Original scientific paper
Mathematical properties and algorithm for fast calculation of full Wick’s contractions in quantum many-body fermion systems
Deni Vale
; Istrian University of Applied Sciences
Nils Paar
; University of Zagreb, Faculty of Science, Department of Physics
Abstract
Wick’s contractions, also related to Wick’s theorem, represent important mathematical technique used in quantum many-body theory to simplify calculations involving creation and annihilation operators. In this work we study the properties of full Wick’s contractions and discuss in details corresponding graph and group theory aspects. We observed isomorphism between graph-like objects which are in fact contained in the full Wick’s contractions and some geometrical objects, such as circle or regular
rectangle with internal structure. We also found isomorphism between two induced groups, one which is related to permutations of one end of Wick’s lines and the second which corresponds to rotations of directed lines inside geometrical object. We present fast and efficient algorithm for calculation of the expectation value of large number of creation and annihilation particle and hole operators in order to achieve different particle-hole or particle-particle terms in many-body theories, from nuclear to solid
state physics or quantum chemistry. The algorithm is based on observed isomorphisms. It simplifies full Wick’s contractions to simple adjacency and geometrical relations, which are also used for sign determination. Also, we presented several illustrative examples of computation, such as calculation of the two-body particle-hole terms in Hartree-Fock’s theory and the Random phase approximation.
Keywords
Wick’s theorem; Wick’s contractions; group theory; graph theory; quantum mechanics; quantum many-body theory; Hartree-Fock; Random phase approximation
Hrčak ID:
309163
URI
Publication date:
24.10.2023.
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