or several approximations. The most common one is probably the mean field approximation, which, as we saw in Complement EXV, is the base of the Hartree-Fock method. In that complement, we showed, in terms of state vectors, how this method could be used to obtain approximate values for the energy levels of a system of interacting particles. As we consider here the more complex problem of thermal equilibrium, which must be treated in terms of density operators, we show how the Hartree-Fock method can be extended to this more general case.
We are going to see that, thanks to this approach, one can obtain compact formulas for an approximate value of the density operator at thermal equilibrium, in the framework of the grand canonical ensemble. The equations to be solved are fairly similar1 to those of Complement EXV. The Hartree-Fock method also gives a value of the thermodynamic grand potential, which leads directly to the pressure of the system. The other thermodynamic quantities can then be obtained via partial derivatives with respect to the equilibrium parameters (volume, temperature, chemical potential, eventually external applied field, etc. – see Appendix VI). It is clearly a powerful method even though it still is an approximation as the particles interactions are treated via a mean field approach where certain correlations are not taken into account. Furthermore, for bosons, it can only be applied to physical systems far from Bose-Einstein condensation; the reasons for this limitation will be discussed in detail in § 4-a of Complement HXV.
Once we have recalled the notation and a few generalities, we shall establish (§ 1) a variational principle that applies to any density operator. It will allow us to search in any family of operators for the one closest to the density operator at thermal equilibrium. We will then introduce (§ 2) a family of trial density operators whose form reflects the mean field approximation; the variational principle will help us determine the optimal operator. We shall obtain Hartree-Fock equations for a non-zero temperature, and study some of their properties in the last section (§ 3). Several applications of these equations will be presented in Complement HXV.
The general idea and the structure of the computations will be the same as in Complement EXV, and we keep the same notation: we establish a variational condition, choose a trial family, and then optimize the system description within this family. This is why, although the present complement is self-contained, it might be useful to first read Complement EXV.
1. Variational principle
In order to use a certain number of general results of quantum statistical mechanics (see Appendix refappend-6 for a more detailed review), we first introduce the notation.
1-a. Notation, statement of the problem
We assume the Hamiltonian is of the form:
which is the sum of the particles’ kinetic energy Ĥ0, their coupling energy
and their mutual interaction
(3)
We are going to use the “grand canonical” ensemble (Appendix VI, § 1-c), where the particle number is not fixed, but takes on an average value determined by the chemical potential μ. In this case, the density operator ρ is an operator acting in the entire Fock space εF (where N can take on all the possible values), and not only in the state space εN for N particles (which is is more restricted since it corresponds to a fixed value of N). We set, as usual:
where kB is the Boltzmann constant and T the absolute temperature. At the grand canonical equilibrium, the system density operator depends on two parameters, β and the chemical potential μ, and can be written as:
with the relation that comes from normalizing to 1 the trace of ρeq:
The function Z is called the “grand canonical partition function” (see Appendix VI, § 1-c). The operator
Because of the particle interactions, these formulas generally lead to calculations too complex to be carried to completion. We therefore look, in this complement, for approximate expressions of ρeq and Z that are easier to use and are based on the mean field approximation.
1-b. A useful inequality
Consider two density operators ρ and ρ′, both having a trace equal to 1:
As we now show, the following relation is always true:
We first note that the function x lnx, defined for x ≥ 0, is always larger than the function x – 1, which is the equation of its tangent at x = 1 (Fig. 1). For positive values of x and y we therefore always have:
(9)
or, after multiplying by y:
the equality occurring only