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Journal of Physics C Solid State Physics

Perturbation theory for the spin-$frac12$ Heisenberg ferromagnet I. Wick's theorem and diagrammatic representation

Anna Kühnel

Published Apr 1, 1969• 8 citations
Publisher

Abstract

Pauli operators are treated as Bose operators fulfilling q -number commutation relations. An analogue of Wick's theorem for Pauli operators is formulated suitable for perturbation theory. The lowest-order contributions to the perturbation series for the temperature dependent one-particle causal Green function are calculated, and a diagrammatic representation is introduced for them. It is not necessary to have a different meaning of the same graph depending on the order of temperatures contained in it. The connected graphs representing the perturbation series up to second order are drawn explicitly.

DOI
10.1088/0022-3719/2/4/314

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Physical Review•Mar 10, 1967•20 citations
Perturbation theory for the spin-frac12frac12frac12 Heisenberg ferromagnet I. Wick's theorem and diagrammatic representation
Pauli operators are treated as Bose operators fulfilling q -number commutation relations. An analogue of Wick's theorem for Pauli operators is formulated suitable for perturbation theory. The lowest-order contributions to the perturbation series for the temperature dependent one-particle causal Green function are calculated, and a diagrammatic representation is introduced for them. It is not necessary to have a different meaning of the same graph depending on the order of temperatures contained in it. The connected graphs representing the perturbation series up to second order are drawn explicitly.
Wick's theorem is combined with a cluster composition to derive a generalized statistical Wick's theorem for a non-equilibrium system with initial correlations. This enables a diagrammatic time-dependent perturbation theory which is a simple extension of the usual one to be developed for time-path causal statistical Green functions and their matrix equivalents. This provides a complete non-equilibrium theory. Reduction of diagrams and equations of motion for the Green functions are also discussed.
This chapter considers a general form of the Wick’s theorem, which leads to a perturbation expansion of the (contour) single- and two-particle Green’s functions, which are expressed in terms of the contour time-ordering operator. The strategy for proving the Wick’s theorem is similar to that adopted within the Matsubara formalism for the Green’s functions at finite temperature and relies on the Gibbs form of the statistical operator in the interaction picture. An extension to superfluid Bose and Fermi systems is also considered.
The possibility of a perturbation treatment for quantum crystals is investigated. It is found that the usual form of the unperturbed Hamiltonian, being a single-particle operator, is not suitable for this situation, since it does not produce correlations necessary in a theory of quantum crystals. These correlations prevent the unperturbed ground state or low excited states from containing multiply occupied lattice sites. An unperturbed Hamiltonian is proposed which generates these correlations but is no longer a single-particle operator. The rules for a diagrammatic representation of a perturbation expansion starting from this unperturbed Hamiltonian are given. They differ from the usual rules for Goldstone diagrams. In particular, the linked-cluster theorem is not valid. Adding and subtracting pairs of mutually cancelling diagrams, in a manner similar to that in which Pauli-principle-violating diagrams are treated in the standard perturbation theory for fermions, we find new rules which, at least at low temperatures, are identical to those for spinless fermions. This result is essentially independent of the statistics of the actual lattice particles. In the case where the lattice particles are actually fermions with spin \textonehalf{}, e.g., 3He^{3}\mathrm{He}3He, we find an expansion formally identical to the problem of spinless fermions interacting among themselves and with a set of spins \textonehalf{} localized at lattice sites.
We study perturbation expansions for quantum systems which are nonlinear in the sense that they may contain derivative couplings. The characteristic feature of these systems is the occurrence of products of noncommuting operators in the interaction Hamiltonian. In order to perform perturbation expansions in the Dyson-Wick manner it is in general necessary to order these operators according to some ordering scheme. We determine the most general scheme of this sort which allows a Dyson-Wick expansion with pair contractions. It involves a set $\ensuremath{\lambda}$ of three dimensionless ordering parameters which are related to the three aaa priori undetermined equal-time contractions. Different $\ensuremath{\lambda}$-ordering schemes are related via an "equal-time" or "static" Wick theorem. In order to be able to contract operators which may be taken either at equal or at different times we introduce a time-ordering operation ${T}_{\ensuremath{\lambda}}$ which extends Dyson's time-ordering operation to equal times in a fashion dependent on the $\ensuremath{\lambda}$ ordering. We arrive at a "dynamic" Wick theorem which enables us to decompose the ${T}_{\ensuremath{\lambda}}$ product of a set of operators taken at arbitrary times into a sum of ${\ensuremath{\lambda}}^{\ensuremath{'}}$ products, where the most important case is that in which ${\ensuremath{\lambda}}^{\ensuremath{'}}$ represents normal ordering. Contrary to the usual Wick theorem ours works also in the case when two or more times coincide. Applying this theorem to the scattering operator expressed as a ${T}_{\ensuremath{\lambda}}$ product we obtain a class of Dyson-Wick expansions for nonlinear systems. Different expansions correspond to different choices of the ordering $\ensuremath{\lambda}$ and lead to different Feynman rules. The resulting diagrammatic technique for nonlinear systems is seen to have the following special features: (i) The interaction vertices include terms which arise from "quantum corrections," which are dependent on the ordering \ensuremath{\lambda} chosen for the noncommuting factors in the interaction Hamiltonian, and (ii) the equal-time contractions which occur in the closed-loop diagrams must be assigned corresponding $\ensuremath{\lambda}$-dependent values. In this way individual diagrams are $\ensuremath{\lambda}$ dependent. Nevertheless their sum is $\ensuremath{\lambda}$ independent, confirming the fact that the complete scattering operator is independent of the particular ordering chosen. The ordering multiplicity is also reflected in functional versions of the Dyson-Wick expansion. In the path-integral version this multiplicity lurks in the different possible discretization prescriptions. We discuss the relevance of our considerations to the question of the invariance of a quantum theory with respect to nonlinear point transformations. For the special case of a reducible system we present explicit calculations which show how this invariance is maintained in perturbation theory, for any ordering scheme belonging to the $\ensuremath{\lambda}$ class. In any particular scheme the invariance is ensured by a more or less intricate cancellation mechanism between the individual diagrams.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
A form of Wick's theorem is derived which is applicable to spin operators of arbitrary magnitude in a manner analogous to the decomposition of fermion and boson operator products. Use of the theorem together with Kubo's cumulant rearrangement of perturbation theory leads to a compact prescription for the calculation of the ground-state energy of spin systems. This method is parallel to Goldstone's linked-cluster expansion for fermions, but the expansion for spins, as well as for bosons, contains cumulant correction terms. The Green functions are shown to be expanded similarly in terms of cumulants. The method is illustrated by the calculation of the antiferromagnetic ground state, and circumvents the involved development in the previous work of Davis and Boon. It is found that our procedure gives better convergence than Davis's expansion and that Boon's final formulas as well as his numerical results contain some errors.