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 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
ZQ-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.
We report some results on the complex-temperature (CT) singularities of
q-state Potts models on the square lattice. We concentrate on the problematic region
Re(a)<0 (where
a=eK) in which CT zeros of the partition function are sensitive to finite lattice artifacts. From analyses of low-temperature series expansions for
3< q< 8, we establish the existence, in this region, of complex-conjugate CT singularities at which the magnetization and susceptibility diverge. From calculations of zeros of the partition function, we obtain evidence consistent with the inference that these singularities occur at endpoints
ae,
ae∗ of arcs protruding into the (complex-temperature extension of the) ferromagnetic phase. Exponents for these singularities are determined; e.g., for
q=3, we find ${\ensuremath{\beta}}_{e}=\ensuremath{-}0.125(1)$, consistent with ${\ensuremath{\beta}}_{e}=\ensuremath{-}\frac{1}{8}$. By duality, these results also imply associated arcs extending into the (CT extension of the) symmetric paramagnetic phase. Analytic expressions are suggested for the positions of some of these singularities; e.g., for
q=5, our finding is consistent with the exact value
ae, ${a}_{e}^{*}=2(\ensuremath{-}1\ensuremath{\mp}i)$. Further discussions of complex-temperature phase diagrams are given.
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
u26 from
u20. The high-temperature series for the zero-field partition function is extended from
v18 to
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.
The distributions of the partition function zeros in the complex a=e2betaJ1 plane of the square-lattice Ising model with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions are investigated as a function of R=J2/J1. Starting from the well-known two-circle distribution of the zeros a=+/-1+sqrt[2]eitheta for R=0 , finally the partition function zeros lie on the unit circle a=eitheta for R=infinity . Between these two ends, the changes in the zero distributions are described. Using the partition function zeros, the critical point ac(R) and the thermal scaling exponent yt(R) are estimated for the Ising ferromagnet (equivalently, antiferromagnet) and superantiferromagnet. For the special case of R=1/2 , the possible implications of the zero distributions are also discussed.