Partial selfconsistency and analyticity in manybody perturbation theory: Particle number conservation and a generalized sum rule
Karlsson, D., & van Leeuwen, R. (2016). Partial selfconsistency and analyticity in manybody perturbation theory: Particle number conservation and a generalized sum rule. Physical Review B, 94 (12), 125124. doi:10.1103/PhysRevB.94.125124
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Physical Review BDate
2016Copyright
© 2016 American Physical Society. Published in this repository with the kind permission of the publisher.
We consider a general class of approximations which guarantees the conservation of particle number in
manybody perturbation theory. To do this we extend the concept of derivability for the selfenergy to a
larger class of diagrammatic terms in which only some of the Green’s function lines contain the fully dressed
Green’s function G. We call the corresponding approximations for partially derivable. A special subclass of
such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of
consistently with G. These approximations are number conserving but do not have to fulfill other conservation
laws, such as the conservation of energy and momentum. From our formalism we can easily deduce whether
commonly used approximations will fulfill the continuity equation, which implies particle number conservation.
We further show how the concept of partial derivability plays an important role in the derivation of a generalized
sum rule for the particle number, which reduces to the LuttingerWard theorem in the case of a homogeneous
electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that
the Green’s function has certain complex analytic properties, which can be guaranteed if the spectral function
is positivesemidefinite. The latter property can be ensured for a subset of partially derivable approximations
for the selfenergy, namely those that can be constructed from squares of socalled half diagrams. For the case
in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts,
pole structure, and multivaluedness. We also show that various schemes of computing the particle number are
consistent for particle number conserving approximations.
...
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