On optimal mean-field descriptions in finite-temperature many-body theories: use of thermal Brillouin and Bruckner conditions

Abstract

Although the structural similarity between the properties of a thermal trace and the zero-temperature expectation values for quantum systems has been known for quite some time, not all the practical computational methods in the thermal field theories exploit this correspondence explicitly. Using a thermal field theory derived by us, which introduces the thermal analogues of normal ordering and Wick's expansion, a very close resemblance between zero-temperature and finite-temperature field theories can be established. We use this apparatus in this paper to derive optimal conditions of mean-field and correlated descriptions of thermally averaged quantities. It is shown that the optimal mean-field conditions for the free energy is equivalent to the minimum value for the average energy. The optimality conditions turn out to be exact thermal analogues of the Brillouin conditions. The optimal mean-field condition to generate the minimum value of the ratio Z/Z_o, where Z and Z_o are the exact and the mean-field partition functions, yields the exact thermal analogue of the Bruckner condition. In a similar vein, we generalize the thermal Brillouin condition to include correlated functions used in the evaluation of the thermal trace. The optimal choice of the correlated ground states leads to many-particle generalizations of the thermal Brillouin conditions. In the context of the path-integral methods for determining Z, we envisage use of a local optimal mean field that depends on each point on the path-leading to local thermal analogues of Brillouin and Bruckner conditions. To derive these conditions, we have used another apparatus of field theory derived recently by us that uses concepts of normal ordering and Wick expansion with respect to a path-integral measure (rather than the measure implied by thermal trace). We hope to demonstrate in our discussions that this way of formulating thermal many-body problems enables us to exploit deep similarities between thermal and zero-temperature situations which are difficult to discern in the traditional methods. Illustrative examples by way of deriving thermal Brillouin and Bruckner conditions for typical Fermionic and Bosonic problems are presented.