{\rm EPF}_{_{\rm 3D}}=\frac{1}{2{π^3}}\int_0^π\int_0^π\int_0^π\log [2(2{{\cosh }^3}2z + 3{{\sinh }^2}2z + 2)^{\frac{1}{2}} -2α\sinh 2z \ (\cosω_1 + \cos ω_2 + \cosω_3)] {\rm{d}}ω_1{\rm{d}}ω_2{\rm{d}}ω_3, α\in[\sqrt{2},\sqrt{3}]EPF3D​​=2π31​∫0π​∫0π​∫0π​log[2(2cosh32z+3sinh22z+2)21​−2αsinh2z (cosω1​+cosω2​+cosω3​)]dω1​dω2​dω3​,α∈[2​,3​] (where z=εkTz=\fracε{kT}z=kTε​, εεε the interaction energy, TTT the temperature, and kkk Boltzmann constant). When α=2α=\sqrt{2}α=2​, this EPF is consistent well numerically with the result from high temperature expansions by Guttmann and Enting (1993). The specific heat from this EPF approaches infinity non-logarithmically at the critical temperature TcT_cTc​. εkTc=cosh⁡−1[14(17−317)]/2≈0.277212\fracε{kT_c}=\cosh^{-1}[\frac{1}{4} (17-3\sqrt{17})]/2\approx 0.277212kTc​ε​=cosh−1[41​(17−317​)]/2≈0.277212, which is greater than 0.221654 from the recent Monte Carlo study.
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices (4×44\times 44×4 and 4×64\times 64×6) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.
There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with NNN lattice sites. It is a sum of 2N2^N2N exponential functions and holds for DDD-dimensional (D=1,2,3,...D=1,2,3,...D=1,2,3,...) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.
The microcanonical transfer matrix is used to evaluate the exact integer values for the number of states $\ensuremath{\Omega}(E,M)$, as a function of energy EEE and magnetization MMM, of the Ising model in an external magnetic field HHH on $L\ifmmode\times\else\texttimes\fi{}2L$ honeycomb lattices (L=6,8,…,14)(L=6,8,\dots{},14)(L=6,8,…,14) and (due to memory limitations) the exact real values for $\overline{\ensuremath{\Omega}}(a,M)={\ensuremath{\sum}}_{E}\ensuremath{\Omega}(E,M){a}^{E}$ (L=16,…,22)(L=16,\dots{},22)(L=16,…,22), where $a={e}^{2\ensuremath{\beta}J}$. Given $\ensuremath{\Omega}(E,M)$ or $\overline{\ensuremath{\Omega}}(a,M)$, the exact partition functions $Z(a,x)={\ensuremath{\sum}}_{E,M}\ensuremath{\Omega}(E,M){a}^{E}{x}^{M}={\ensuremath{\sum}}_{M}\overline{\ensuremath{\Omega}}(a,M){x}^{M}$, where $x={e}^{\ensuremath{-}2\ensuremath{\beta}H}$, are obtained. From the exact partition functions, the precise distributions of the partition function zeros in the complex xxx plane of the honeycomb-lattice Ising antiferromagnet in a uniform external magnetic field, where the famous circle theorem for the partition function zeros of the Ising ferromagnet is not applicable, are determined. In addition, the critical points xcp(a){x}_{cp}(a)xcp​(a) and the magnetic scaling exponents yh(a){y}_{h}(a)yh​(a) of the honeycomb-lattice Ising antiferromagnet in a uniform magnetic field are estimated using its partition function zeros in the complex xxx plane. Our results for the critical points are also compared to those of the two different closed-form approximations.
Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.
The partition functions of ferromagnetic Ising models of square lattices in a finite magnetic field is deduced using topological considerations within a heuristic graph-theoretical approach. These equations are derived separately for low and high temperature regimes while the exact solution of Onsager is obtained therefrom when the magnetic field is zero. The derived partition function equations here are almost similar to those given by Onsager, thus indicating a straight-forward protocol, even when the magnetic field is present. The spontaneous magnetization derived here using the Helmholtz free energy is identical with that arising from the exact solution. The partition functions lead to the known series expansions of the magnetization and zero-field susceptibility.
The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.