Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Combinatorial Approach to the Ferroelectric Problem

R. W. Gibberd

Published Jul 10, 1968• 3 citations
Publisher

Abstract

The partition function of the ferroelectric problem is expressed as a vacuum-to-vacuum expectation value of a time-ordered product of Fermi operators. The expression obtained is typical of those that arise in treating interacting Fermi systems. Apart from evaluating the critical temperature as a function of the electric field, we have not been able to obtain Lieb's exact solution from this formalism. However, the first-order approximation is found to be closely related to Lieb's solution, giving some of the critical behavior correctly.

DOI
10.1103/physrev.171.563

Related papers

A reformulation of the dimer problem

R. W. Gibberd

It is shown that the partition function of the generalized dimer problem can be formulated in terms of a vacuum-to-vacuum expectation value of time-ordered operators. This expression is then evaluated by using Green's function technique, which has already been used in conjunction with the Ising model and ferroelectric problem.

Canadian Journal of Physics•Aug 1, 1968•4 citations

Application of many-fermion techniques to lattice statistical problems

R. W. Gibberd

The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.

Canadian Journal of Physics•Apr 1, 1969•2 citations

Ising-Model Reformulation. I. Fundamentals

Frank H. Stillinger

The partition function ($Z$) for the classical Ising model of cooperative phenomena in lattice of arbitrary dimensionality and with interactions of unspecified range is expressed as a vacuum-state expectation value of a product of two operators, each constructed from boson annihilation and creation operators. In the absence of external fields, $Z$, and similarly the spin-pair correlation function $\ensuremath{\psi}(\mathrm{r})$, are expanded thereupon into a series of Feynman diagrams. In the case of $\ensuremath{\psi}(\mathrm{r})$, a formally exact diagram summation (1) shows how the spherical model may be recovered in low-order approximation, (2) suggests a way of introducing systematic corrections to this approximation, and (3) leads to a generalized criterion for suppression of antiferro-magnetic order in "nonfitting" lattices. A full topological-diagram reduction to restricted sets of "elementary" subdiagrams is carried out.

Physical Review•Sep 14, 1964•8 citations

Exact partition functions of interacting lattice fermions

Yves Grandati, Alain Bérard, P. Grangé

Exact expressions are obtained for Grassmann integrals of exponentials corresponding to the partition functions of systems of lattice fermions with nearest-neighbor interactions. It is achieved by relating to the dimer problem and use of its combinatorial techniques.

Journal of Mathematical Physics•Mar 1, 1992•4 citations

Form of the exact partition function for the generalized Ising Model

T. R. S. Prasanna

The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.

arXiv:cond-mat/0404550•Bulletin of the American Physical Society•Apr 22, 2004•0 citations

A Note on the Partition Function for Systems of Independent Particles

Douglas Ike Ford

The N-particle Bose-Einstein and Fermi-Dirac canonical partition functions are expressed explicitly in terms of the single-particle partition function. The nature of the approximations made in using corrected Maxwell-Boltzmann statistics is thereby made more obvious. A recurrence relation is developed to facilitate calculations using the exact partition function.

American Journal of Physics•Feb 1, 1971•15 citations
The partition function of the ferroelectric problem is expressed as a vacuum-to-vacuum expectation value of a time-ordered product of Fermi operators. The expression obtained is typical of those that arise in treating interacting Fermi systems. Apart from evaluating the critical temperature as a function of the electric field, we have not been able to obtain Lieb's exact solution from this formalism. However, the first-order approximation is found to be closely related to Lieb's solution, giving some of the critical behavior correctly.
The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.
The partition function (ZZZ) for the classical Ising model of cooperative phenomena in lattice of arbitrary dimensionality and with interactions of unspecified range is expressed as a vacuum-state expectation value of a product of two operators, each constructed from boson annihilation and creation operators. In the absence of external fields, ZZZ, and similarly the spin-pair correlation function $\ensuremath{\psi}(\mathrm{r})$, are expanded thereupon into a series of Feynman diagrams. In the case of $\ensuremath{\psi}(\mathrm{r})$, a formally exact diagram summation (1) shows how the spherical model may be recovered in low-order approximation, (2) suggests a way of introducing systematic corrections to this approximation, and (3) leads to a generalized criterion for suppression of antiferro-magnetic order in "nonfitting" lattices. A full topological-diagram reduction to restricted sets of "elementary" subdiagrams is carried out.
The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.