On the One Variational Principle in Quantum Statistical Mechanics

Albrecht Huber

P. D�rre, H. Haug, M. Heise +1 more

H.H. Rosenbrock

Gerald Speisman

The variational principle of Bohm and Pines for the ground-state energy of the electron gas with uniform neutralizing positive-charge background, employing auxiliary variables, is reviewed as an illustration of the past use of auxiliary variables and as an example of the type of physical system to which the variational principle of this paper can be applied. We then develop this variational principle for the logarithm of the partition function of a physical system at nonzero temperatures, employing auxiliary variables. The variational principle contains a trial Hamiltonian H′ in an extended Hilbert space. For H′ equal to its optimal value H, the variational expression ln Q′ is equal to the logarithm of the partition function ln Q. For H′ ≠ H, it is shown that ln Q ≥ ln Q′ for H′ – H sufficiently small or temperatures sufficiently high, or for sufficiently low temperatures when an additional assumption is made, which reduces to one made by Bohm and Pines when applied to the electron gas. The variational expression ln Q′ contains more complicated trace formulas than are usually encountered in quantum statistical mechanics; one possible method of evaluation is sketched leading to a simpler approximate formula for ln Q′. Corrections to the variational approximation for ln Q are provided by the second- and higher-order terms in a certain perturbation expansion of ln Q. The variational principle developed here implies the variational principle of Bohm and Pines in the zero-temperature limit; in the ``no auxiliary variable'' limit it reduces to a modified form of Peierls' variational theorem. It is shown how the variational principle can be applied to any physical system containing charged particles in which the long-range collective effects of the Coulomb interaction are important.

R. Giachetti, V. Tognetti, Ruggero Vaia

Claude Bloch, James S. Langer

The Ising model has been employed in a study of the variational principles which are associated with renormalized diagram expansions in statistical mechanics. For this model the variational functional is expressed in terms of the renormalized semi-invariants, which play the role of the one-particle density or the one-particle Green's function in quantum statistics. We review the derivation of this functional and discuss some of its properties. In order to examine the mathematical content of the variational principle in more detail, we specialize to the exactly soluble infinite-dimensional (infinite-range) model. We find that, by virtue of the Dyson relation, the variational theorem is not valid over all possible values of the renormalized semi-invariants but applies only within a restricted domain. Within this domain the variational functional is multiple valued. The renormalized expansion is asymptotically convergent to the branch of this functional which describes a single phase with uniform magnetization.