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Lecture notes in physics

Linked Cluster Theorem

Moritz Helias, David Dahmen

Published Jan 1, 2020• 2 citations
Publisher
DOI
10.1007/978-3-030-46444-8_5

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The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.
The Wick reduction theorem is used in a linked-cluster-expansion calculation to facilitate the evaluation of the multi-integrals of the ordered products encountered in the expansion. The method resolves a major problem in the general linked-cluster expansion for quantum spin systems. We apply it to the evaluation of the perpendicular susceptibility of the Ising model. An eighth-order linked-cluster series is obtained for a general lattice. The temperature behavior of the perpendicular susceptibility of the three-dimensional Ising model (fcc lattice) is discussed.