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Chapman & Hall/CRC eBooks

Chapter 10

Resummation of Perturbation Series

Published Aug 30, 2004• 0 citations
Publisher

Abstract

In the preceding chapters we wrote down the perturbation series for the partition function and for various correlation functions of the many-electron system, and we identified the large contributions of the perturbation series. These contributions have the effect that, as usual, the lowest order terms of the naive perturbation expansion are not a good approximation.

DOI
10.1201/9781482286052-9

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In the preceding chapters we wrote down the perturbation series for the partition function and for various correlation functions of the many-electron system, and we identified the large contributions of the perturbation series. These contributions have the effect that, as usual, the lowest order terms of the naive perturbation expansion are not a good approximation.
At the end of the last chapter, we considered the electron-electron interaction. Here, we found two terms: the Hartree energy and the exchange energy. These work fine if we can solve for the energy exactly. However, as was remarked, this can only be done for a few electron systems. For more meaningful systems, we need to develop an approximation technique for the interacting Green’s functions. The standard approach to this is via the interaction representation, which gives us a prescription for developing the perturbation series. As one might infer from our previous discussion, this will lead to multiple, messy integrals. Fortunately, the Feynman diagrams give us an elegant and pictorial method of looking at the various terms in the perturbation series. As a result, they also give us a much more intuitive method of examining the various terms.
A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​ for large rrr. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.
A new formalism of the thermodynamic perturbation theory for the quantum-anharmonic oscillator is proposed. By this method, for each order of the perturbation series, the partition function and the free energy of the system can be obtained in a compact form which is expressed by the temperature Green's function of the harmonic quanta. Based upon a unified method, the various expressions of the perturbation series for the partition function and the free energy are also given. It is shown that the perturbation series for the free energy can be naturally formulated in terms of a cluster expansion.