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

Journal of Mathematical Physics

Exact partition functions of interacting lattice fermions

Yves Grandati, Alain Bérard, P. Grangé

Published Mar 1, 1992• 4 citations
Publisher

Abstract

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.
DOI
10.1063/1.529770

Related papers

Grassmann Variables and Exact Solutions for Two-Dimensional Dimer Models

R. Hayn, V. N. Plechko

We discuss some aspects of a new noncombinatorial fermionic approach to the two-dimensional dimer problem in statistical mechanics based on the integration over anticommuting Grassmann variables and factorization ideas for dimer density matrix. The dimer partition function can be expressed as a Gaussian fermionic integral. For regular lattices, the analytic solution then follows by passing to the momentum space for fermions.

arXiv:cond-mat/9711156•PAN•Nov 17, 1997•2 citations

Evaluating Grassmann Integrals

Michael J. Creutz

I discuss a simple numerical algorithm for direct evaluation of multiple Grassmann integrals. The approach is exact, suffers no fermion sign problems, and allows arbitrarily complicated interactions. Memory requirements grow exponentially with the interaction range and the transverse size of the system. Low dimensional systems with 1000 Grassmann variables can be evaluated on a workstation. The technique is illustrated with a spinless fermion hopping along a one dimensional chain.

arXiv:hep-lat/9806037•Physical Review Letters•Oct 26, 1998•17 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

Grassmann algebra and fermions in the lattice: A simple example

S. M. de Souza, M. T. Thomaz

Using the exact results for integrals of exponentials of polynomials of Grassmann variables and the Feynman construction of path integrals, an approximate method is presented to calculate the thermodynamic quantities of a fermionic system in equilibrium with a reservoir at temperature T and chemical potential μ. To exemplify, the quantum anharmonic fermionic oscillator is considered in a lattice with N points, where N=2, 3, and 4.

Journal of Mathematical Physics•Dec 1, 1991•4 citations

Finite Lattice Calculations on Discrete Models

Michael J. Velgakis

A new method is proposed for the calculation of the exact partition function for Ising-like systems, based on finite-lattice arguments. In a way, the method is originated with the finite scaling theories presented by Bhanot [J. Stat. Phys. 60, 55 (1990)] and earlier by Binder [Physica 62, 508 (1972)]. The method is tested on a 2D Ising model with nearest- and next-nearest-neighbors interactions. An asset of the method is the low computer power demand.

Journal of Computational and Theoretical Nanoscience•Apr 1, 2006•0 citations

1 Grassmann integrals by machine

Michael J. Creutz

I present a numerical algorithm for direct evaluation of multiple Grassmann integrals. The approach is exact and suffers no Fermion sign problems. Memory requirements grow exponentially with the interaction range and the transverse size of the system. Low dimensional systems of order a thousand Grassmann variables can be evaluated on a workstation. In quantum field theory fermions are usually treated via integrals over anti-commuting Grassmann variables [2], providing an elegant framework for the formal establishment of Feynman perturbation theory. With non-perturbative approaches, such as Monte Carlo studies on the lattice, these objects are more problematic. Essentially all approaches formally integrate the fermionic fields in terms of determinants depending only on bosonic fields. When a background

•Jan 1, 1998•2 citations
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.
I present a numerical algorithm for direct evaluation of multiple Grassmann integrals. The approach is exact and suffers no Fermion sign problems. Memory requirements grow exponentially with the interaction range and the transverse size of the system. Low dimensional systems of order a thousand Grassmann variables can be evaluated on a workstation. In quantum field theory fermions are usually treated via integrals over anti-commuting Grassmann variables [2], providing an elegant framework for the formal establishment of Feynman perturbation theory. With non-perturbative approaches, such as Monte Carlo studies on the lattice, these objects are more problematic. Essentially all approaches formally integrate the fermionic fields in terms of determinants depending only on bosonic fields. When a background