Partition Function Of Ising Model Research Articles

Zeros of Partition Function and High Temperature Expansion for the Two-Dimensional Ising Models

Distribution of zeros of partition function Z without magnetic field is studied for some two-dimensional Ising models with nearst-neighbor interactions. The distributions are presented graphically for the honeycomb, triangular diced and Kagomé lattices. It is shown that an asymptotic form of high temperature expansion for ln Z is closely related with the distribution of zeros. The expansion coefficients are derived up to large orders by computer for the honeycomb and Kagomé lattices. It turns out that their oscillatory behaviors are understood very well by studying the zeros off the positive real axis, in particular the period of oscillation for the Kagomé lattice is proved to be about 5.25.

The partition function of the 2D Ising model with magnetic fields on the boundaries and [formula omitted] Virasoro characters

The partition function of a planar Ising model on a finite lattice with magnetic fields on the boundaries is represented through an anticommuting functional integral with Gaussian distribution. In particular, the previously unknown solution for the case of fields of opposite direction is obtained. It is shown also that the partition of the model at the critical point in the continuum limit can be expressed through certain characters of the highest weight irreducible representations of the Virasoro algebra with c= 1 2 .

A numerical method to compute exactly the partition function with application toZ(n) theories in two dimensions

I present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions. The time for the computation for ann-state model on anL d lattice scales like\(n^{L^{d - 1} } nL^d \). I show examples of the use of this method by computing the partition function of the 2D Ising and 3-state Potts models for maximum lattice sizes 10×10 and 8×8, respectively. The critical exponentsv andα and the critical temperature one obtains from these are very near the exactly known values. The distribution of zeros of the partition function of the Potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity.

Partition function zeros for the two-dimensional Ising model VI

Special features of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices with all “odd” interactions, such as one with interaction ratios 5:3:1, are investigated. A bifurcation of one of the boundary lines at the imaginary axis occurs. An analytical explanation of this bifurcation is provided, and the algebraic reduction of the bifurcation eliminant to a symmetrical form is reported. Also, the end-points of lines of pure imaginary zeros are related to the critical point and disorder point equations.

Partition function zeros for the two-dimensional Ising model VII

The density of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices is investigated near real and complex critical points. The non-uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically, and numerically for a lattice with interactions in the ratios 3:2:1. The limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane. In the neighbourhood of interior critical points one finds only a single layer of zeros. But generally there are two layers or superposed sets of zeros with different distributions, and different limiting densities at boundary critical points. On anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical, by symmetry of the partition function. The divergence of the (two- dimensional) density of zeros at real critical points is discussed briefly in relation to scaling theory.

An exact functional relation for the partition function of a 2D Ising model with magnetic field

An exact functional relation is established for the partition function of the Ising model with magnetic field formulated on a Kagome lattice of arbitrary size N. An exact solution found in a previous work, see ibid., vol.21, p.31, (1988), is a particular case of the result presented in this letter.

Partition function zeros for the two-dimensional Ising model V

The boundaries of regions containing complex temperature zeros of the partition function of the two-dimensional Ising model on partially anisotropic triangular lattices are investigated in detail. A “diagonal” symmetry of one of the boundaries on lattices with one “odd” and two equal “even” interactions is accounted for. A bifurcation of the other boundary at the imaginary axis is observed (for the two examples considered) on lattices with all “odd” interactions. An explanation of this bifurcation is provided. The locations of pure imaginary boundary points, and the end-points of lines of pure imaginary zeros, when present (on lattices with all “odd” interactions), have been related to the critical and disorder point equations.

Partition function zeros for the two-dimensional using model IV

The boundaries of regions containing complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular and quadratic lattices are investigated in detail. Numerical solutions of the boundary equations are presented for triangular lattices with interactions in the ratios 3 : 2 : 1 and 4 : 2 : 1. For general anisotropic triangular and quadratic lattices, the origins of “pinch-points”, at which the boundary lines intersect, is elucidated, and the relation of critical and disorder points to pure imaginary zeros, and their role in the classification of zeros, is analyzed. An explanation is given of why interior (non-boundary) complex solutions of the critical point equations can occur for triangular lattices.

Accurate estimate of nu for the three-dimensional Ising model from a numerical measurement of its partition function.

We give results from an accurate determination of the partition function of the three-dimensional Ising model on lattices of size up to ${10}^{3}$. We compute the two complex zeros of Z closest to the real temperature axis. From a finite-size scaling analysis of these, we get the estimates \ensuremath{\nu}=0.6295(10) for the correlation-length exponent and \ensuremath{\varphi}=52.2(7) for the angle between the first two zeros and the real u=${e}^{\mathrm{\ensuremath{-}}4\ensuremath{\beta}}$ axis.

Zeros of the partition function of Ising models on fractal lattices.

Exact results are presented for the density of zeros of the partition function for Ising models on fractal lattices. For nearest-neighbor ferromagnetic interactions and real temperatures T, the zeros lie on the unit circle in the complex activity plane but leave the circle at high temperatures when four-spin interactions are included. The density of zeros exhibits a scaling form near the Yang-Lee edge and nontrivial values of the edge exponent \ensuremath{\sigma} are found which are independent of T for all T>0 and independent of the spin magnitude. Some results for the position of the zeros in the complex T plane are also given.

On the boundaries and density of partition function temperature zeros for the two-dimensional Ising model

Explicit expressions, which determine the boundaries of the regions containing temperature zeros of the partition function of the two-dimensional Ising model on an anisotropic triangular lattice, are obtained, along with a general formula for the density of zeros valid everywhere in regions containing zeros.

Fermion representation of the three-dimensional Ising model

The construction of the sign factor ensuring the cancellation of self-intersecting two-dimensional surfaces in the partition function of the three-dimensional Ising model is reviewed. It is shown that the three-dimensional Ising model is exactly equivalent to a model of surfaces with fermionic structure on them. In the naive continuum limit the action of the resulting string theory is defined by the Dirac action induced on a two-dimensional surface.

Partition function of a finite Ising model on a torus

The detailed proof presented in an earlier paper, justifying Vdovichenko's method (1965) which gives an exact expression of the free energy of the Ising model in a plane, is extended to the system on a torus. This justifies the exact expression of the free energy in the form given by Vdovichenko for the Ising model on the general two-dimensional lattice in the thermodynamic limit.

The transverse Ising model: fermionic representation and the functional integral approach

The authors derive a functional integral representation for the partition function of the transverse Ising model. The spin operators are replaced by fermion operators and special care is taken to achieve complete equivalence of the spin model and the fermion model. They discuss several approximations for the functional integral and show that in a recent paper on spin glasses, where a similar method was used, the original spin model and the fermion model, which was obtained by replacing the spin operators by fermion operators, are not equivalent and have different critical temperatures.

Yang-Lee edge singularity on fractals.

Exact results are presented for the density of zeros of the partition function for Ising models on fractal lattices. For nearest-neighbor ferromagnetic interactions, the zeros lie on the unit circle in the complex activity plane but leave the circle at high temperatures when four spin interactions are included. The density of zeros exhibits a scaling form near the edge, and the value of the edge exponent $\ensuremath{\sigma}$ is found to be independent of $T$ for all $T>0$.

The sign factor of the three-dimensional Ising model and the quantum fermionic string

The sign factor ensuring the cancellation of self-intersecting two-dimensional surfaces in the partition function of the three-dimensional Ising model is defined. The possible equivalence of the continuum limit of the three-dimensional Ising gauge model to the Dirac action induced on two-dimensional surfaces is pointed out.

Partition function zeros for the two-dimensional Ising model III

Complex temperature zeros are calculated numerically from a polynomial form of the partition function of the two-dimensional Ising model in the absence of a magnetic field. In terms of a suitably chosen variable, z, the distribution of zeros is symmetrical under reflection in both the real and imaginary axes. The numerical results confirm the analytical theory in the neighbourhood of the critical point(s), and extend into the complex plane, permitting an analysis of the symmetries of zero distribution for a variety of selected cases of interaction ratios.

Partition function zeros for the two-dimensional Ising model II

The distribution of complex temperature zeros of the partition function of the two-dimensional Ising model in the absence of a magnetic field is investigated in detail for a general anisotropic triangular lattice. When the three Ising interactions are all different the explicit expression for the distribution function obtained is valid near both the ferro- and antiferromagnetic critical points, and has the same mathematical form as for the anisotropic quadratic lattice. In the special case when the strongest interaction is antiferromagnetic and the two weaker interactions are equal, the distribution of zeros is radially symmetric around the antiferromagnetic zero temperature point (in the appropriate complex variable plane), except for a narrow cusp-like region near the real axis, and yields an unusual specific heat behaviour in (inevitable) agreement with the (novel?) thermodynamic results.

Disorder solutions for Ising and Potts models with a field

The exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard Ising and Potts models with a magnetic field are obtained. Different specialisations and extensions of these exact results are examined.

A disorder solution for a cubic Ising model

The exact expression of the partition function of a three-dimensional cubic Ising model, with nearest-neighbour interactions, is given, when a certain relation between the coupling constants of the model is satisfied. This disorder solution is then compared with a partially resummed high-temperature expansion of the partition function of the model. The constraints on this expansion, which result from the existence of the disorder solution and from the inversion relation are discussed.