Generalized Mean Fields and Variational Principle of Bogoliubov

Giampaolo Co’

A. L. Kuzemsky

The approach to the theory of many-particle interacting systems from a unified standpoint, based on the variational principle for free energy is reviewed. A systematic discussion is given of the approximate free energies of complex statistical systems. The analysis is centered around the variational principle of Bogoliubov for free energy in the context of its applications to various problems of statistical mechanics. The review presents a terse discussion of selected works carried out over the past few decades on the theory of many-particle interacting systems in terms of the variational inequalities. It is the purpose of this paper to discuss some of the general principles which form the mathematical background to this approach and to establish a connection of the variational technique with other methods, such as the method of the mean (or self-consistent) field in the many-body problem. The method is illustrated by applying it to various systems of many-particle interacting systems, such as Ising, Heisenberg and Hubbard models, superconducting (SC) and superfluid systems, etc. This work proposes a new, general and pedagogical presentation, intended both for those who are interested in basic aspects and for those who are interested in concrete applications.

J.-B. Bru, W. de Siqueira Pedra

Diego da Cunha Carvalho

In this work, we have analytically studied, by means of the Bogoliubov Variational Principle (BVP), the pure Heisenberg and XY with bond dilution models, both ferromagnetic, classical (continuous spins), anisotropic, three-dimensional and in the presence of a crystalline field. Furthermore, we have shown the equivalence between the BVP and theSelf-Consistent Harmonic Approximation (SCHA) a trial harmonic Hamiltonian. In the study of anisotropic Heisenberg model, the magnetization and phase diagrams are obtained as a function of parameters of the Hamiltonian. Limiting cases, such as the isotropicHeisenberg, XY and Planar Rotor models in two and three dimensions are analyzed and compared to previous results obtained from analytical approximations and Monte Carlo simulations. In the study of anisotropic XY model with bond dilution, we have obtained the critical concentration and compared with the exact result as well as from other approximation methods, besides reviewing the Planar Rotor diluted system as a limitingcase. By showing the equivalence between the BVP and SCHA, we arrive at a system of coupled equations for the variational parameters, which arise by employing the BVP, and the system of coupled equations for the coupling constants that appear employing the SCHA, and we found that such systems are identical.

H. Flocard

Volker Bach, Sébastien Breteaux, Hans Konrad Knörr +1 more

In its original formulation, Lieb's variational principle holds for fermion systems with purely repulsive pair interactions. As a generalization we prove for both fermion and boson systems with semi-bounded Hamiltonian that the infimum of the energy over quasifree states coincides with the infimum over pure quasifree states. In particular, the Hamiltonian is not assumed to preserve the number of particles. To shed light on the relation between our result and the usual formulation of Lieb's variational principle in terms of one-particle density matrices, we also include a characterization of pure quasifree states by means of their generalized one-particle density matrices.